Book LIE · English

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The 1703 statements of Lie Groups and Lie Algebras, by chapter and §. Pick a kind, filter by a word, or put the most cited first.

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I. Lie Algebras

§ 1. Definition of Lie algebras

Definition 1 An algebra g over K is called a Lie algebra over K if its multiplication (denoted by (x, y) ↦ [x, y]) satisfies the identities: 09TJ
Example 1 Let L be an associative algebra over K. The bracket [x, y] = xy - yx is a bilinear function of x and y. It is easily verified that the law of composition (x, y) ↦ [x, y] on the K-m… ↩ 3 09TK
Example 2 In Example 1 choose L to be the associative algebra of endomorphisms of a K-module E. We obtain the Lie algebra of endomorphisms of E, denoted by gl(E). (If E = Kⁿ, the Lie algebra… ↩ 1 09TL
Example 3 The set Mₙ(K) of square matrices of order n can be considered as a Lie algebra over K canonically isomorphic to gl(n, K). Let (Eᵢⱼ) be the canonical basis of Mₙ(K) (Algebra, Chapte… 09TM
Definition 2 Let g be a Lie algebra and x an element of g. The linear mapping y ↦ [x, y] of g into g is called the adjoint linear mapping of x and is denoted by ad_g x or ad x. 09TN
Proposition 1 Let g be a Lie algebra. For all x ∈ g, ad x is a derivation. The mapping x ↦ ad x is a homomorphism of the Lie algebra g into the Lie algebra 𝔡 of derivations of g. If D ∈ 𝔡 and x… 09TO
Definition 3 Two elements x, y of a Lie algebra are said to be permutable if [x, y] = 0. g is said to be commutative if any two of its elements are permutable. 09TP
Example 1 Let L be an associative algebra and g the Lie algebra defined by it (no. 2, Example 1). Two elements x, y are permutable in g if and only if xy = yx in L. 09TQ
Example 2 If a real Lie group G is commutative, its Lie algebra is commutative. 09TR
Example Let G be a Lie group, g its Lie algebra and H a Lie subgroup of G. Every left invariant vector field on H defines canonically a left invariant vector field on G, whence there is a… 09TS
Definition 4 A submodule of g which is stable under every derivation of g is called a characteristic ideal of g. 09TT
Proposition 2 Let g be a Lie algebra, a an ideal (resp. a characteristic ideal) of g and b a characteristic ideal of a. Then b is an ideal (resp. a characteristic ideal) of g. ↩ 1 09TU
Proposition 3 If a and b are ideals (resp. characteristic ideals) of g, [a, b] is an ideal (resp. a characteristic ideal) of g. 09TV
Proposition 4 Let g and h be two Lie algebras over K and f a homomorphism of g onto h. Then f(Dᵖg) = Dᵖf, f(Cᵖg) = Cᵖh. ↩ 4 09TW
Corollary Let g be a Lie algebra and a an ideal of g. For the Lie algebra g/a to be commutative, it is necessary and sufficient that a ⊃ Dg. 09TX
Proposition 5 Let g be a Lie algebra and a an ideal (resp. a characteristic ideal) of g. The centralizer a' of a in g is an ideal (resp. a characteristic ideal) of g. 09TY
Definition 5 Let a and b be two Lie algebras over K. An extension of b by a is a sequence: 09TZ
Proposition 6 Let ↩ 1 09U0
Definition 6 Let ↩ 1 09U1
Proposition 7 Let a and b be two Lie algebras over K, 09U2
Example 1 Let g be a Lie algebra over K and D a derivation of g. Let h be the commutative Lie algebra K. The mapping λ ↦ λ D (λ ∈ K) is a homomorphism of h into the Lie algebra of derivation… ↩ 2 09U3
Example 2 Let g be a Lie algebra over K, M a K-module and ρ a homomorphism of g into 𝔤𝔩(M). If M is considered as a commutative Lie algebra, the Lie algebra of derivations of M is 𝔤𝔩(M). We… ↩ 2 09U4

§ 2. Enveloping algebra of a Lie algebra

Definition 1 Let g be a Lie algebra over K, T the tensor algebra of the K-module g and J the two-sided ideal of T generated by the tensors x ⊗ y - y ⊗ x - [x, y] where x ∈ g, y ∈ g. The associa… ↩ 1 09UW
Proposition 1 Let σ be an α-mapping of g into the associative algebra L with unit element. There exists one and only one homomorphism τ of U into L, mapping 1 to 1, such that σ = τ ∘ σ₀, where σ… ↩ 6 09UX
Proposition 2 The homomorphism φ is an algebra isomorphism ↩ 1 09UY
Proposition 3 Let h be an ideal of g, p the canonical homomorphism of g onto g/h and W the enveloping algebra of g/h. The homomorphism: tildep: U → W defined canonically by p is surjective and i… 09UZ
Proposition 4 The homomorphism φ is an isomorphism of V onto U⁰. 09V0
Proposition 5 The mapping φ of T onto G is an algebra homomorphism mapping 1 to 1 and is zero on the two-sided ideal generated by the tensors x ⊗ y - y ⊗ x (x ∈ g, y ∈ g). 09V1
Proposition 6 If K is Noetherian and g is a finitely generated module, the ring U is right and left Noetherian. ↩ 1 09V2
Corollary Suppose that K is a field and that g is finite-dimensional over K. Let I₁, ..., Iₘ be right (resp. left) ideals of finite codimension in U. Then the product ideal I₁I₂...Iₘ is of f… ↩ 1 09V3
Remark Let g' be another Lie algebra over the ring K, U' its enveloping algebra, U'n the set of elements of U' of filtration ≤ n and U^n (resp. U'^n) the set of canonical images in U (res… 09V4
Theorem 1 Let g be a Lie K-algebra, U its enveloping algebra, G the graded algebra associated with the filtered algebra U and S the symmetric algebra of the K-module g. If g is a free K-modu… ↩ 6 09V5
Lemma 1 For every integer p ≥ 0, there exists a unique homomorphism fₚ of the K-module g ⊗_K Pₚ into the K-module P satisfying the following conditions: beginalign* (Aₚ) & fₚ(x_λ ⊗ z_M) =… ↩ 1 09V6
Lemma 2 There exists an α-mapping σ of g into L_K(P) such that: ↩ 1 09V7
Lemma 3 Let t be a tensor in Tₙ ∩ J. The homogeneous component tₙ of t of order n is in the kernel I of the canonical homomorphism T → S. 09V8
Corollary 1 Suppose that g is a free K-module. Let W be a sub-K-module of Tⁿ. ↩ 3 09V9
Corollary 2 If g is a free K-module, the canonical mapping of g into its enveloping algebra is injective. 09VA
Corollary 3 If g admits a totally ordered basis (x_λ)_λ ∈ Λ, the elements x_λ₁ x_λ₂ … x_λₙ of the enveloping algebra U, where (λ₁, …, λₙ) is an arbitrary increasing finite sequence of elements… ↩ 4 09VB
Corollary 4 Let S'ⁿ ⊂ Tⁿ be the set of homogeneous symmetric tensors of order n. Suppose that K is a field of characteristic 0. Then the composite mapping of the canonical mappings ↩ 2 09VC
Corollary 5 Let h be a subalgebra of the Lie algebra g and U' its enveloping algebra. Suppose that the K-modules h and g/h are free (for example if K is a field). Let (x_α)_α ∈ L be a basis of… ↩ 1 09VD
Corollary 6 Suppose that the K-module g is the direct sum of subalgebras g₁, g₂, …, gₙ and that each gᵢ is a free K-module. Let Uᵢ be the enveloping algebra of gᵢ (1 ≤ i ≤ n). Let φ be the K-l… ↩ 2 09VE
Corollary 7 If K is an integral domain and g is a free K-module, the algebra U has no divisors of zero. ↩ 2 09VF
Lemma 4 Let V be a K-module and T the tensor algebra of V. Let u be an endomorphism of V. There exists one and only one derivation of T which extends u. This derivation commutes with the s… ↩ 1 09VG
Proposition 7 Let g be a Lie algebra, U its enveloping algebra, σ the canonical mapping of g into U and D a derivation of g. (a) There exists one and only one derivation D_U of U such that σ ∘ D… ↩ 1 09VH

§ 3. Representations

Definition 1 Let g be a Lie algebra over K and M a K-module. A homomorphism of g into the Lie Algebra gl(M) is called a representation of g on the module M. An injective representation is calle… ↩ 1 09VS
Example Let G be a real Lie group, g its Lie algebra and θ an analytic representation of G on a finite-dimensional real vector space E. Then the corresponding homomorphism of g into gl(E)… 09VT
Definition 2 Let g be a Lie algebra over K and U its enveloping algebra. A unitary left module over U is called a left g-module, or simply a g-module. 09VU
Proposition 1 Let g be a Lie algebra over K and a an ideal of g. Let M be a g-module and N a simple a-module. Consider M as an a-module and suppose that the set of sub-a-modules of M which are p… ↩ 1 09VV
Corollary Let g be a Lie algebra over K and a an ideal of g. Let M be a simple g-module, of finite length as a K-module. There exists a simple a-module N such that M is a pure a-module of sp… ↩ 1 09VW
Proposition 2 Let g be a Lie algebra over K and Mᵢ a g-module (1 ≤ i ≤ n). On the tensor product M₁ ⊗_K M₂ ⊗ ⋯ ⊗ Mₙ, there exists one and only one g-module structure such that ↩ 1 09VX
Proposition 3 Let g be a Lie algebra over K and Mᵢ a g-module (1 ≤ i ≤ n + 1). Let N be the K-module L_K(M₁, …, Mₙ; Mₙ₊₁) of multilinear mappings of ∏ᵢ₌₁ⁿ Mᵢ into Mₙ₊₁. There exists one and only… ↩ 1 09VY
Proposition 4 Let M₁, M₂ be two g-modules. The canonical K-linear mappings (Algebra, Chapter II, § 4, no. 2, Proposition 2 and no. 1, Proposition 1): ↩ 6 09VZ
Example 1 Let g be a Lie algebra over K and M a g-module. The g-module structure on M and the trivial g-module structure on K define a g-module structure on the K-module N = L(M, M; K) of bi… ↩ 1 09W0
Example 2 We preserve the notation of Example 1. 09W1
Definition 3 Let g be a Lie algebra and M a g-module. An element m ∈ M is called invariant (with respect to the g-module structure on M or with respect to the corresponding representation of g)… ↩ 1 09W2
Example 1 Let M, N be two g-modules and P = L_K(M, N). For an element f of P to be invariant, it is necessary and sufficient, by (6), that f be a homomorphism of the g-module M into the g-mo… ↩ 2 09W3
Example 2 Let M be a K-module with a finite basis. If M has a g-module structure, L(M, M) and M^* ⊗ M have g-module structures and the canonical mapping of M^* ⊗ M into L(M, M) is a g-module… ↩ 1 09W4
Example 3 Let M be a g-module. Let β be a bilinear form on M and f the corresponding element of L(M, M^*). For β to be invariant, it is necessary and sufficient that f be a g-module homomorp… ↩ 1 09W5
Proposition 5 Let g be a Lie K-algebra, h an ideal of g, ρ a representation of g on M and ρ' the restriction of ρ to h. Then the set N of elements of M invariant with respect to ρ' is stable und… ↩ 3 09W6
Proposition 6 Let M be a semi-simple g-module. Then the submodule M₀ of invariant elements of M admits one and only one supplement stable under the x_M, namely the submodule M₁ generated by the… ↩ 5 09W7
Proposition 7 Let g be a Lie algebra, β an invariant symmetric bilinear form on g and a an ideal of g. ↩ 2 09W8
Definition 4 Let g be a Lie K-algebra and M a g-module. Suppose that M, considered as a K-module, admits a finite basis. The bilinear form associated with the g-module M (or with the correspond… 09W9
Proposition 8 Let g be a Lie algebra and M a g-module. Suppose that M, considered as a K-module, admits a finite basis. The bilinear form associated with M is invariant. 09WA
Proposition 9 Suppose that K is a field and that the Lie algebra g is finite-dimensional over K. Let a be an ideal of g, β the Killing form of g and β' the Killing form of a. Then β' is the rest… ↩ 1 09WB
Proposition 10 Suppose that K is a field and that the Lie algebra g is finite-dimensional over K. The Killing form β of g is completely invariant. ↩ 1 09WC
Proposition 11 Let g be a Lie algebra over a field K, U its enveloping algebra, h a finite-dimensional ideal of g and β an invariant bilinear form on g, whose restriction to h is non-degenerate… ↩ 1 09WD
Proposition 12 Let g be a Lie algebra over a field K, h an ideal of g of finite dimension n and M a g-module of finite dimension over K. Let c be the Casimir element (assumed to exist) associated… ↩ 3 09WE
Proposition 13 Let K be a field, K₁ an extension of K, g a Lie K-algebra and M, N two g-modules of finite dimension over K. If M_(K₁) and N_(K₁) are isomorphic g_(K₁)-modules, M and N are isomorp… ↩ 1 09WF

§ 4. Nilpotent Lie algebras

Definition 1 A Lie algebra g is called nilpotent if there exists a decreasing finite sequence of ideals (gᵢ)_1 ≤ i ≤ p of g with g₀ = g, gₚ = 0, such that [g, gᵢ] ⊂ gᵢ₊₁ for 0 ≤ i < p. ↩ 3 09WS
Proposition 1 Let g be a Lie algebra. The following conditions are equivalent: (a) g is nilpotent; (b) Cᵏ g = 0 for sufficiently large k; (c) Cₖ g = g for sufficiently large k; (d) there exists… ↩ 5 09WT
Corollary 1 The centre of a non-zero nilpotent Lie algebra is non-zero. 09WU
Corollary 2 The Killing form of a nilpotent Lie algebra is zero. For all x and y in a nilpotent Lie algebra ad x ∘ ad y is nilpotent and hence of zero trace. 09WV
Proposition 2 Subalgebras, quotient algebras and central extensions of a nilpotent Lie algebra are nilpotent. A finite product of nilpotent Lie algebras is a nilpotent Lie algebra. Let g be a Li… ↩ 1 09WW
Proposition 3 Let g be a nilpotent Lie algebra and 𝔥 a subalgebra of g distinct from g. The normalizer of 𝔥 in g is distinct from 𝔥. Let k be the greatest integer such that Cᵏ g + 𝔥 ≠ 𝔥. Then [C… ↩ 3 09WX
Lemma 1 Let V be a vector space over K. If x is a nilpotent endomorphism of V, the mapping y ↦ [x, y] of L(V) into L(V) is nilpotent. If f denotes this mapping, fᵐ(y) is a sum of terms of… ↩ 2 09WY
Theorem 1 (Engel) Let V be a vector space over K and g a finite-dimensional subalgebra of 𝔤𝔩(V) whose elements are nilpotent endomorphisms of V. If V ≠ 0, there exists u ≠ 0 in V such that x.u = 0 f… ↩ 5 09WZ
Corollary 1 For a Lie algebra g to be nilpotent, it is necessary and sufficient that, for all x ∈ g, ad\ x be nilpotent. ↩ 2 09X0
Corollary 2 Let g be a Lie algebra and h an ideal of g. Suppose that g/h is nilpotent and that, for all x ∈ g, the restriction of ad\ x to h is nilpotent. Then g is nilpotent. 09X1
Corollary 3 Let V be a vector space and g a finite-dimensional subalgebra of 𝔤𝔩(V) whose elements are nilpotent endomorphisms of V. Then g is a nilpotent Lie algebra. ↩ 1 09X2
Example The algebra 𝔫(n, K) (§ 1, no. 2, Example 2 (3)) is nilpotent. 09X3
Lemma 2 Let g be a Lie algebra, a an ideal of g and M a simple g-module. If, for all x ∈ a,\ x_M is nilpotent, then x_M = 0 for all x ∈ a. ↩ 3 09X4
Lemma 3 Let g be a Lie algebra, a an ideal of g, M a g-module of finite dimension over ↩ 2 09X5
Proposition 4 Let g be a Lie algebra, M a g-module of finite dimension over K and A the associative algebra generated by 1 and the set of x_M (x ∈ g). ↩ 5 09X6
Definition 2 The ideal n of Proposition 4 is called the largest nilpotency ideal for the g-module M or the largest nilpotency ideal of the corresponding representation. ↩ 1 09X7
Proposition 5 Let V be a vector space of finite dimension n over K and g a Lie sub-algebra of gl(V) whose elements are nilpotent endomorphisms of V. Then there exists a decreasing sequence of ve… 09X8
Proposition 6 Let g be a Lie algebra and E the associative subalgebra of L(g) generated by 1 and the ad_g x (x ∈ g). Let R be the Jacobson radical of E. (a) The set n of y ∈ g such that ad_g y ∈… ↩ 1 09X9

§ 5. Solvable Lie algebras

Definition 1 A Lie algebra g is called solvable if its kth derived algebra Dᵏ g is zero for sufficiently large k. 09Y1
Proposition 1 Subalgebras and quotient algebras of a solvable Lie algebra are solvable. Every extension of a solvable algebra by a solvable algebra is solvable. Every finite product of solvable… ↩ 3 09Y2
Proposition 2 Let g be a Lie algebra. The following conditions are equivalent: ↩ 2 09Y3
Definition 2 The radical of a Lie algebra is its largest solvable ideal. ↩ 1 09Y4
Proposition 3 The radical r of a Lie algebra g is the smallest ideal of g such that g/r has radical 0. ↩ 2 09Y5
Proposition 4 Let g₁, …, gₙ be Lie algebras. The radical r of the product of the gᵢ is the product of the radicals rᵢ of the gᵢ. ↩ 1 09Y6
Definition 3 Let g be a Lie algebra. The nilpotent radical of g is the intersection of the kernels of the finite-dimensional simple representations of g. 09Y7
Remark (1) Let s be the nilpotent radical of g. As every decreasing sequence of vector subspaces of g is stationary, there exists a finite number of finite-dimensional simple representati… 09Y8
Lemma 1 Let V be a finite-dimensional vector space over K, g a subalgebra of gl(V) such that V is a simple g-module and a a commutative ideal of g. Then a ∩ Dg = 0. Let S be the subalgebra… ↩ 1 09Y9
Theorem 1 Let g be a Lie algebra, r its radical and s its nilpotent radical. Then s = Dg ∩ r. ↩ 10 09YA
Corollary 1 Let g be a solvable Lie algebra. The nilpotent radical of g is Dg. If ρ is a finite-dimensional simple representation of g, ρ(g) is commutative and the associative algebra L genera… ↩ 2 09YB
Corollary 2 (Lie’s Theorem) Let g be a solvable Lie algebra; suppose that K is algebraically closed. Let M be a g-module of finite dimension over K and let (Mᵢ)_0 ≤ i ≤ r be a Jordan-Hölder series of M. Then… ↩ 2 09YC
Remark (1) If (Mᵢ)_0 ≤ i ≤ r is replaced by another Jordan-Hölder series of M, the sequence (λ₁, …, λᵣ) is replaced by a sequence of the form (λ_π(1), …, λ_π(r)), where π is a permutation… ↩ 2 09YD
Corollary 3 Suppose that K is algebraically closed. If g is an r-dimensional solvable Lie algebra, every ideal of g is a term of a decreasing sequence of ideals of dimensions r, r-1, …, 0. 09YE
Corollary 4 Suppose that K = R. Let g be a solvable Lie algebra. Every simple representation of g is of dimension ≤ 2. Every ideal of g is a term of a decreasing sequence (gᵢ)_0 ≤ i ≤ m of ide… 09YF
Corollary 5 For a Lie algebra g to be solvable, it is necessary and sufficient that Dg be nilpotent. ↩ 2 09YG
Corollary 6 Let φ be a finite-dimensional representation of a Lie algebra g. Let r be the radical of g. Every element x ∈ r such that ρ(x) is nilpotent belongs to the largest nilpotency ideal… ↩ 3 09YH
Corollary 7 Let g be a Lie algebra and r its radical. The following four sets are identical: (a) the largest nilpotent ideal of g; (b) the largest nilpotent ideal of r; (c) the set of x ∈ r su… 09YI
Lemma 2 Let x be an endomorphism of a finite-dimensional vector space V and s (resp. n) its semi-simple (resp. nilpotent) component (cf. Algebra, Chapter VIII, § 9, no. 4, Definition 4). L… ↩ 2 09YJ
Lemma 3 Let M be a finite-dimensional vector space, A and B two vector subspaces of 𝔤𝔩(M) such that B ⊂ A and T the set of t ∈ 𝔤𝔩(M) such that [t, A] ⊂ B. If z ∈ T is such that Tr(zu) = 0… ↩ 2 09YK
Theorem 2 (Cartan's criterion) Let g be a Lie algebra, M a finite-dimensional vector space, ρ a representation of g on M and β the bilinear form on g associated with ρ. Then ρ(g) is solvable if and only if Dg is… ↩ 1 09YL
Proposition 5 Let g be a Lie algebra and r its radical. (a) If ρ is a finite-dimensional representation of g and β is the associated bilinear form, r and Dg are orthogonal with respect to β. (b)… ↩ 4 09YM
Corollary 1 Let g be a Lie algebra. Then g is solvable if and only if Dg is orthogonal to g with respect to the Killing form. 09YN
Corollary 2 The radical r of a Lie algebra g is a characteristic ideal. ↩ 2 09YO
Corollary 3 Let g be a Lie algebra, r its radical and a an ideal of g. Then the radical of a is equal to r ∩ a. ↩ 3 09YP
Proposition 6 Let g be a Lie algebra, r its radical and n its largest nilpotent ideal. Every derivation of g maps r into n. 09YQ
Corollary The largest nilpotent ideal of a Lie algebra is a characteristic ideal. 09YR
Remark To summarize some of the above results, note that, if r, n, s, t denote respectively the radical of g, the largest nilpotent ideal of g, the nilpotent radical of g and the orthogon… 09YS

§ 6. Semi-simple Lie algebras

Definition 1 Let g be a Lie algebra. g is called semi-simple if and only if the only commutative ideal of g is {0}. ↩ 1 09ZA
Remark (1) The algebra {0} is semi-simple. An algebra of dimension 1 or 2 is not semi-simple (cf. § 5, no. 1, Example 1). There exist semi-simple algebras of dimension 3 (cf. no. 7). 09ZB
Theorem 1 Let g be a Lie algebra. The following conditions are equivalent: ↩ 5 09ZC
Corollary Let g be a semi-simple Lie algebra and ρ a representation of g on a finite-dimensional vector space V. Then ρ(g) ⊂ 𝔰𝔩(V). 09ZD
Proposition 1 Let g be a semi-simple Lie algebra and φ a finite-dimensional faithful representation of g. Then the bilinear form on g associated with φ is non-degenerate. ↩ 9 09ZE
Corollary 1 Let g be a Lie algebra, β its Killing form and a a semi-simple sub-algebra of g. The orthogonal h of a with respect to β is a supplementary subspace of a in g and [a, h] ⊂ h. If a… ↩ 4 09ZF
Corollary 2 Every extension of a semi-simple Lie algebra by a semi-simple Lie algebra is semi-simple and trivial. ↩ 1 09ZG
Corollary 3 If g is semi-simple, every derivation of g is inner. ↩ 1 09ZH
Lemma 1 Let g be a semi-simple Lie algebra. The adjoint representation of g is semi-simple. Every ideal and every quotient algebra of g is semi-simple. ↩ 2 09ZI
Lemma 2 Let g be a Lie algebra. Then the following two conditions are equivalent: (a) All finite-dimensional linear representations of g are semi-simple. (b) Given a linear representation… ↩ 2 09ZJ
Lemma 3 Let g be a semi-simple Lie algebra, ρ a linear representation of g on a finite-dimensional vector space V and W a subspace of V of codimension 1 such that ρ(x)(V) ⊂ W for all x ∈ g… ↩ 2 09ZK
Theorem 2 (H. Weyl) Every finite-dimensional linear representation of a semi-simple Lie algebra is completely reducible. ↩ 5 09ZL
Definition 2 A Lie algebra g is called simple if the only ideals of g are {0} and g and if further g is non-commutative. ↩ 1 09ZM
Proposition 2 For a Lie algebra g to be semi-simple, it is necessary and sufficient that it be a product of simple algebras. 09ZN
Corollary 1 A semi-simple Lie algebra is the product of its simple ideals gᵢ. Every ideal of g is the product of certain of the gᵢ. ↩ 1 09ZO
Corollary 2 Let g, g' be two Lie algebras, r and r' their radicals and f a homomorphism of g onto g'. Then r' = f(r). 09ZP
Remark (1) Theorem 2 admits a converse: if every finite-dimensional representation of g is semi-simple, g is semi-simple. For since the adjoint representation is semi-simple, every ideal… 09ZQ
Proposition 3 Let M be a finite-dimensional vector space over K and g a semi-simple subalgebra of 𝔤𝔩(M). Then g contains the semi-simple and nilpotent components of its elements. ↩ 2 09ZR
Corollary An element x of g is a semi-simple (resp. nilpotent) endomorphism of M if and only if ad_g x is a semi-simple (resp. nilpotent) endomorphism of g. ↩ 2 09ZS
Definition 3 Let g be a semi-simple Lie algebra. An element x of g is called semi-simple (resp. nilpotent) if, for every g-module M of finite dimension over K, x_M is a semi-simple (resp. nilpo… ↩ 1 09ZT
Proposition 4 Let g, g' be semi-simple Lie algebras and f a homomorphism of g into g'. If x ∈ g is semi-simple (resp. nilpotent), so is f(x). If f is surjective, every semi-simple (resp. nilpote… ↩ 3 09ZU
Theorem 3 Let g be a semi-simple Lie algebra. ↩ 5 09ZV
Definition 4 A Lie algebra is called reductive if its adjoint representation is semi-simple. ↩ 1 09ZW
Proposition 5 Let g be a Lie algebra and r its radical. The following conditions are equivalent: (a) g is reductive. (b) Dg is semi-simple. (c) g is the product of a semi-simple algebra and a co… ↩ 8 09ZX
Remark If a Lie algebra g can be decomposed as a product a × b of a commutative Lie algebra a and a semi-simple Lie algebra b, this decomposition is unique. More precisely, the centre of… ↩ 1 09ZY
Corollary (a) Every finite product of reductive algebras is a reductive algebra. (b) If g is a reductive Lie algebra of centre c, every ideal of g is a direct factor, the product of its inte… ↩ 2 09ZZ
Proposition 6 Let g be a Lie algebra, r its radical and s its nilpotent radical. (a) s = [g, r] = Dg ∩ r. (b) s is the intersection of the orthogonals of g with respect to the bilinear forms ass… ↩ 5 0A00
Corollary Let g, g' be Lie algebras, s (resp. s') the nilpotent radical of g (resp. g') and f a homomorphism of g onto g'. (a) Then s' = f(s). (b) g' is reductive if and only if the kernel o… ↩ 1 0A01
Theorem 4 Let g be a Lie algebra, r its radical, ρ a finite-dimensional representation of g, g' = ρ(g) and r' = ρ(r). Then the following conditions are equivalent: (a) ρ is semi-simple; (b)… ↩ 10 0A02
Corollary 1 Let g be a Lie algebra and ρ and ρ' two finite-dimensional semi-simple representations of g. Then the tensor product of ρ and ρ' is semi-simple. ↩ 4 0A03
Corollary 2 Let g be a Lie algebra, ρ a semi-simple representation of g on a finite-dimensional vector space V, T and S the tensor and symmetric algebras of V and σ_T, σ_S the representations… ↩ 1 0A04
Corollary 3 Let g be a Lie algebra and ρ and ρ' two finite-dimensional semi-simple representations of g on spaces M and M'. Then the representation of g on L_K(M, M') canonically derived from… ↩ 2 0A05
Corollary 4 Let g be a Lie algebra, a an ideal of g and ρ a semi-simple representation of g. (a) The restriction ρ' of ρ to a is semi-simple. (b) If ρ is simple, ρ' is a sum of simple represen… 0A06
Definition 5 Let g be a Lie algebra and h a Lie subalgebra of g. h is called reductive in g if the representation x ↦ ad_g x of h is semi-simple. 0A07
Proposition 7 Let g be a Lie algebra, h a subalgebra reductive in g, ρ a representation of g on a vector space V and W the sum of the finite-dimensional subspaces of V which are simple h-modules… ↩ 3 0A08
Corollary 1 Let g be a Lie algebra, 𝔥 a subalgebra reductive in g and ρ a finite-dimensional semi-simple representation of g. Then the restriction of ρ to 𝔥 is semi-simple. ↩ 3 0A09
Corollary 2 Let g be a Lie algebra, 𝔥 a subalgebra reductive in g and 𝔨 a subalgebra of 𝔥 reductive in 𝔥. Then 𝔨 is reductive in g. ↩ 1 0A0A
Proposition 8 Let V be a finite-dimensional vector space. Then gl(V) is reductive. Its centre is the set of homotheties of V, its derived algebra is sl(V) and the latter is semi-simple. ↩ 1 0A0B
Example We identify sl(K²) with the Lie algebra of matrices of order 2 and trace zero. We write ↩ 1 0A0C
Proposition 9 Let V be a vector space of finite dimension n over K and β a non-degenerate symmetric (resp. alternating) bilinear form on V. Let g be the Lie algebra consisting of the x ∈ gl(V) s… ↩ 2 0A0D
Definition 6 A special automorphism of g is an automorphism of g of the form e^ad x, where x is in the nilpotent radical of g. ↩ 2 0A0E
Definition 7 Let g be a Lie algebra and r its radical. A Levi subalgebra of g is any subalgebra of g supplementary to r. ↩ 2 0A0F
Theorem 5 (Levi–Malcev) A Lie algebra g always has a Levi subalgebra s. Every Levi subalgebra of g is the image of s under a special automorphism. ↩ 7 0A0G
Corollary 1 Let s be a Levi subalgebra of g and h a semi-simple subalgebra of g. (a) There exists a special automorphism of g mapping h onto a subalgebra of s. (b) h is contained in a Levi sub… ↩ 2 0A0H
Corollary 2 For a subalgebra h of g to be a Levi subalgebra of g, it is necessary and sufficient that h be a maximal semi-simple subalgebra of g. 0A0I
Corollary 3 Let g be a Lie algebra and m an ideal of g such that g/m is semi-simple. Then g contains a subalgebra supplementary to m in g. In other words, every extension of a semi-simple Lie… ↩ 1 0A0J
Corollary 4 Let g be a Lie algebra, r its radical, s a Levi subalgebra of g and m an ideal of g. Then m is the direct sum of m ∩ r which is its radical and m ∩ s which is a Levi subalgebra of… 0A0K
Lemma 4 Let ρ, σ, τ be representations of g on vector spaces M, N, P. Suppose that we are given a K-bilinear mapping (m, n) ↦ m.n of M × N into P such that ↩ 1 0A0L
Lemma 5 Suppose further that σ and τ are semi-simple and hence N (resp. P) is the direct sum of the N_δ (resp. P_δ). For all n ∈ N (resp. p ∈ P), let n^natural (resp. p^natural) be its com… ↩ 2 0A0M
Theorem 6 Let g be a Lie algebra, V a semi-simple g-module of finite dimension over K, S the symmetric algebra of V and x_S the derivation of S which extends x_V (so that x ↦ x_S is a repres… ↩ 1 0A0N

§ 7. Ado’s Theorem

Definition 1 Let g be a Lie algebra, g' a subalgebra of g and ρ' a representation of g' on V'. A representation ρ of g on V is called an extension of ρ' to g if there exists an injective homomo… 0A1F
Theorem 1 (Zassenhaus) Let g = g' + h be a Lie algebra which is the direct sum of an ideal g' and a subalgebra h and ρ' a finite-dimensional representation of g' whose largest nilpotency ideal contains [… ↩ 4 0A1G
Lemma 1 Let g = g' + h be a Lie algebra which is the sum of an ideal g' and a subalgebra h. Let σ be a finite-dimensional representation of g. Suppose that σ(x) is nilpotent for all x ∈ g'… 0A1H
Proposition 1 Let g be a Lie algebra, n its largest nilpotent ideal, a a nilpotent ideal of g and ρ a finite-dimensional representation of a such that every element of ρ(a) is nilpotent. Then ρ… ↩ 1 0A1I
Theorem 2 Every Lie algebra has a finite-dimensional faithful linear representation. ↩ 2 0A1J
Theorem 3 (Ado) Let g be a Lie algebra and n its largest nilpotent ideal. There exists a finite-dimensional faithful representation φ of g such that every element of φ(n) is nilpotent. 0A1K

II. Free Lie algebras

§ 1. Enveloping bigebra of a Lie algebra

Definition 1 An element x of E is called u-primitive if ↩ 1 0A1Q
Proposition 1 Every u-primitive element of E belongs to E⁺. (2) implies x = ε(x).u + ε(u).x = ε(x).u + x, whence ε(x) = 0. ↩ 2 0A1R
Remark If x ∈ E and c(x) = x' ⊗ u + u ⊗ x'', where x', x'' are in E⁺, then x = ε(x').u + ε(u).x'' = x''; similarly x = x' and x is u-primitive. 0A1S
Proposition 2 We have ↩ 2 0A1T
Proposition 3 If the cogebras (E, c) is coassociative (resp. cocommutative) (Algebra, Chapter III, § 11, no. 2), so is the cogebras (E⁺, cᵤ⁺). ↩ 1 0A1U
Lemma 1 Let π : E → E' be a surjective cogebras morphism. If E is coassociative (resp. cocommutative), so is E'. 0A1V
Proposition 4 The set P(E) of primitive elements of E is a Lie subalgebra of E. ↩ 2 0A1W
Proposition 5 Let f : E → E' be a bigebra morphism. If x is a primitive element of E, then f(x) is a primitive element of E' and the restriction of f to P(E) is a Lie algebra homomorphism P(f)… 0A1X
Remark (1) Let p be a prime number such that p · 1 = 0 in K. The binomial formula and the congruences binompi ≡ 0 pmodp for 1 ≤ i ≤ p - 1 imply that 0A1Y
Definition 2 Let E be a bigebra with coproduct c. A filtration compatible with the bigebra structure on E is an increasing sequence (Eₙ)_n ≥ 0 of submodules of E such that ↩ 1 0A1Z
Example Let E be a graded bigebra (Algebra, Chapter III, § 11, no. 4, Definition 3) and (Eⁿ)_n ≥ 0 its graduation. We write Eₙ = ∑ᵢ₌₀ⁿ Eⁱ. The sequence (Eₙ) is a filtration compatible with… 0A20
Proposition 6 Let E be a filtered bigebra and (Eₙ)_n ≥ 0 its filtration. For every integer n ≥ 0, let Eₙ⁺ = Eₙ ∩ E⁺. Then E₀⁺ = 0 and ↩ 2 0A21
Corollary The elements of E₁⁺ are primitive. If x ∈ E₁⁺, then c⁺(x) = 0 by (7), whence (5). ↩ 1 0A22
Proposition 7 There exists on the algebra U one and only one coproduct c which makes U into a bigebra such that the elements of σ(g) are primitive. The bigebra (U, c) is cocommutative; its couni… 0A23
Definition 3 The bigebra (U, c) is called the enveloping bigebra of the Lie algebra g. 0A24
Proposition 8 Let E be a bigebra with coproduct denoted by cE and let h be a Lie algebra homomorphism of g into P(E) (no. 2, Proposition 4). The unital algebra homomorphism f : U → E such that f… ↩ 2 0A25
Corollary Let gi (i = 1, 2) be a Lie algebra, U(gi) its enveloping bigebra and σi : gi → U(gi) the canonical mapping. For every Lie algebra homomorphism h : g1 → g2, the unital algebra homom… 0A26
Proposition 9 For every integer n ≥ 0, let Uⁿ be the vector subspace of U generated by the σ(x)ⁿ for x ∈ 𝔤. 0A27
Corollary The canonical mapping σ defines an isomorphism of 𝔤 onto the Lie algebra P(U) of primitive elements of U. ↩ 2 0A28
Remark (1) Uₙ = ∑ᵢ₌₀ⁿ Uⁱ (Chapter I, § 2, no. 7, Corollary 4 to Theorem 1). 0A29
Proposition 10 The mapping λ ↦ f_λ is an isomorphism of the algebra U' onto the algebra of formal power series K[[Xᵢ]]_i ∈ I. ↩ 2 0A2A
Theorem 1 Let E be a cocommutative bigebra. (a) The bigebra morphism f_E : U(P(E)) → E is injective. (b) If there exists on E a filtration compatible with its bigebra structure (no. 3, Defin… ↩ 4 0A2B
Lemma 2 Let V be a vector space, E a cogebra and f : S(V) → E a cogebra morphism. If the restriction of f to S⁰(V) + S¹(V) is injective, then f is injective. Let n ≥ 0; we write Sₙ = ∑_i ≥… ↩ 1 0A2C
Lemma 3 If a family of scalars (λ_α, β) of finite support (with α, β in N^(Ω) - 0) satisfies relations (20) and (21), there exists a family (μ_α)_|α| ≥ 2 of finite support such that μ_α +… 0A2D

§ 2. Free Lie algebras

Definition 1 The free Lie algebra over the set X is the quotient algebra L(X) = Lib(X)/a, where a is the two-sided ideal of Lib(X) generated by the elements of one of the forms (1) Q(a) = a.a f… ↩ 1 0A2Q
Proposition 1 Let ψ be the canonical mapping of Lib(X) onto L(X) and φ the restriction of ψ to X. For every mapping f of X into a Lie algebra g, there exists one and only one homomorphism F: L(X… ↩ 7 0A2R
Corollary 1 The family (φ(x))_x ∈ X is free over K in L(X). ↩ 3 0A2S
Corollary 2 Let a be a Lie algebra. Every extension of L(X) by a is inessential. ↩ 1 0A2T
Remark When X is empty, M(X) is empty and hence L(X) = 0. If X consists of a single element x, the submodule K.x of L(X) is a Lie subalgebra of 0A2U
Proposition 2 Let X and Y be two sets. Every mapping u : X → Y can be extended uniquely to a Lie algebra homomorphism L(u) : L(X) → L(Y). For every mapping v : Y → Z, L(v ∘ u) = L(v) ∘ L(u). ↩ 1 0A2V
Corollary If u is injective (resp. surjective, bijective), so is L(u). 0A2W
Proposition 3 Let K' be a non-zero commutative ring and u : K → K' a ring homomorphism. For every set X there exists one and only one Lie K'-algebra homomorphism ↩ 1 0A2X
Remark Let 𝔥' be a Lie K'-algebra and 𝔥 the Lie K-algebra derived from 𝔥' by restricting the ring of scalars. If P ∈ L_K(X), we can define tildeP : 𝔥^X → 𝔥 (no. 4). We see immediately tha… 0A2Y
Lemma 1 The ideal a of Definition 1 is graded. For a, b in Lib(X), let B(a, b) = a.b + b.a. The formulae (10) B(a, b) = Q(a + b) - Q(a) - Q(b) (11) Q(λ₁.w₁ + ⋯ + λₙ.wₙ) = ∑ᵢ λᵢ² Q(wᵢ) + ∑_… 0A2Z
Proposition 4 Let S be a subset of X. If N^(S) is identified with its canonical image in N^(X) (Algebra, Chapter I, § 7, no. 7), then L(S) = ∑_α ∈ N^(S) L^α(X). Further, for all α ∈ N^(S), the h… ↩ 3 0A30
Corollary For every family (Sᵢ)_i ∈ I of subsets of X, L(⋂_i ∈ I Sᵢ) = ⋂_i ∈ I L(Sᵢ). This follows from Proposition 4 and the obvious formula N^(S) = ⋂_i ∈ I N^(Sᵢ) where we have written S =… 0A31
Proposition 5 Let g be a Lie algebra and P a submodule of g. We define the submodules Pₙ of g by the formulae P₁ = P and Pₙ₊₁ = [P, Pₙ] for n ≥ 1. Then [Pₘ, Pₙ] ⊂ Pₘ₊ₙ, Pₙ = ∑ₚ₌₁ⁿ⁻¹ [Pₚ, Pₙ₋ₚ] f… ↩ 1 0A32
Proposition 6 Let g be a Lie algebra and (Cⁿ g)_n ≥ 1 the lower central series of g. Then [Cᵐ g, Cⁿ g] ⊂ Cᵐ⁺ⁿ g for m ≥ 1 and n ≥ 1. 0A33
Proposition 7 Let X be a set and n an integer ≥ 1. (a) Lⁿ⁺¹(X) = [L¹(X), Lⁿ(X)]. (b) The module Lⁿ(X) is generated by the elements [x₁, [x₂, …, [xₙ₋₁, xₙ] …]] where (x₁, …, xₙ) runs through the… ↩ 2 0A34
Corollary Let g be a Lie algebra and (xᵢ)_i ∈ I a generating family of g. The n-th term Cⁿ g of the lower central series of g is the module generated by the iterated brackets [x_i₁, [x_i₂, …… 0A35
Proposition 8 Let X be a set, let M be an L(X)-module and let d be a mapping of X into M. There exists one and only one linear mapping D of L(X) into M extending d and satisfying the relation: D… 0A36
Corollary Every mapping of X into L(X) can be extended uniquely to a derivation of L(X). ↩ 1 0A37
Proposition 9 Let S₁ and S₂ be two disjoint sets and d a mapping of S₁ × S₂ into L(S₂). Let g be the quotient Lie algebra of L(S₁ ∪ S₂) by the ideal generated by the elements [s₁, s₂] - d(s₁, s₂… ↩ 1 0A38
Proposition 10 (elimination theorem) Let X be a set, S a subset of X and T the set of sequences (s₁, …, sₙ, x) with n ≥ 0, s₁, …, sₙ in S and x in X - S.† ↩ 1 0A39
Corollary Let y ∈ X. The free Lie algebra L(X) is the direct sum of the free submodule K.y and the Lie subalgebra admitting as basic family the family of ((ad\ y)ⁿ, z) where n geqslant 0 and… ↩ 2 0A3A
Definition 2 A Hall set relative to X is any totally ordered subset H of M(X) satisfying the following conditions: (A) If u ∈ H, v ∈ H and l(u) < l(v), then u < v. (B) X ⊂ H and H ∩ M²(X) consi… ↩ 3 0A3B
Proposition 11 There exists a Hall set relative to X. ↩ 2 0A3C
Proposition 12 Let H be a Hall set relative to X and let x, y be in X. (a) H ∩ M(x) = x. (b) Suppose that x < y and let d_y be the homomorphism of M(X) into N such that d_y(y) = 1 and d_y(z) = 0… ↩ 2 0A3D
Corollary If Card X ≥ 2, then H ∩ Mⁿ(X) ≠ ∅ for every integer n ≥ 1. ↩ 1 0A3E
Proposition 13 Let X be a finite set with at least two elements. Let H denote a Hall set relative to X. Then there exist a strictly increasing bijection p ↦ wₚ of N onto H and a sequence (Pₚ)_p ∈… ↩ 2 0A3F
Example Suppose that X has two elements x, y; let X be ordered such that x < y. The construction given in the proof of Proposition 11 gives a set H with 14 elements of length ≤ 5 given in… 0A3G
Theorem 1 Let H be a Hall set relative to X and Ψ the canonical mapping of M(X) into the free Lie algebra L(X). The restriction of Ψ to H is a basis of the module L(X). ↩ 4 0A3H
Lemma 2 For every integer p ≥ 0, the module Lₚ admits the family (barwᵢ)_0 ≤ i < p as basis, the Lie algebra gₚ admits (baru)_u ∈ Pₚ as basic family and the module L(X) is the direct sum o… 0A3I
Corollary The module L(X) is free and so is each of the submodules L^α(X) where α ∈ N^(X) and Lⁿ(X) where n ∈ N. The modules L^α(X) are of finite rank and so are the modules Lⁿ(X) if X is fi… ↩ 2 0A3J
Definition 3 A Hall basis of a free Lie algebra L(X) is any basis of L(X) which is the canonical image of a Hall set relative to X. 0A3K
Remark Suppose that X consists of two distinct elements x and y and let L^(·,1) be the submodule of L(X) the sum of the L^α(X) where α ∈ N^X and α(y)=1. It follows immediately from Theore… 0A3L

§ 3. Enveloping algebra of the free Lie algebra

Theorem 1 Let α : L(X) → A(X) be the unique Lie algebra homomorphism extending the canonical injection of X into A(X) (§ 2, no. 2, Proposition 1). Let σ : L(X) → U(L(X)) be the canonical map… ↩ 4 0A42
Corollary 1 There exists on the algebra A(X) a unique coproduct making A(X) into a bigebra such that the elements of X are primitive. Further, β is an isomorphism of the bigebra U(L(X)) onto A… ↩ 2 0A43
Corollary 2 If K is a field of characteristic 0, L(X) is the Lie algebra of primitive elements of A(X). ↩ 2 0A44
Remark (1) Let K' be a commutative ring containing K. If A(X), L(X) and L_K'(X) are identified with subsets of A_K'(X), we deduce from part (a) of Theorem 1 the relation 0A45
Proposition 1 (a) The restriction π₀ of π to L(X) is a derivation of L(X). (b) For every integer n ≥ 1 and all u in Lⁿ(X), π(u) = n.u. ↩ 1 0A46
Corollary Suppose that K is a Q-algebra. Let P be the linear mapping of A⁺(X) into itself such that ↩ 1 0A47
Remark Suppose that K is a field of characteristic zero and let Q be the projector of A(X) = U(L(X)) onto L(X) associated with the canonical graduation of U(L(X)), cf. § 1, no. 5. For α ∈… 0A48
Lemma 1 Let n be an integer 0, T₁, …, Tₙ indeterminates and u₁, …, uₙ elements of Z. Let (c(α))_α ∈ Nⁿ - 0 be a family of elements of Z such that ↩ 1 0A49
Theorem 2 Let X be a finite set and n = Card(X). ↩ 1 0A4A
Example We have beginalign* c(1) &= n, & c(2) &= 1/2(n² - n), & c(3) &= 1/3(n³ - n), \\ c(4) &= 1/4(n⁴ - n²), & c(5) &= 1/5(n⁵ - n), & c(6) &= 1/6(n⁶ - n³ - n² + n). endalign 0A4B
Remark Let X be a set and let α ∈ N^(X); the rank of the free K-module L^α(X) is also given by formula (16). This follows immediately from Theorem 2 (b) and Proposition 4 of § 2, no. 6. 0A4C

§ 4. Central filtrations

Definition 1 Let G be a group. A real filtration on G is a family (G_α)_α ∈ R of subgroups of G such that G_α = ⋂_β < α G_β for all α ∈ R. 0A4H
Remark Let (Gₙ)_n ∈ Z be a decreasing sequence of subgroups of G. It is a decreasing filtration in the sense of Commutative Algebra, Chapter III, § 2, no. 1, Definition 1. For every integ… 0A4I
Remark If v(x) is not everywhere equal to +∞, conditions (7) and (8) imply v(1) = 0. 0A4J
Remark When the filtration (G_α) is integral, gr_α(G) = 0 for non-integral α and grₙ(G) = Gₙ / Gₙ₋₁ for every integer n. The definition of the associated graded group therefore coincides… 0A4K
Definition 2 Let G be a group. A real filtration (G_α) on G is called central if G = ⋃_α > 0 G_α and the commutator (x, y) = x⁻¹y⁻¹xy of an element x of G_α and an element y of G_β belongs to G… ↩ 1 0A4L
Proposition 1 (i) Let α, β be in R. There exists a biadditive mapping φ_α β : gr_α(G) × gr_β(G) → gr_α + β(G) 0A4M
Proposition 2 The set Γ is a subgroup of A^* and (Γ_α) is a central filtration on Γ. ↩ 2 0A4N
Remark Let α ≥ 0, β ≥ 0 and x ∈ Γ_α, y ∈ Γ_β. By formulae (21), (22) and (23), (24) x⁻¹ - 1 ≡ -(x - 1) pmodA_2α (25) xy - 1 ≡ (x - 1) + (y - 1) pmodA_α + β (26) (x, y) - 1 ≡ [(x - 1), (y… 0A4O
Proposition 3 (i) For all α ∈ R, there exists a unique group homomorphism g_α : gr_α(G) → gr_α(A) which maps the class modulo G_α⁺ of an element a ∈ G_α to the class modulo A_α⁺ of ρ(a) - 1. ↩ 2 0A4P
Proposition 4 (i) The lower central series of G is an integral central filtration on G. ↩ 1 0A4Q
Proposition 5 Let G be a group and gr(G) the graded Lie Z-algebra associated with the lower central filtration on G. Then gr(G) is generated by gr₁(G) = G/(G, G). ↩ 2 0A4R
Remark Let k be a ring, n an integer > 0 and A the set of lower triangular matrices with n rows and n columns and elements in k. For p ≥ 0, let Aₚ be the set of (xᵢⱼ) ∈ A such that xᵢⱼ =… 0A4S

§ 5. Magnus algebras

Proposition 1 Let B be a unital associative algebra with a real filtration (B_α)_α ∈ R such that B is Hausdorff and complete (§ 4, nos. 1 and 2). Let f be a mapping of X into B such that there e… ↩ 4 0A5B
Lemma 1 For an element a of hatA(X) to be invertible, it is necessary and sufficient that its constant term be invertible in K. 0A5C
Theorem 1 Let r be a mapping of X into hatA(X) such that ω(r(x)) ≥ 2 for all x ∈ X. The unique homomorphism g of the free group F(X) into the Magnus group Γ(X) such that g(x) = 1 + x + r(x)… ↩ 3 0A5D
Lemma 2 Let n be a non-zero rational integer. In the ring of formal power series K[[t]] we write (1+t)ⁿ = ∑_j ≥ 0 c_j,n tʲ. There exists an integer j ≥ 1 such that c_j,n ≠ 0. ↩ 1 0A5E
Lemma 3 Let x₁, …, xₛ be elements of X such that s ≥ 1 and xᵢ ≠ xᵢ₊₁ for 1 ≤ i ≤ s-1; let n₁, …, nₛ be non-zero rational integers. Then the element ∏ᵢ₌₁ˢ (1+xᵢ)^nᵢ of hatA(X) is ≠ 1. 0A5F
Lemma 4 Let σ be the continuous endomorphism of hatA(X) such that σ(x) = x + r(x) for x ∈ X (no. 1, Proposition 1). Then σ is an automorphism and σ(hatAₘ(X)) = hatAₘ(X) for all m ∈ N. ↩ 1 0A5G
Theorem 2 Suppose that in the ring K the relation n.1 = 0 implies n = 0 for every integer n. Let r be a mapping of X into hatA(X) such that ω(r(x)) ≥ 2 for x ∈ X and let g be the homomorphis… ↩ 3 0A5H
Theorem 3 For all x ∈ X, let c(x) be the canonical image of x in F(X)/(F(X), F(X)). Let g be the graded Lie Z-algebra associated with the filtration (Cⁿ F(X))_n ≥ 1 of F(X) (§ 4, no. 6). The… ↩ 1 0A5I
Corollary ⋂_n geqslant 1 Cⁿ F(X) = e. 0A5J
Remark Let H be a Hall set relative to X (§ 2, no. 10). Let M be the magma defined by the law of composition (x, y) \mapsto (x, y) = x^{-1}y^{-1}xy on F(X) and let φ be the homomorphism o… 0A5K
Example Suppose that X is a set with two elements x, y and let H1 = \{x, y\}, H2 = \{xy\}. Every element w of F(X) can therefore be written w ≡ xᵃ yᵇ (x, y)ᶜ mod. C³ with a, b, c in Z. For… 0A5L
Proposition 2 Suppose that X is finite. For every integer n \geqslant 1, the group F(X)/Fn^{(p)}(X) is a finite p-group of nilpotency class \leqslant n. ↩ 1 0A5M
Proposition 3 For all w ≠ 1 in F(X), there exist a finite p-group G and a homomorphism f of F(X) into G such that f(w) ≠ 1. ↩ 1 0A5N
Corollary The intersection of the normal subgroups of finite index in F(X) is 1. 0A5O

§ 6. The Hausdorff series

Proposition 1 The exponential mapping is a homeomorphism of m onto 1 + m and the logarithmic mapping is the inverse homeomorphism. ↩ 2 0A61
Remark (1) If x ∈ m, y ∈ m and x and y commute, then ↩ 1 0A62
Theorem 1 The restriction of the exponential mapping of hatA(X) to hatL(X) is a bijection of hatL(X) onto a closed subgroup of the Magnus group Γ(X). 0A63
Remark The above proof shows that exp(hatL) is the subgroup Δ of Γ consisting of the b satisfying (4). 0A64
Lemma 1 Let 𝔤 be a filtered Lie algebra (§ 4, no. 1), (𝔤_α)_α ∈ R its filtration and let α ∈ R. Let P be a homogeneous Lie polynomial of degree n in the indeterminates (Γᵢ)_i ∈ I (§ 2, no… ↩ 1 0A65
Proposition 2 The homomorphism fₜ : L(I) → g such that fₜ(Tᵢ) = tᵢ (§ 2, no. 4) can be extended by continuity to one and only one continuous homomorphism hatfₜ of hatL(I) into g. ↩ 1 0A66
Definition 1 The element H = U circledast V = log(exp U · exp V) (no. 2) of the Lie algebra hatL_Q(U, V) is called the Hausdorff series in the indeterminates U and V. ↩ 4 0A67
Theorem 2 If r and s are two positive integers such that r + s ≥ 1, then H_r,s = H'_r,s + H''_r,s, where 0A68
Remark (1) We have defined (§ 3, no. 2, Remark) a projector Q of A(X) onto L(X) such that Q(aᵐ) = 0 for a ∈ L(X) and m geqslant 2 and Q(1) = 0. Then H = Q(exp H) = Q(exp U.exp V), whence… 0A69
Proposition 3 If a ∈ m, b ∈ m, then exp H(a, b) = exp a . exp b. ↩ 4 0A6A
Remark 1 If a and b commute, then H_r,s(a, b) = 0 for r + s ≥ 2, for every homogeneous Lie polynomial of degree ≥ 2 is zero at (a, b). Then H(a, b) = a + b and Proposition 3 recovers the fo… 0A6B
Proposition 4 Let g be a complete Hausdorff filtered Lie algebra such that g = ⋃_α > 0 g_α. The mapping (a, b) ↦ H(a, b) is a group law on g compatible with the topology on g under which 0 is th… ↩ 3 0A6C
Remark (2) Take g to be the Lie algebra hatL(X). The group law introduced in the above proposition coincides with the law defined in no. 2. In other words, a mathbinh b = H(a, b) for a, b… ↩ 1 0A6D
Proposition 5 Let K_r,s be the component of H(U + V, -U) of multidegree (r, s). Then K_n,1(U, V) = 1/((n+1)!) (ad U)ⁿ(V) for n ≥ 0. ↩ 2 0A6E

§ 7. Convergence of the Hausdorff series (real or complex case)

Lemma 1 | tildeH_r,s | ≤ M^r+s-1 η_r,s. Let rᵢ, sᵢ be in N for 1 ≤ i ≤ m, with sₘ = 1; we write r = ∑ᵢ rᵢ, s = ∑ᵢ sᵢ and consider the following element of L(U, V): Z = ( ( ∑ᵢ₌₁ᵐ⁻¹ (ad\ U)^… ↩ 1 0A6L
Proposition 1 The formal power series tildeH is a convergent series (Differentiable and Analytic Manifolds, R, 3.1.1); its domain of absolute convergence (Differentiable and Analytic Manifolds… ↩ 2 0A6M
Proposition 2 Let 0A6N
Proposition 3 For |x| + |y| < 1/2 log 2, (23) exp_A x · exp_A y = exp_A h(x, y). It follows from (18) and the relation eᵘ eᵛ = eᵘ⁺ᵛ that m ∘ (1 + tildee, 1 + tildee) = (1 + tildee) ∘ tildeH in h… 0A6O

§ 8. Convergence of the Hausdorff series (ultrametric case)

Lemma 1 Let n be an integer geqslant 0 and let n = n₀ + n₁ p + ⋯ + nₖ pᵏ, with 0 leqslant nᵢ leqslant p - 1, be the p-adic expansion of n. Let S(n) = n₀ + n₁ + ⋯ + nₖ. Then vₚ(n!) = (n - S… ↩ 2 0A6Q
Lemma 2 v(n) leqslant v(n!) leqslant (n - 1)θ and v(n) leqslant (log n)/(log p) for every integer n geqslant 1. v(n!) = vₚ(n!) = (n - S(n))θ leqslant (n - 1)θ by Lemma 1. On the other hand… ↩ 3 0A6R
Proposition 1 Let r and s be two integers geqslant 0. If H_r,s = ∑_b ∈ B λ_b e_b, where λ_b ∈ Q, is the decomposition of H with respect to the basis (e_b)_b ∈ B, then vₚ(λ_b) geqslant -(r + s -… ↩ 3 0A6S
Definition 1 A normed Lie algebra over K is a Lie algebra with a norm such that |x + y| ≤ sup(|x|, |y|) |[x, y]| ≤ |x| · |y| for all x, y in g. 0A6T
Lemma 3 | tildeH_r,s | ≤ a^-(r+s-1)θ. ↩ 2 0A6U
Proposition 2 The formal power series tildeH is a convergent series (Differentiable and Analytic Manifolds, R, 4.1.1). If G is the ball x ∈ g | | x | < a^θ , the domain of absolute convergence o… ↩ 3 0A6V
Proposition 3 Let G be the ball x ∈ g | | x | < a^θ . The analytic mapping h : G × G → G makes G into a group in which 0 is the identity element and -x is the inverse of x for all x ∈ G. Moreove… ↩ 1 0A6W
Proposition 4 Let R be a real number such that 0 < R ≤ a^θ. The mapping exp_A defines an analytic isomorphism of G_R onto 1 + G_R and the inverse isomorphism is the restriction of log_A to 1 + G… 0A6X
Proposition 5 For x, y in G, exp_A · exp_A y = exp_A h(x, y). e^U eⱽ = e^H(U, V) and hence m ∘ (1 + tildeε, 1 + tildeε) = (1 + tildeε) ∘ tildeH in tildeH(A × A; A) (where m denotes multiplicatio… 0A6Y

App. 0. Möbius function

Proposition (i) The function μ is the unique mapping of N^* into Z such that μ(1) = 1 and ∑_d|n μ(d) = 0 for every integer n > 1. 0A73

III. Lie groups

§ 1. Lie groups

Definition 1 A Lie group over K is a set G with a group structure and an analytic K-manifold structure such that these two structures are compatible. 0A76
Lemma 1 Let G be a Lie group, U an open neighbourhood of e, E a complete normed space and φ : U → E a chart of the manifold G. There exists a neighbourhood W of e contained in U such that… ↩ 1 0A77
Proposition 1 A Lie group is a complete metrizable topological group. ↩ 2 0A78
Proposition 2 Let G be a Lie group. (i) If K = R or C, G is locally connected. (ii) If K is distinct from R and C, G is zero-dimensional (General Topology, Chapter IX, § 6, Definition 5). (iii)… 0A79
Corollary If K = R or C and G is connected and finite-dimensional, then G is locally connected and locally compact and its topology admits a countable base. 0A7A
Lemma 2 Let X be a manifold of class C^r, e a point of X, U and V open neighbourhoods of e and m a mapping of class C^r of U × U into X satisfying the following conditions: (a) m(e, x) = m… ↩ 1 0A7B
Proposition 3 Let X be an analytic manifold and m an analytic associative law of composition on X admitting an identity element. The set G of invertible elements of X is open in X and G is a Lie… 0A7C
Definition 2 Let G and H be Lie groups. A Lie group morphism of G into H (or simply a morphism of G into H if no confusion can arise) is a mapping of G into H which is a group homomorphism and… 0A7D
Example (1) Let G be a Lie group. For all x ∈ G, Int(x) is an automorphism of the Lie group G. 0A7E
Proposition 4 Let G and H be Lie groups and f a homomorphism of the group G into the group H. For f to be analytic, it is necessary and sufficient that there exist a non-empty open subset U of G… ↩ 1 0A7F
Remark If f is an immersion at e (resp. a submersion at e), clearly f is an immersion (resp. a submersion). 0A7G
Definition 3 Let G be a Lie group. A subset H of G is called a Lie subgroup if H is a subgroup and a submanifold of G. ↩ 1 0A7H
Proposition 5 Let G be a Lie group and H a Lie subgroup of G. (i) H is closed in G. (ii) The canonical injection of H into G is a Lie group morphism. (iii) Let L be a Lie group and f a mapping o… ↩ 1 0A7I
Proposition 6 Let G be a Lie group and H a subgroup of G. For H to be a Lie subgroup of G, it is necessary and sufficient that there exist a point h ∈ H and an open neighbourhood U of h in G suc… ↩ 3 0A7J
Remark If "submanifold" is replaced by "quasi-submanifold" in Definition 3, we obtain the definition of Lie quasi-subgroups of G. (For finite-dimensional G, the Lie quasi-subgroups are ju… 0A7K
Proposition 7 If the mapping (m, l) ↦ σ(l)m of M × L into M is analytic, the group S, with the product manifold structure of M and L, is a Lie group. 0A7L
Example Let E be a normable space, G = GL(E), T the translation group of E and A the permutation group of E generated by G and T. The group A is algebraically the semi-direct product of G… ↩ 1 0A7M
Proposition 8 Let G and H be Lie groups, p: G → H and s: H → G Lie group morphisms such that p ∘ s = id_H and N = Ker\ p. Then N is a Lie subgroup of G, s is an isomorphism of H onto a Lie subgr… ↩ 2 0A7N
Proposition 9 Let x ∈ X and g₀ ∈ G. ↩ 3 0A7O
Corollary Let x ∈ X. If K is of characteristic 0 and X is finite-dimensional, ρ(x) is a subimmersion. If further ρ(x) is injective, ρ(x) is an immersion. ↩ 1 0A7P
Proposition 10 Let G be a Lie group and X a manifold of class Cʳ with a law of left operation of class Cʳ of G on X. Suppose that: ↩ 4 0A7Q
Remark With the above hypotheses, further let H be a manifold of class Cʳ and (x, h) ↦ m(x, h) a mapping of class Cʳ of X × H into X such that m(gx, h) = gm(x, h) for x ∈ X,\ g ∈ G,\ h ∈… 0A7R
Proposition 11 Let X be a Lie group and G a Lie subgroup of X. ↩ 1 0A7S
Proposition 12 Let X be a Lie group and Y a non-empty analytic manifold with a law of analytic left operation of X on Y. For all y ∈ Y, let φ(y) be the orbital mapping by y and X_y the stabilizer… 0A7T
Example Let G be a Lie group. We make G × G operate on G on the left by (g₁, g₂)x = g₁xg₂⁻¹. Let ρ be the orbital mapping of e. Then the restrictions of T_(e, e)(ρ) to T_(e, e)(G × e) = Tₑ… 0A7U
Proposition 13 Let G be a Lie group, H a normal Lie subgroup of G, X a manifold of class Cʳ and (g, x) ↦ gx a law of left operation of class Cʳ of G on X. Suppose that conditions (a) and (b) of P… 0A7V
Corollary Let G be a Lie group and H and L normal Lie subgroups of G with L ⊂ H. Then H/L is a normal Lie subgroup of G/L and the canonical bijection of G/H onto (G/L)/(H/L) is a Lie group i… 0A7W
Proposition 14 Let G be a Lie group, X an analytic manifold and (g, x) ↦ gx a law of analytic left operation of G on X. Let x ∈ X. Suppose that the corresponding orbital mapping ρ(x) is a subimme… ↩ 6 0A7X
Remark Let G be a finite-dimensional Lie group, X a manifold of class C^r and (g, x) ↦ gx a law of left operation of class C^r of G on X. Then Proposition 14 remains valid. The only point… 0A7Y
Corollary Let G be a Lie group whose topology admits a countable base and X a non-empty finite-dimensional analytic manifold with a law of analytic left operation of G on X. Suppose that G o… 0A7Z
Definition 4 If ψ is a φ-morphism of vector bundles of class C^r, E is called a vector G-bundle of class C^r. ↩ 3 0A80
Proposition 15 If E is a vector G-bundle of class C^r, τE is a vector G-bundle of class C^r. ↩ 1 0A81
Proposition 16 Let G be a Lie group, X a manifold of class C^r (r ≥ 2) and (g, x) ↦ gx a law of left operation of class C^r of G on X, whence, by transporting the structure, there is a law of lef… ↩ 2 0A82
Corollary If τ is a vector functor of class C^r for isomorphisms, τ(TX) is a vector G-bundle of class C^{r-1}. ↩ 2 0A83
Remark 1 If τ is a vector functor of class C^r for isomorphisms in finite dimension and E is of finite rank, τE is defined similarly and Proposition 15 remains valid; the Corollary to Propo… 0A84
Example With the hypotheses and notation of Proposition 16, let F be a complete normable space. Then L((TX)ᵖ; F) is a vector G-bundle of class C^{r-1}; so is Alt^p(TX; F) if K is of charac… 0A85
Proposition 17 Let G be a Lie group, X a left Lie homogeneous space of G, x0 a point of X, G0 the stabilizer of x0 in G, E and E' left vector G-bundles of class C^r and base space X, E0 (resp. E'… ↩ 2 0A86
Corollary 1 Let E0^{G0} be the set of elements of E0 which are invariant under G0. For all u ∈ E0^{G0}, let σu be the mapping of X into E defined by σu(gx0) = gu for all g ∈ G. (i) The G-invar… ↩ 3 0A87
Corollary 2 Suppose that G₀ operates trivially on E₀. Let E' be the trivial G-bundle with base space X and fibre E₀. There exists one and only one isomorphism of E onto E' compatible with the… ↩ 1 0A88
Remark 2 In this no., the laws of left operation can be replaced throughout by laws of right operation. 0A89
Proposition 18 Let G be a group and U and V two subsets of G containing e. Suppose that U has an analytic manifold structure satisfying the following conditions: (i) V = V⁻¹, V² ⊂ U, V is open in… ↩ 8 0A8A
Proposition 19 Let G be a topological group, H a Lie group and f a homomorphism of the group G into the group H. Suppose that there exist an open neighbourhood of eG in G, a chart (V, \phi, E) of… 0A8B
Corollary Let G be a topological group, N a discrete normal subgroup of G and \pi the canonical mapping of G onto G/N. Suppose that an analytic manifold structure is given on G/N, compatible… 0A8C
Remark Let H be a connected real or complex Lie group, \tilde{H} its universal covering† and \pi the canonical mapping of \tilde{H} onto H. When we speak of \tilde{H} 0A8D
Definition 5 A Lie group germ over K is a system (G, e, θ, m) satisfying the following conditions: (i) G is an analytic manifold over K; (ii) e ∈ G; (iii) θ is an analytic mapping of G into G… 0A8E
Proposition 20 Let G be a Lie group germ and g ∈ G. There exist an open neighbourhood U of e and an open neighbourhood V of g with the following properties: (a) ug is defined for all u ∈ U; (b) v… ↩ 1 0A8F
Proposition 21 Let G, H be two Lie group germs and φ a morphism of G into H. The following conditions are equivalent: ↩ 1 0A8G
Proposition 22 Let H be a Lie group, U a Lie subgroup germ of H and N the set of g ∈ H such that U and gUg⁻¹ have the same germ at e (General Topology, Chapter I, § 6, no. 10). Then N is a subgro… ↩ 2 0A8H
Corollary Preserving the notation of Proposition 22, let G be the subgroup of H generated by U. Then G is an open subgroup of N. There exists one and only one Lie group structure on G such t… ↩ 2 0A8I
Remark Preserving the notation of Proposition 22 and its corollary, suppose that K is of characteristic 0, that H is finite-dimensional and that the topology on U admits a countable base… 0A8J
Definition 6 A law chunk of left operation of class Cʳ of G on X is a mapping ψ defined on an open subset Ω of G × X containing e × X, with values in X and with the following properties: (i) ψ… ↩ 1 0A8K
Proposition 23 Let (G, e, θ, m) be a Lie group germ, X a manifold of class Cʳ, x₀ a point of X, Ω an open neighbourhood of (e, x₀) in G × X and ψ a mapping of Ω into X with the following properti… ↩ 1 0A8L
Lemma 3 Let X be a normal space and (Xᵢ)_i ∈ I a locally finite open covering of X. For all (i, j) ∈ I × I and all x ∈ Xᵢ ∩ Xⱼ, let Vᵢⱼ(x) be a neighbourhood of x contained in Xᵢ ∩ Xⱼ. The… ↩ 1 0A8M
Proposition 24 Let G be a Lie group germ, X a manifold of class C^r and (Xi){i \in I} a locally finite open covering of X. For all i \in I, let ψᵢ be a law chunk of left operation of class C^r of… ↩ 1 0A8N

§ 2. Group of tangent vectors to a Lie group

Proposition 1 Let X be a manifold of class C^r and m a law of composition of class C^r on X. If m is associative (resp. commutative), then T(m) is associative (resp. commutative). ↩ 2 0A8Y
Proposition 2 Let X be a manifold of class Cʳ, m a law of composition of class Cʳ on X and e an identity element for m. ↩ 4 0A8Z
Proposition 3 Let X₁, X₂, …, Xₚ, Y be manifolds of class Cʳ, i an integer of 1, p, mᵢ (resp. n) a law of composition of class Cʳ on Xᵢ (resp. Y) and u a mapping of class Cʳ of X₁ × X₂ × ⋯ × Xₚ i… ↩ 1 0A90
Proposition 4 Let G be a Lie group. Then T(G), with the law of composition tangent to the multiplication of G, is a Lie group. The identity element of T(G) is the vector 0ₑ. 0A91
Proposition 5 Let G and H be Lie groups and f a morphism of G into H. Then T(f) is a morphism of the Lie group T(G) into the Lie group T(H). 0A92
Corollary Let G₁, …, Gₙ be Lie groups. The canonical isomorphism of the manifold T(G₁ × ⋯ × Gₙ) onto the manifold T(G₁) × ⋯ × T(Gₙ) is a Lie group isomorphism. 0A93
Proposition 6 Let G be a Lie group. (i) The canonical projection p : T(G) → G is a Lie group morphism. (ii) The kernel of p is Tₑ(G). It is a Lie subgroup of T(G). The Lie group structure induce… ↩ 1 0A94
Proposition 7 Let G be a Lie group, M a manifold of class Cʳ and f and g mappings of class Cʳ of M into G, so that fg is a mapping of class Cʳ of M into G. Let m ∈ M, x = f(m), y = g(m), u ∈ Tₘ(… ↩ 2 0A95
Corollary Let n ∈ Z. The tangent mapping at e to the mapping g ↦ gⁿ of G into G is the mapping x ↦ nx of Tₑ(G) into Tₑ(G). ↩ 1 0A96
Proposition 8 Let G1 and G2 be Lie groups, X1 and X2 manifolds of class C^r and fi a law of left operation of class C^r of Gi on Xi (i = 1, 2). Let \phi be a morphism of G1 into G2 and \psi a \p… 0A97

§ 3. Passage from a Lie group to its Lie algebra

Definition 1 Let G be a Lie group, g and g' two points of G and let t ∈ T_g^(∞)(G) t' ∈ T_g'^(∞)(G) be two point distributions at g and g' on G (Differentiable and Analytic Manifolds, R, 13.2.1… ↩ 3 0A98
Proposition 1 (i) If t ∈ T_g⁽ˢ⁾(G) and t' ∈ T_g'^(s')(G), then t * t' ∈ T_gg'^(s+s')(G). ↩ 2 0A99
Proposition 2 The algebra \mathscr{T}^{(\infty)}(G) is associative. It is commutative if and only if G is commutative. ↩ 4 0A9A
Proposition 3 If t \in \mathscr{T}^{(\infty)}(G) and g \in G, then γ(g)_* t = ε_g * t, δ(g)_* t = t * ε_g⁻¹, (Int g) t = ε_g * t * ε_g⁻¹. In particular, εₑ is the unit element of \mathscr{T}^{(\… ↩ 2 0A9B
Definition 2 Let G be a Lie group. The subalgebra of T^(∞)(G) consisting of the distributions with support contained in e is denoted by U(G). ↩ 1 0A9C
Example Let E be a complete normable space considered as a Lie group. Then the vector space U(E) is canonically identified with the vector space TS(E) (Differentiable and Analytic Manifold… 0A9D
Proposition 4 Consider the bilinear mapping (u, v) ↦ u * v (resp. (u, v) ↦ v * u) of U(G) ⊗ K⁽ᴳ⁾ into T^(∞)(G). The corresponding linear mapping of U(G) ⊗ K⁽ᴳ⁾ into T^(∞)(G) is a vector space is… 0A9E
Proposition 5 The isomorphism i_G,e : gr\ U(G) → TS(L) is an algebra isomorphism. ↩ 1 0A9F
Proposition 6 Let G, H be Lie groups and φ a morphism of G into H. For t, t' in T^(∞)(G), φ_*(t * t') = φ_*(t) * φ_*(t'). ↩ 3 0A9G
Example Suppose that G is the Lie group defined by a complete normable space E. Then U(G) is identified with TS(E) and the restriction θ_* to U(G) is identified with TS(Tₑ(θ)) (Differentia… 0A9H
Proposition 7 Let G be a Lie group. Let t, t' be in T^(∞)(G). ↩ 2 0A9I
Proposition 8 Let G, H be Lie groups and φ a morphism of G into H. If t ∈ T^(∞)(G), then φ_*(t^∨) = (φ_*(t))^∨. 0A9J
Proposition 9 Let G₁, …, Gₙ be Lie groups and G = G₁ × ⋯ × Gₙ. If the vector spaces T^(∞)(G) and T^(∞)(G₁) ⊗ ⋯ ⊗ T^(∞)(Gₙ) are canonically identified, the algebra T^(∞)(G) is the tensor product… 0A9K
Proposition 10 Let H be a Lie subgroup of G and i : H → G the canonical injection. Then i_* is an injective homomorphism of the algebra T^(∞)(H) into the algebra T^(∞)(H) and i_*(t^∨) = (i_*(t))^… ↩ 1 0A9L
Remark Proposition 10 remains valid if H is a Lie quasi-subgroup. 0A9M
Proposition 11 Let G be a Lie group. (i) The cogebra T^(∞)(G), with convolution, is a bigebra (Algebra, Chapter III, § 11, no. 4). (ii) Let c be the coproduct on T^(∞)(G). Let t ∈ T^(∞)(G) and wr… 0A9N
Proposition 12 Let G, H be two Lie groups and φ a morphism of G into H. Then φ_* is a bigebra morphism of T^(∞)(G) into T^(∞)(H). 0A9O
Proposition 13 Let t ∈ T⁽ˢ⁾(G), t' ∈ T^(s')(G), u ∈ T^(s'')(X), such that ↩ 1 0A9P
Proposition 14 (i) Let g₀ ∈ G and τ(g₀) be the mapping x ↦ f(g₀, x) of X into X. If u ∈ T⁽ʳ⁾(X), then τ(g₀)*u = ε_g₀*u. ↩ 4 0A9Q
Proposition 15 Let G (resp. G') be a Lie group and X (resp. X') a manifold of class C^r. Suppose that a law of left operation of class C^r of G (resp. G') on X (resp. X') is given. Let φ be a mor… ↩ 2 0A9R
Remark Let f be a law of right operation of class C^r of G on X. If t ∈ T⁽ˢ⁾(G) and u ∈ T^(s')(X), with s + s' ≤ r, we denote by u * t the image of u ⊗ t under f_*. Propositions 13, 14, 1… 0A9S
Proposition 16 Let G, G' be Lie groups, X a manifold of class C^r and suppose that G (resp. G') operates on X on the left (resp. right), with (gx)g' = g(xg') for all x ∈ X, g ∈ G, g' ∈ G'. Let t… 0A9T
Definition 3 Let t ∈ T⁽ˢ⁾(G) with s ≤ r. Let f : X → F be a function of class C^r with values in a Hausdorff polynormed space (for example F = K). The convolution of t and f, denoted by t * f… 0A9U
Proposition 17 Let t ∈ T⁽ˢ⁾(G), t' ∈ T^(s')(X) and f : X → F a function of class Cʳ with s + s' ≤ r. Then ⟨ t', t * f ⟩ = ⟨ t^∨ * t', f ⟩. beginalign* ⟨ t', t * f ⟩ &= ⟨ t', x ↦ ⟨ t, g ↦ f(g⁻¹x)… ↩ 2 0A9V
Proposition 18 Let t ∈ T⁽ˢ⁾(G), t' ∈ T^(s')(G) and f : X → F a function of class Cʳ, with s + s' ≤ r. Then (t * t') * f = t * (t' * f). For all x ∈ X, beginalign* ⟨ εₓ, (t * t') * f ⟩ &= ⟨ ((t *… ↩ 1 0A9W
Proposition 19 Let t ∈ T⁽ˢ⁾(G), with s ≤ r. Let f (resp. f') be a function of class Cʳ on X with values in a Hausdorff polynormed space F (resp. F'). Let (u, u') ↦ uu' be a continuous bilinear ma… 0A9X
Remark 1 Let G be a Lie group, X a manifold of class Cʳ and (x, g) ↦ xg a law of right operation of class Cʳ of G on X. If t ∈ T⁽ˢ⁾(G) with s ≤ r and f : X → F is a function of class Cʳ on… 0A9Y
Proposition 20 Let G, G' be Lie groups, X a manifold of class Cʳ and (g, x) ↦ gx (resp. (x, g') ↦ xg') a law of left (resp. right) operation of class Cʳ of G (resp. G') on X. Suppose that (gx)g'… ↩ 1 0A9Z
Remark 2 Let (g, x) ↦ gx be a law of left operation of class Cʳ of G on X. Let t ∈ Uₛ(G) with s ≤ r, Ω be an open subset of X and f : Ω → F a function of class Cʳ. t * f can also be defined… 0AA0
Proposition 21 Let ψ be an automorphism of X commuting with the operations of G. Then Dₜ is invariant under ψ. ↩ 2 0AA1
Proposition 22 If g ∈ G, the transform of Dₜ under g is D_ε_g * t * ε_g⁻¹. 0AA2
Definition 4 Let G be a Lie group. A field of distributions on G is called left (resp. right) invariant if it is invariant under left (resp. right) translations of G. 0AA3
Definition 5 Let G be a Lie group and t ∈ U(G). Let Lₜ denote the field of distributions g ↦ ε_g * t on G and Rₜ the field of distributions g ↦ t * ε_g on G. ↩ 1 0AA4
Proposition 23 (i) The mapping t ↦ Lₜ (resp. t ↦ Rₜ) is an isomorphism of the vector space U(G) onto the vector field of left (resp. right) invariant distributions on G. (ii) For t, t' in U(G), L… ↩ 3 0AA5
Remark 1 It is the action of G on itself by right translation which defines the fields of left invariant distributions. ↩ 1 0AA6
Remark 2 Suppose that G is finite-dimensional. The mapping 0AA7
Lemma 1 Let X and X' be complete normable spaces, X₀ an open neighbourhood of 0 in X and f an analytic mapping of X₀ into X' such that f(0) = 0. Let f = f₁ + f₂ + f₃ + ⋯ be the expansion o… ↩ 1 0AA8
Proposition 24 Let G be a Lie group and (U, φ, E) a chart on G such that φ(e) = 0. Let V be an open neighbourhood of e such that V² ⊂ U. Let m be the analytic mapping (a, b) ↦ φ(φ⁻¹(a)φ⁻¹(b)) of… ↩ 1 0AA9
Corollary The normable space Tₑ(G) together with the bracket, is a normable Lie algebra. 0AAA
Definition 6 The normable space Tₑ(G), together with the bracket, is called the normable Lie algebra of G, or simply the Lie algebra of G, and is denoted by L(G). 0AAB
Proposition 25 Let G be a Lie group and E(G) the enveloping algebra of L(G). The canonical injection of L(G) into E(G) defines a homomorphism θ of the algebra E(G) into the algebra U(G). If K is… 0AAC
Proposition 26 Suppose that K is of characteristic p > 0. For all a ∈ L(G), aᵖ ∈ L(G) and ad(aᵖ) = (ad a)ᵖ (the power aᵖ being calculated in U(G)). 0AAD
Definition 7 Let X be a manifold of class Cʳ (r ≥ 2) and g a complete normable Lie algebra. A law of infinitesimal left (resp. right) operation of class Cʳ⁻¹ of g on X is a mapping a ↦ Dₐ of g… ↩ 2 0AAE
Remark Let X be a manifold of class Cʳ, g a finite-dimensional Lie algebra and a ↦ Dₐ a linear mapping of g into the vector space of vector fields of class Cʳ⁻¹ on X. Then condition (a) o… 0AAF
Proposition 27 Let G be a Lie group and X a manifold of class Cʳ. Suppose that a law of left (resp. right) operation of class Cʳ of G on X is given. For all a ∈ L(G), let Dₐ be the field of point… ↩ 2 0AAG
Proposition 28 Let G and H be Lie groups and φ a morphism of G into H. Suppose that K is of characteristic 0 and that H is finite-dimensional. ↩ 3 0AAH
Proposition 29 Let G and H be Lie groups and φ a morphism of G into H. Suppose that K is of characteristic 0 and that H is finite-dimensional. If H' is a Lie subgroup of H, then G' = φ⁻¹(H') is a… ↩ 3 0AAI
Corollary 1 Let G, H be Lie groups and φ₁ and φ₂ morphisms of G into H. Suppose that K is of characteristic 0 and that H is finite-dimensional. The set of g ∈ G such that φ₁(g) = φ₂(g) is a Li… ↩ 3 0AAJ
Corollary 2 Let G be a finite-dimensional Lie group and G₁ and G₂ two Lie subgroups of G. Suppose that K is of characteristic 0. Then G₁ ∩ G₂ is a Lie subgroup of G with Lie algebra L(G₁) ∩ L(… ↩ 3 0AAK
Corollary 3 Let G, G', H be Lie groups and φ : G → H and φ' : G' → H Lie group morphisms. Suppose that K is of characteristic 0 and that H is finite-dimensional. Let F be the set of (g, g') ∈… 0AAL
Proposition 30 Let G be a finite-dimensional Lie group with a countable base and ↩ 2 0AAM
Proposition 31 Let G be a finite-dimensional Lie group with countable base, H a normal Lie subgroup of G and A a Lie subgroup of G. Suppose that K is of characteristic 0 and that AH is closed. Le… 0AAN
Proposition 32 Let G and H be Lie groups, k a non-discrete closed subfield of K and φ a morphism of G into H for the Lie group structures over k. Suppose that K is of characteristic 0. If L(φ) is… ↩ 1 0AAO
Proposition 33 The mapping t ↦ ⟨ t, f ⟩ is a morphism of the algebra T^(∞)(G) into the algebra A. ↩ 2 0AAP
Corollary The canonical mapping ζ of L(A^*) into A is an isomorphism of the Lie algebra, L(A^*) onto the Lie algebra A. In other words, ζ([a, b]) = ζ(a)ζ(b) - ζ(b)ζ(a) for all a, b in L(A^*)… ↩ 2 0AAQ
Proposition 34 Let H be a Lie group, A a unital complete normable associative algebra and φ : H → A^* a Lie group morphism. The associated morphism φ' of T^(∞)(H) into A is obtained by composing… ↩ 2 0AAR
Proposition 35 Let E be a finite-dimensional vector space. Let φ be the morphism g ↦ det g of the Lie group GL(E) into the Lie group K^*. The mapping L(φ) of L(E) into K is the mapping x ↦ Tr x… 0AAS
Proposition 36 Let I be a subset of 1, 2, …, n and G the subgroup of GL(E) consisting of the g = (gᵢⱼ)_1 ≤ i, j ≤ n ∈ GL(E) such that gᵢⱼ = 0 for i < j and gᵢᵢ = 1 for i ∈ I. Then G is a Lie subg… ↩ 2 0AAT
Proposition 37 Let A be a complete normable unital associative algebra and x ↦ xᵗ a continuous linear mapping of A into A such that (xᵗ)ᵗ = x and (xy)ᵗ = yᵗ xᵗ for all x, y in A. Suppose that K i… ↩ 2 0AAU
Corollary 1 Suppose that K is of characteristic ≠ 2. Let E be a finite-dimensional vector space over K and φ a non-degenerate symmetric (resp. alternating) bilinear form on E. For all u ∈ L(E)… ↩ 1 0AAV
Remark Let B be a basis of E and J the matrix of φ with respect to B. Then L(G) is the set of elements of L(E) whose matrix X with respect to B satisfies the equation 0AAW
Corollary 2 Let E be a complex (resp. real) Hilbert space and U the unitary group of E. Then U is a real subgroup of GL(E) and L(U) is the set of x ∈ L(E) such that x^* = -x. ↩ 1 0AAX
Corollary 3 Let E be a finite-dimensional complex vector space, φ a non-degenerate Hermitian sesquilinear form on E and U the unitary group of φ. Then U is a real Lie subgroup of GL(E) and L(U… 0AAY
Proposition 38 Consider G as operating on E on the left by the mapping (g, x) ↦ π(g)x. Let b ∈ E and φ(b) be its orbital mapping. Let T_b(E) be canonically identified with E. For all t ∈ L(G), ↩ 1 0AAZ
Proposition 39 Suppose that K is of characteristic 0. Let G be a Lie group, E a finite-dimensional vector space and π an analytic linear representation of G on E. Let E₁, E₂ be vector subspaces o… ↩ 2 0AB0
Corollary 1 In the notation of Proposition 39, the set of g ∈ G such that π(g)(E₁) ⊂ E₁ is a Lie subgroup of G and its Lie algebra is the set of a ∈ L(G) such that L(π)a maps E₁ into E₁. ↩ 1 0AB1
Corollary 2 Let G, E, π be as in Proposition 39. Let F be a subset of E. The set of g ∈ G such that π(g)x = x for all x ∈ F is a Lie subgroup of G and its Lie algebra is the set of a ∈ L(G) su… ↩ 3 0AB2
Proposition 40 Let G be a Lie group, E a complete normable space, π an analytic linear representation of G on E and F a closed vector subspace of E stable under π(G). Suppose that K is of charact… ↩ 2 0AB3
Proposition 41 Let G be a Lie group and π₁, π₂, …, πₙ, π analytic linear representation of G on complete normable spaces E₁, E₂, …, Eₙ, E. Let (x₁, x₂, …, xₙ) ↦ x₁ x₂ … xₙ be a continuous multili… ↩ 3 0AB4
Corollary 1 Let G be a Lie group, E₁, …, Eₙ₊₁ complete normable spaces and π₁, …, πₙ₊₁ analytic linear representations of G on E₁, …, Eₙ₊₁. Let E = L(E₁, …, Eₙ; Eₙ₊₁) the complete normable spa… ↩ 6 0AB5
Corollary 2 Let G be a Lie group and π an analytic linear representation of G on a complete normable space E. Then g ↦ ᵗπ(g)⁻¹ is an analytic linear representation ρ of G on the complete norma… 0AB6
Corollary 3 Let G be a Lie group and π₁, …, πₙ analytic linear representations of G on finite-dimensional vector spaces E₁, …, Eₙ. Then the representation π₁ ⊗ ⋯ ⊗ πₙ of G (Appendix) is analyt… ↩ 2 0AB7
Corollary 4 Let G be a Lie group and π an analytic linear representation of G on a finite-dimensional vector space E. Then the representations Tⁿ(π), Sⁿ(π) and ∧ⁿ(π) of G (Appendix) are analyt… 0AB8
Corollary 5 Let A be a finite-dimensional algebra. Suppose that K is of characteristic 0. The automorphism group Aut(A) of A is a Lie subgroup of GL(A) and L(Aut(A)) is the Lie algebra of deri… ↩ 1 0AB9
Remark We apply Corollary 1 with G = GL(F) (F a complete normable space), π₁ = π₂ = Id_G and π₃ the trivial representation of G on K. We obtain an analytic representation π of GL(F) on L(… 0ABA
Proposition 42 Let G be a Lie group, X an analytic manifold, (g, x) ↦ gx (resp. xg) a law of analytic left (resp. right) operation of G on X and x₀ a point of X which is invariant under G. For al… ↩ 2 0ABB
Proposition 43 Let t, u be in T^(∞)(G). Let ∑ᵢ₌₁ⁿ tᵢ ⊗ t'ᵢ be the image of t under the coproduct. Then t top u = ∑ᵢ₌₁ⁿ tᵢ * u * t'ᵢ^∨. 0ABC
Corollary 1 Let u ∈ L(G) and u' ∈ T^(∞)(G). Then u top u' = u * u' - u' * u. ↩ 1 0ABD
Corollary 2 Let t ∈ T^(∞)(G) and g ∈ G. Then ε_g top t = ε_g * t * ε_g⁻¹. If t ∈ L(G), then ε_g top t = gtg⁻¹ (where the latter product is evaluated in the group T(G)). 0ABE
Corollary 3 Let a ∈ L(G). The vector field defined by a and the left operation g ↦ Int g of G on G is the field Rₐ - Lₐ. ↩ 1 0ABF
Definition 8 The representation Ad of G on L(G) is called the adjoint representation of G. ↩ 2 0ABG
Proposition 44 For all a ∈ L(G), (L(Ad))(a) = ad_L(G)a. Let b ∈ L(G). By Proposition 42 (ii) of no. 11 and Corollary 3 to Proposition 43, (L(Ad))(a).b = -[Rₐ - L_a₁L_b](e). Now Rₐ ∘ L_b = L_b ∘ R… ↩ 7 0ABH
Proposition 45 Suppose that G is finite-dimensional and that K is of characteristic 0. Let s be an integer geqslant 0. Then the mapping π : g ↦ Ad_Uₛ(G)(g) is an analytic linear representation of… ↩ 1 0ABI
Proposition 46 Let h ∈ G, x ∈ Tₕ(G) and a ∈ L(G). Let φ be the mapping (g, g') ↦ gg'g⁻¹ of G × G into G. The image y of (a, x) ∈ Tₑ(G) × Tₕ(G) under T_(e, h)(φ) is y = x + h((Ad h⁻¹)a - a). We ha… ↩ 2 0ABJ
Proposition 47 Let G be a Lie group, H and E Lie subgroups of G and suppose that hEh⁻¹ = E for all h \in H. Then T^(∞)(H) ⊂ T^(∞)(E) ⊂ T^(∞)(E). In particular, Ad(H)(L(E)) ⊂ L(E) and [L(H), L(E)]… ↩ 2 0ABK
Proposition 48 Let G be a Lie group and H and E Lie subgroups of G. Suppose that G is, as a Lie group, the semi-direct product of H by E. Let φ be the linear representation g ↦ (Ad g) | L(E) of t… ↩ 1 0ABL
Corollary Let G be a Lie group. Let Tₑ(G) be given its unique commutative Lie algebra structure. Let τ be an adjoint representation of L(G). Then the Lie algebra of T(G) is the semi-direct p… 0ABM
Proposition 49 Let A be a complete normable unital associative algebra. We identify A with L(A^*). Then, if g ∈ A^* and y ∈ A, (Ad g)y = gyg⁻¹. 0ABN
Corollary For all g ∈ A^*, let i(g) be the automorphism y ↦ gyg⁻¹ of A, so that i is an analytic linear representation of A^* on A. For all z ∈ L(A^*) = A, L(i)z is the inner derivation y ↦… 0ABO
Proposition 50 Let G be a Lie Group (assumed to be finite-dimensional if K is of characteristic 0). Let E be the vector space of continuous alternating multilinear forms of degree k on Tₑ(G). For… ↩ 2 0ABP
Proposition 51 Let G be a Lie group, of finite dimension if K is of characteristic > 0, and let a₁, …, aₚ be elements of L(G), F a complete normable space and α a differential form of degree p-1… ↩ 3 0ABQ
Corollary 1 Let G be a Lie group, of finite dimension if K is of characteristic 0, and ω and ω' the left and right canonical differential forms of G. Then dω + [ω]² = 0, dω' - [ω']² = 0. By Pr… ↩ 1 0ABR
Corollary 2 Suppose that G is finite-dimensional. Let (e₁, …, eₙ) be a basis of L(G), (e₁^*, …, eₙ^*) the dual basis, (cᵢⱼₖ) the constants of structure of L(G) relative to the basis (e₁, …, eₙ… 0ABS
Lemma 2 Let G be a Lie group, U a symmetric open neighbourhood of e in G, E a complete normable space and φ : U² → E an analytic mapping. For all g ∈ U, let ω_g be the differential at the… ↩ 2 0ABT
Proposition 52 Let n be an integer 0, G an n-dimensional Lie group, U a symmetric open neighbourhood of e in G and ψ : U² → Kⁿ a chart of G such that ψ(e) = 0. If (x₁, …, xₙ) are the coordinates… ↩ 1 0ABU
Proposition 53 Let G be a Lie group, A a complete normable algebra and φ a Lie group morphism of G into A^*. For all g ∈ G, let ω_g = φ(g)⁻¹ d_g φ. Then ω is the left invariant differential form… 0ABV
Proposition 54 Let G be a Lie group of finite dimension n, ω a left invariant differential form of degree n on G and φ an endomorphism of G. Then ↩ 1 0ABW
Corollary For all g ∈ G, ↩ 1 0ABX
Proposition 55 Suppose that K is locally compact. Let μ be a Haar measure on the additive group of K. Let G be a Lie group of finite dimension n. ↩ 2 0ABY
Corollary For all g ∈ G, Δ_G(g) = (mod det Ad g)⁻¹. In particular, for G to be unimodular, it is necessary and sufficient that mod det Ad g = 1 for all g ∈ G. ↩ 2 0ABZ
Remark Preserving the hypotheses and notation of Proposition 52, suppose that K is locally compact. Let μ be the measure mod det(Dₙ₊ᵢmₖ(x₁, …, xₙ, x₁, …, xₙ))_1 ≤ i, k ≤ n dx₁ … dxₙ on ψ(… 0AC0
Proposition 56 Let G be a Lie group of finite dimension n, H a p-dimensional Lie subgroup and X the Lie homogeneous space G/H. Suppose that det Ad_L(G) h = det Ad_L(H) h for all h ∈ H. Then: (i)… ↩ 1 0AC1
Proposition 57 Let G be a Lie group of finite dimension n. Choose a basis for bigwedgeⁿ Tₑ(G)^*; by means of the right (resp. left) trivialization of bigwedgeⁿ T(G)^*, we can identify this vector… ↩ 1 0AC2
Corollary Let G be a finite-dimensional real Lie group, μ (resp. ν) a left (resp. right) Haar measure on G, k an integer geqslant 0, u ∈ Uₖ(G) and f and g real-valued functions of class Cᵏ o… 0AC3
Definition 9 Let G be a Lie group, M a manifold of class Cʳ and f a mapping of class Cʳ of M into G. The left (resp. right) differential of f is the differential form of degree 1 on M with valu… 0AC4
Example (1) If G is the additive group of a complete normable space and T₀(E) is canonically identified with E, f⁻¹.df is the differential df defined in Differentiable and Analytic Manifol… 0AC5
Proposition 58 Let G and H be two Lie groups, M a manifold of class Cʳ, f a mapping of class Cʳ of M into G and h a morphism of G into H. Then 0AC6
Proposition 59 Let G be a Lie group, M a manifold of class Cʳ, f and g mappings of class Cʳ of M into G and p the canonical surjection of TM onto M. ↩ 2 0AC7
Corollary 1 Let s ∈ G and sg be the mapping x ↦ sg(x) of M into G. Then 0AC8
Corollary 2 If the mappings f and g of M into G have the same left differential, the tangent mapping to fg⁻¹ is everywhere zero. If further K is of characteristic 0, then fg⁻¹ is locally const… ↩ 1 0AC9
Proposition 60 Let G be a Lie group, of finite dimension if K is of characteristic 0, M a manifold of class Cʳ, f a mapping of class Cʳ of M into G and α the left differential of f. Then dα + [α]… 0ACA

§ 4. Passage from Lie algebras to Lie groups

Lemma 1 Let G be a Lie group germ and 𝔥 a Lie subalgebra of L(G) admitting a topological supplement. The union of the g𝔥 (resp. 𝔥g) for g ∈ G is an integrable vector subbundle of T(G). ↩ 1 0ACK
Theorem 1 Let G and H be Lie group germs and f a continuous morphism of L(G) into L(H). (i) There exist an open Lie subgroup germ G' of G and a morphism φ of G' into H such that f = L(φ). (i… ↩ 8 0ACL
Corollary 1 Let G and H be two Lie group germs. If L(G) and L(H) are isomorphic, G and H are locally isomorphic. ↩ 4 0ACM
Corollary 2 Let G be a Lie group germ. If L(G) is commutative, G is locally isomorphic to the additive Lie group L(G). ↩ 1 0ACN
Corollary 3 Let G be a Lie group. If L(G) is commutative, G contains a commutative open subgroup. 0ACO
Lemma 2 Let L be a complete normed Lie algebra over R or C. Let G be the set of x ∈ L such that |x| < 1/3 log 3/2. Let θ be the mapping x ↦ -x of G into G. Let H be the restriction to G ×… ↩ 1 0ACP
Lemma 3 Let L be a complete normed Lie algebra over K. Let G be the set of x ∈ L such that |x| < λ. Let H : G × G → G be the Hausdorff function of L (Chapter II, § 8, no. 3). (i) With the… ↩ 3 0ACQ
Theorem 2 Let L be a complete normable Lie algebra. There exists a Lie group germ G such that L(G) is isomorphic to L. Two such Lie group germs are locally isomorphic. ↩ 4 0ACR
Corollary 1 Let G be a Lie group. There exists a neighbourhood of e which contains no finite subgroup distinct from {e}. If K = R or C, there exists an open neighbourhood of e which contains n… ↩ 2 0ACS
Corollary 2 Let k be a non-discrete closed subfield of K, G a Lie group over k and L = L(G). Suppose that L has a normable Lie K-algebra structure L', compatible with the normable Lie k-algebr… 0ACT
Theorem 3 Let G be a Lie group germ and 𝔥 a Lie subalgebra of L(G) admitting a topological supplement. There exists a Lie subgroup germ H of G such that L(H) = 𝔥. If H₁ and H₂ are Lie subgro… ↩ 5 0ACU
Proposition 1 Let G be a Lie group germ and H and H' two Lie subgroup germs. In order that L(H) ⊃ L(H'), it is necessary and sufficient that H ∩ H' be open in H'. 0ACV
Proposition 2 Let G be a Lie group over K, k a non-discrete closed subfield of K and H a Lie subgroup of the Lie k-group G. Suppose that L(H) is a vector sub-K-space of L(G) which admits a topol… ↩ 2 0ACW
Theorem 4 Let G be a Lie group germ, L its Lie algebra, V an open neighbourhood of 0 in L, φ an analytic mapping of V into G such that φ(0) = 0 and T0(φ) = IdL. The following conditions are… ↩ 4 0ACX
Definition 1 Let G be a Lie group germ and L its Lie algebra. An exponential mapping of G is any analytic mapping φ defined on an open neighbourhood of 0 in L, with values in G and satisfying t… ↩ 1 0ACY
Example (1) Let G be the additive group of a complete normable space E. The canonical isomorphism of L(G) onto E satisfies condition (i) of Theorem 4 and is therefore an exponential mappin… 0ACZ
Proposition 3 Let G be a Lie group germ, L its Lie algebra and φ an exponential mapping of G. Let L₁, …, Lₙ be vector subspaces of L such that L is the topological direct sum of L₁, …, Lₙ. The m… ↩ 1 0AD0
Proposition 4 Let G be a Lie group germ and φ an injective exponential mapping of G. For all x, y in L(G), ↩ 3 0AD1
Proposition 5 Let G be a Lie group, k a non-discrete closed subfield of K, G' the group G considered as a Lie group over k and φ (resp. φ') an exponential mapping of G (resp. G'). Then φ and φ'… ↩ 1 0AD2
Proposition 6 Let G be a Lie group germ, L its Lie algebra and φ : V → G an exponential mapping of G. For all x ∈ V, let Tₓ(L) be identified with L, so that the right differential varpi(x) of φ… ↩ 1 0AD3
Proposition 7 (i) If t ∈ Z, a t-th power mapping coincides on a neighbourhood of e with the mapping g ↦ gᵗ. ↩ 1 0AD4
Proposition 8 Let G and H be Lie group germs, h a morphism of G into H and φ_G and φ_H exponential mappings of G and H. There exists a neighbourhood V of 0 in L(G) such that h ∘ φ_G and φ_H ∘ L(… ↩ 3 0AD5
Corollary 1 Let G be a Lie group germ, G' a Lie subgroup germ of G and φ an exponential mapping of G. ↩ 1 0AD6
Corollary 2 Let G be a Lie group, ρ an analytic linear representation of G and φ an exponential mapping of G. There exists a neighbourhood V of 0 in L(G) such that ρ(φ(x)) = exp(L(ρ)x) for all… ↩ 2 0AD7
Corollary 3 Let G be a Lie group and φ an exponential mapping of G. ↩ 1 0AD8
Lemma 4 Let G be a finite-dimensional Lie group, Ω a symmetric open neighbourhood of e in G and H a subset of Ω containing e such that the conditions x ∈ H, y ∈ H, xy⁻¹ ∈ Ω imply xy⁻¹ ∈ H… ↩ 1 0AD9
Definition 2 𝔥 is called the subalgebra tangent to H at e. ↩ 1 0ADA
Proposition 9 Let G be a finite-dimensional Lie group and H a subgroup of G. ↩ 2 0ADB
Definition 3 The Lie group structure on H defined in Proposition 9 is called the structure induced on H by the Lie group structure on G. ↩ 1 0ADC
Lemma 5 Let X be a manifold of class Cʳ, F and F' vector bundles of class Cʳ with base space X and φ a morphism of F into F'. For all x ∈ X, let Sₓ be the set of (a, φ(a)) ∈ Fₓ ⊕ F'ₓ for a… ↩ 2 0ADD
Lemma 6 Let G be a Lie group germ, ω the canonical left differential form of G (§ 3, no. 18.9), M a manifold of class Cʳ (r ≥ 2) and α a differential form of class Cʳ⁻¹ and degree 1 on M w… ↩ 1 0ADE
Theorem 5 Let G be a Lie group germ, M a manifold of class Cʳ (r ≥ 2) and α a differential form of class Cʳ⁻¹ and degree 1 on M with values in L(G), such that dα + [α]² = 0. For all x ∈ M an… ↩ 2 0ADF
Proposition 10 Let M be an analytic manifold, 𝔤 a complete normable Lie algebra and α an analytic differential form of degree 1 on M, with values in 𝔤, with the following properties: (a) for all… ↩ 1 0ADG
Corollary Let M be an analytic manifold of finite dimension n. Let ω₁, …, ωₙ be analytic differential forms of degree 1 on M, with scalar values, which are linearly independent at each point… 0ADH
Proposition 11 Let G₁ and G₂ be Lie group germs and X₁ and X₂ manifolds of class Cʳ (r ≥ 2). For i = 1, 2, let ψᵢ be a law chunk of left operation of class Cʳ of Gᵢ on Xᵢ and Dᵢ the associated la… 0ADI
Corollary Let G be a Lie group germ and X a manifold of class Cʳ. Consider two law chunks of left operation of class Cʳ of G on X. Suppose that, for all a ∈ L(G), the corresponding vector fi… ↩ 3 0ADJ
Theorem 6 Let G be a Lie group germ, X a manifold of class Cʳ (r ≥ 2) and x₀ a point of X. Let a ↦ Dₐ be a law of left infinitesimal operation of class Cʳ⁻¹ of L(G) on X. ↩ 1 0ADK
Corollary 1 Let G be a Lie group germ and X a paracompact manifold of class Cʳ (r ≥ 2). Let a ↦ Dₐ be a law of left infinitesimal operation of class Cʳ⁻¹ of L(G) on X. ↩ 2 0ADL
Corollary 2 Let X be a paracompact manifold of class Cʳ (r ≥ 2) and ξ a vector field of class Cʳ⁻¹ on X. There exists a law chunk of operation ψ of class Cʳ⁻¹ of K on X such that for all x ∈ X… ↩ 1 0ADM
Remark Laws of left operation can of course be replaced, throughout this no., by laws of right operation. 0ADN

§ 5. Formal calculations in Lie groups

Proposition 1 (i) x^[-1] ≡ -x + B(x, x) mod deg 3 (ii) x.y.x^[-1] ≡ y + [x, y] mod deg 3 (iii) y^[-1].x.y ≡ x + [x, y] mod deg 3 (iv) x^[-1].y^[-1].x.y ≡ [x, y] mod deg 3 (v) x.y.x^[-1].y^[-1] ≡… ↩ 2 0AE1
Proposition 2 (i) ψⱼ ≡ 0 mod deg j. (ii) If t ∈ K, then x^[t] = ∑ᵢ₌₁⁸ binomti ψᵢ(x) where the formal power series on the right is meaningful because of (i). ↩ 2 0AE2
Remark (1) We write condition (ii) of Proposition 2 for t an integer geqslant 0: 0AE3
Proposition 3 (27) L = ∑ₚ₌₁^∞ frac(-1)ᵖ⁻¹p ψₚ (28) E = ∑ₚ₌₁^∞ 1/p! ψ_p,p (Recall that ψ_p,p is the homogeneous component of ψₚ of degree p.) ↩ 2 0AE4
Proposition 4 For the chart of G used to be canonical, it is necessary and sufficient that ψⱼ = 0 for j ≥ 2. 0AE5

§ 6. Real and complex Lie groups

Lemma 1 Let G be a simply connected† topological group, W a symmetric connected open neighbourhood of e, H a group and f a mapping of W³ into H such that ↩ 2 0AE8
Theorem 1 Let G and H be Lie groups and h a continuous morphism of L(G) into L(H). Suppose that G is simply connected. Then there exists one and only one Lie group morphism φ of G into H suc… ↩ 5 0AE9
Corollary Let G be a finite-dimensional simply connected Lie group. There exists a finite-dimensional analytic linear representation of G whose kernel is discrete. ↩ 2 0AEA
Remark (1) There exist finite-dimensional simply connected Lie groups which have no finite-dimensional injective analytic linear representation (Exercise 2). 0AEB
Definition 1 Let G be a Lie group. An integral subgroup of G is a subgroup H with a connected Lie group structure such that the canonical injection of H into G is an immersion. 0AEC
Example (1) A connected Lie subgroup of G is an integral subgroup of G. 0AED
Theorem 2 Let G be a Lie group. ↩ 3 0AEE
Remark 1 Let H be an integral subgroup of G. Let Y be the left foliation of G associated with L(H). If g ∈ G, let gH be given the manifold structure derived from that on H by γ(g). By Diffe… ↩ 1 0AEF
Proposition 1 Let G and M be Lie groups, H an integral subgroup of G and φ a morphism of M into G such that L(φ)(L(M)) ⊂ L(H). Suppose that M is connected. Then φ(M) ⊂ H and φ, considered as a m… ↩ 5 0AEG
Corollary 1 Let G and H be Lie groups, φ a Lie group morphism of G into H, N the kernel of φ and h = L(φ). Suppose that G is connected and that H is finite-dimensional. (i) N is a Lie subgroup… ↩ 3 0AEH
Corollary 2 Let G be a Lie group and H1 and H2 integral subgroups of G. If L(H₂) ⊂ L(H₁), then H2 is an integral subgroup of H1. ↩ 2 0AEI
Corollary 3 Let G be a finite-dimensional Lie group and (Hᵢ)_i ∈ I a family of Lie subgroups of G. Then H = ⋂_i ∈ I Hᵢ is a Lie subgroup of G and L(H) = ⋂_i ∈ I L(Hᵢ). There exists a finite su… ↩ 1 0AEJ
Corollary 4 Let G be a finite-dimensional connected Lie group. The following conditions are equivalent: (i) G is unimodular (Integration, Chapter VII, § 1, no. 3, Definition 3); (ii) det Ad g… 0AEK
Proposition 2 Let G be a finite-dimensional Lie group and H an integral subgroup of G. The following conditions are equivalent: (i) H is closed; (ii) the topology on H is induced by that on G; (… ↩ 1 0AEL
Proposition 3 Let G be a Lie group, H an integral subgroup of G, M a non-empty connected analytic manifold, f a mapping of M into G and r \in NK. Consider the following conditions: (i) f is of c… ↩ 5 0AEM
Corollary 1 Let G be a finite-dimensional Lie group and H an integral subgroup of G. Then the Lie subalgebra tangent to H at e (§ 4, no. 5, Definitions 2 and 3) is L(H) and the Lie group struc… 0AEN
Corollary 2 Let G be a Lie group and H1 and H2 integral subgroups of G. Suppose that the topology on H1 admits a countable base. Then ↩ 3 0AEO
Corollary 3 Let G be a Lie group and H1 and H2 integral subgroups of G whose topology admits a countable base. If H1 and H2 have the same underlying set, the Lie group structures on H1 and H2… ↩ 1 0AEP
Remark 2 Let G be a finite-dimensional Lie group. Let H be a subgroup of G. We shall say, by an abuse of language, that H is an integral subgroup of G if there exists a Lie group structure… ↩ 1 0AEQ
Remark 3 Let V be a manifold of class C^r. Let M be a subset of V and x and y elements of M. Consider the following property: ↩ 1 0AER
Proposition 4 Let G be a finite-dimensional Lie group and H a subgroup of G. Let r \in \mathbf{N}\mathbf{K}. The following conditions are equivalent: (i) H is an integral subgroup of G; (ii) wit… 0AES
Remark 4 If K = R, the integral subgroups of G can also be characterized as the subgroups which, with the topology induced by that on G, are arcwise connected (§ 8, Exercise 4). However, su… 0AET
Corollary Let G be a finite-dimensional Lie group and H1 and H2 two integral subgroups of G. The subgroup of G generated by H1 and H2 and the subgroup (H1, H2) of G are integral subgroups of… ↩ 1 0AEU
Definition 2 Let a be a finite-dimensional Lie algebra. Let Ad(a) or Int(a) denote the integral subgroup of Aut(a) with Lie algebra ad(a). The elements of this group are called inner automorphi… ↩ 4 0AEV
Theorem 3 (i) If L is a finite-dimensional Lie algebra, there exists a simply connected Lie group G such that L(G) is isomorphic to L. ↩ 2 0AEW
Proposition 5 Let G be a connected real Lie group. Suppose that L(G) has a complex normable Lie algebra structure L' compatible with its real normable Lie algebra structure. There exists on G on… 0AEX
Proposition 6 Let G be a connected complex Lie group. If G is compact, G is commutative. ↩ 2 0AEY
Theorem 4 Let G be a Lie group. There exists one and only one exponential mapping of G defined on L(G). 0AEZ
Example Let A be a complete normed unital associative algebra. Then exp_A^* is the exponential mapping defined in Chapter II, § 7, no. 3. 0AF0
Proposition 7 Let G be a Lie group and a an element of L(G). The mapping λ ↦ exp(λ a) of K into G is the unique morphism φ of the Lie group K into G such that (T₀ φ)1 = a. ↩ 1 0AF1
Proposition 8 Let G be a Lie group. For all x, y in L(G) and n an integer, ↩ 5 0AF2
Proposition 9 Let G be a complex Lie group and G' the underlying real Lie group. Then exp_G = exp_G'. 0AF3
Proposition 10 Let G and H be Lie groups and φ a morphism of G into H. (i) φ ∘ exp_G = exp_H ∘ L(φ). (ii) If G is an integral subgroup of H, then exp_G = exp_H | L(G). ↩ 3 0AF4
Corollary 1 Let G be a Lie group, G' a Lie subgroup of G and a ∈ L(G). The following conditions are equivalent: (i) a ∈ L(G'); (ii) exp(λ a) ∈ G' for all λ ∈ K and |λ| sufficiently small; (iii… 0AF5
Corollary 2 Let G be a Lie group, H an integral subgroup of G and a ∈ L(G). Consider the following conditions: (i) a ∈ L(H); (ii) exp_G(λ a) ∈ H for all λ ∈ K. Then (i) ⇒ (ii). If the topology… 0AF6
Corollary 3 Let G be a Lie group, ρ an analytic linear representation of G, x ∈ L(G) and g ∈ G. (i) ρ(exp x) = exp L(ρ)x; (ii) Ad(exp x) = exp ad x; (iii) g(exp x)g⁻¹ = exp(Ad g.x). The argume… ↩ 5 0AF7
Corollary 4 Let G be a finite-dimensional connected Lie group. (i) Int(L(G)) = Ad(G). (ii) Let Z be the centre of G. Then Z is a Lie subgroup of G whose Lie algebra is the centre of L(G). The… ↩ 5 0AF8
Definition 3 Let G be a finite-dimensional connected Lie group. The Lie group Int(L(G)) = Ad(G) is called the adjoint group of G. 0AF9
Proposition 11 Let G be a connected commutative Lie group. (i) exp is an étale morphism of the additive Lie group L(G) onto G. ↩ 1 0AFA
Proposition 12 Let G be a Lie group and L = L(G). For all x ∈ L, let Tₓ(L) be identified with L, so that the right differential varpi(x) of exp at x is a linear mapping of L into L. For all x ∈ L… ↩ 2 0AFB
Remark varpi(x) · (ad\ x) = exp ad\ x - 1. We write, by an abuse of notation, 0AFC
Corollary Let G be a complex Lie group and x ∈ L(G). The tangent mapping at x to exp_G has kernel ⨁_n ∈ Z - 0 Ker(ad\ x - 2iπ n). 0AFD
Lemma 2 Let E be a complex Banach space, u an element of L(E), S the spectrum of u in L(E) (Spectral Theories, Chapter I, § 1, no. 2) and f a holomorphic complex function on an open neighb… ↩ 1 0AFE
Proposition 13 Let G be a connected Lie group and φ an analytic linear representation of G on a complete normable space E. Let E₁, E₂ be two closed vector subspaces of E such that E₂ ⊂ E₁. The fo… ↩ 3 0AFF
Corollary 1 For E₁ to be stable under φ, it is necessary and sufficient that E₁ be stable under L(φ). ↩ 2 0AFG
Corollary 2 Suppose that φ is finite-dimensional. For φ to be simple (resp. semi-simple), it is necessary and sufficient that L(φ) be simple (resp. semi-simple). ↩ 3 0AFH
Corollary 3 Let x ∈ E. For x to be invariant under φ(G), it is necessary and sufficient that x be annihilated by L(φ)(L(G)) (that is that x be invariant under L(φ) in the sense of Chapter I, §… ↩ 4 0AFI
Corollary 4 Let φ' be another analytic linear representation of G on a complete normable space E'. Let T ∈ L(E, E'). The following conditions are equivalent: ↩ 1 0AFJ
Corollary 5 Suppose that ρ and ρ' are finite-dimensional. For ρ and ρ' to be equivalent, it is necessary and sufficient that L(ρ) and L(ρ') be equivalent. 0AFK
Corollary 6 Suppose that G is finite-dimensional. Let t ∈ U(G). For Lₜ (resp. Rₜ) to be right (resp. left) invariant, it is necessary and sufficient that t belong to the centre of U(G). 0AFL
Lemma 3 Let G be a Lie group, H₁ and H₂ integral subgroups whose topology admits a countable base and g ∈ G. Then ↩ 1 0AFM
Proposition 14 Let G be a Lie group and H an integral subgroup whose topology admits a countable base. The following conditions are equivalent: (i) H is normal in G; (ii) L(H) is invariant under… ↩ 6 0AFN
Corollary 1 Let G be a finite-dimensional simply connected Lie group. Let m, h be Lie subalgebras of L(G) such that L(G) is the semi-direct product of m by h. Let M, H be the corresponding int… ↩ 3 0AFO
Corollary 2 Let G be a finite-dimensional simply connected Lie group, H a normal connected Lie subgroup of G and π the canonical morphism of G onto G/H. ↩ 1 0AFP
Corollary 3 Let G be a finite-dimensional connected Lie group and H a normal connected Lie subgroup of G. The canonical morphism of π₁(H) into π₁(G) is injective. 0AFQ
Proposition 15 Let G be a Lie group, M a manifold of class Cʳ (r ≥ 2) and α a differential form of class Cʳ⁻¹ and degree 1 on M with values in L(G), such that dα + [α]² = 0. Suppose that M is sim… 0AFR
Lemma 4 Let G be a connected topological group, X a Hausdorff topological space and f₁, f₂ laws of left (resp. right) operation of G on X such that, for all x ∈ X, the mappings s ↦ f₁(s, x… ↩ 3 0AFS
Proposition 16 Let G be a connected Lie group, X a Hausdorff manifold of class C^r and f1, f2 laws of left (resp. right) operation of class C^r of G on X. If the laws of infinitesimal operation a… ↩ 1 0AFT
Lemma 5 Let G be a simply connected topological group, X a Hausdorff topological space, U an open neighbourhood of e in G and \psi a continuous mapping of U \times X into X such that \psi(… ↩ 1 0AFU
Theorem 5 Let G be a simply connected Lie group, X a compact manifold of class C^r (r \geq 2) and a \mapsto Da a law of left (resp. right) infinitesimal operation of class C^{r-1} of L(G) on… ↩ 2 0AFV
Proposition 17 Let Δ be the set of z ∈ C such that −π < J(z) < π and Δ' the set of z ∈ C which are not real ≤ 0. Let E be a complete normable space over C and A (resp. A') the set of x ∈ L(E) who… ↩ 4 0AFW
Theorem 6 Let E be a real or complex Hilbert space and U the unitary group of E. ↩ 1 0AFX
Proposition 18 Let E be a complete normable space over C, v ∈ L(E) and g = exp v. Suppose that Sp(v) contains none of the points 2iπn with n ∈ Z − {0}. Then, for all x ∈ E, the conditions vx = 0… ↩ 1 0AFY
Corollary 1 Let E be a complete normable space over C and F the space of continuous n-linear mappings of Eⁿ into E. For all v ∈ L(E), let σ(v) be the element of L(F) defined by ↩ 1 0AFZ
Lemma 6 If v ∈ L(E), then Sp σ(v) ⊂ Sp v + Sp v + ⋯ + Sp v, where the sum comprises n+1 terms. ↩ 1 0AG0
Corollary 2 Let E be a complete normable algebra over C and w ∈ L(E). Suppose that every z ∈ Sp w satisfies |J(z)| < 2π/3. The following conditions are equivalent: (i) w is a derivation of E… ↩ 1 0AG1
Proposition 19 Let E be a complete normable space over C, v ∈ L(E) and g = exp v. Suppose that every z ∈ Sp v satisfies -π < J(z) < π. Then, for every closed vector subspace E' of E, the conditio… ↩ 1 0AG2
Corollary Let E be a complete normed space over C, v ∈ L(E) and g = exp v. Suppose that every z ∈ Sp v satisfies -π/2 < J(z) < π/2. Then, for every closed vector subspace M of L(E), the cond… 0AG3
Lemma 7 Let B be a group, A a normal subgroup of B, C the group B/A and i : A → B and p : B → C the canonical morphisms. Let A' be a group and f a homomorphism of A into A'. Let ω be a mor… ↩ 1 0AG4
Proposition 20 Let G be a finite-dimensional real Lie group. ↩ 2 0AG5
Definition 4 (G, γ), or simply tildeG, is called the universal complexification of G. 0AG6
Remark (1) Let (tildeG, γ) be the universal complexification of G. Let G₀ (resp. tildeG₀) be the identity component of G (resp. tildeG). By the proof of Proposition 20, (tildeG₀, γ|G₀) is… 0AG7

§ 7. Lie groups over an ultrametric field

Proposition 1 Let G be a Lie group germ with identity element e. There exists a fundamental system of open neighbourhoods of e in G consisting of Lie subgroups of G. ↩ 4 0AH2
Theorem 1 Let L be a complete normable Lie algebra. There exists a Lie group G such that L(G) is isomorphic to L. Two such groups are locally isomorphic. ↩ 2 0AH3
Theorem 2 Let G be a Lie group and h a Lie subalgebra of L(G) admitting a topological supplement. There exists a Lie subgroup H of G such that L(H) = h. If H1 and H2 are Lie subgroups of G s… 0AH4
Theorem 3 Let G and H be Lie groups and h a continuous morphism of L(G) into L(H). ↩ 1 0AH5
Proposition 2 Let G be a Lie group and h a Lie subalgebra of L(G) admitting a topological supplement. The following conditions are equivalent: 0AH6
Proposition 3 Let G be a Lie group. There exists an exponential mapping φ of G with the following properties: (i) φ is defined on an open subgroup U of the additive group L(G); (ii) φ(U) is an o… ↩ 4 0AH7
Proposition 4 Let G be a Lie group and φ an injective exponential mapping of G. Suppose that p > 0. For all x, y in L(G), 0AH8
Definition 1 Let r be an integer ≥ 0. A standard group of dimension r over K is a Lie group G with the following properties: (i) the underlying analytic manifold of G is m × m × ⋯ × m (r factor… ↩ 2 0AH9
Example Let G = 1 + Mn(m), which is an open subset of Mn(K). If x ∈ G, then det x ∈ 1 + m and hence G ⊂ GL(n, K). Clearly GG ⊂ G. If x = 1 + y with y ∈ Mn(m), the calculation of the invers… 0AHA
Theorem 4 Let G be a finite-dimensional Lie group. There exists an open subgroup of G isomorphic to a standard group. ↩ 3 0AHB
Proposition 5 Let G be a standard group. ↩ 1 0AHC
Corollary Suppose that K is locally compact and let q = Card(A/m). 0AHD
Proposition 6 Let a, b, c, c' be non-zero ideals of A contained in m such that ↩ 1 0AHE
Proposition 7 (i) The family (G(a_λ)) is a central filtration on G (Chapter II, § 4, no. 4, Definition 2). (ii) For λ ∈ R₊^*, G(a_λ) = x ∈ G | ω(x) ≥ λ , G(a_λ⁺) = x ∈ G | ω(x) > λ . (ii) is obv… 0AHF
Proposition 8 Let n ∈ Z and hₙ be the mapping x ↦ xⁿ of G into G. Let a be a non-zero ideal of A contained in m, such that n ∉ a. Then hₙ|G(a) is an isomorphism of the analytic manifold G(a) ont… ↩ 1 0AHG
Corollary If n is invertible in A, hₙ is an isomorphism of the analytic manifold G onto itself. For every non-zero ideal a of A contained in m, hₙ(G(a)) = G(a). For all x ∈ G, ω(xⁿ) = ω(x). 0AHH
Proposition 9 Suppose that p ≠ 0. (i) Let a, b be non-zero ideals of A such that b ⊂ a ⊂ m. In the group G(a)/G(b), every element has order a power of p. (ii) Suppose that v(p) = 1. If x ∈ G is… ↩ 2 0AHI
Lemma 1 Suppose that p ≠ 0. Let G be a Lie group, G₁ an open subgroup of G which is isomorphic to a standard group and x ∈ G. The following conditions are equivalent: (i) there exists a po… ↩ 3 0AHJ
Proposition 10 Suppose that p ≠ 0. Let G be a finite-dimensional Lie group. Let G_f be the set of x ∈ G for which there exists a strictly increasing sequence (nᵢ) of integers such that x^nᵢ tends… ↩ 1 0AHK
Definition 2 The mapping ψ of Proposition 10 is called the logarithmic mapping of of G and is denoted by log_G or simply log. 0AHL
Proposition 11 Suppose that p ≠ 0. Let x, y be two permutable elements of G_f. Then xy ∈ G_f and log(xy) = log x + log y. 0AHM
Proposition 12 Suppose that p ≠ 0. Let x ∈ G_f. The following conditions are equivalent: (i) log x = 0; (ii) x is of finite order in G. 0AHN
Proposition 13 Suppose that p ≠ 0. If G is compact or standard, then G_f = G. ↩ 2 0AHO
Corollary Suppose that K is locally compact. Then G_f is the union of the compact subgroups of G. 0AHP
Example Suppose that K is locally compact. Let U be the set of invertible elements of A; it is a compact open subgroup of the Lie group K^*. Then U ⊂ (K^*)_f by Proposition 13; on the othe… 0AHQ
Proposition 14 Suppose that p ≠ 0 and that v is chosen such that v(p) = 1. Let G be a standard group and E(X) (resp. L(X)) the expansion of the exponential function of G (resp. the logarithmic fu… 0AHR

§ 8. Lie groups over R or Qp

Theorem 1 Let G and H be two Lie group germs over R or Qₚ. Let f be a continuous morphism of G into H. Then f is analytic. ↩ 5 0AI2
Corollary 1 Let G be a topological group. There exists on G at most one analytic manifold structure over R (resp. Qₚ) compatible with the topological group structure on G. ↩ 1 0AI3
Definition 1 A topological group G is called a real (resp. p-adic) Lie group if there exists on G a real (resp. p-adic) Lie group structure compatible with its topology. 0AI4
Corollary 2 Let G be a topological group and V an open neighbourhood of e. Suppose that V has an analytic manifold structure which makes it into a real (resp. p-adic) Lie group germ. Then G is… ↩ 1 0AI5
Remark (1) Theorem 1 and its corollaries are no longer true if R (resp. Qₚ) is replaced by C (Exercise 1). 0AI6
Proposition 1 Let G, G' be topological groups and f a continuous morphism of G into G'. Suppose that one of the following three cases holds: (a) G is a real Lie group and G' is a p-adic Lie grou… 0AI7
Theorem 2 Let G be a finite-dimensional Lie group over R or Qₚ. Every closed subgroup of G is a Lie subgroup of G. More generally, let U be a symmetric open neighbourhood of e in G and H a n… ↩ 8 0AI8
Corollary 1 Let G' be a locally compact group, G a finite-dimensional Lie group over R (resp. Qₚ) and f a continuous morphism of G' into G. If the kernel of f is discrete, G' is a finite-dimen… 0AI9
Corollary 2 Let G be a finite-dimensional Lie group over K, H a subgroup of G, V an open neighbourhood of e in G and (Mᵢ)_i ∈ I a family of analytic manifolds over K; for all i ∈ I, let fᵢ be… ↩ 2 0AIA

§ 9. Commutators, centralizers and normalizers in a Lie group

Proposition 1 Let G be a topological group and A and B subgroups of G. Then (overlineA, overlineB) = (overlineA, overlineB), overlineDⁱA = overlineDⁱA, overlineCⁱA = overlineCⁱA. ↩ 5 0AII
Corollary 1 If G is Hausdorff, the following conditions are equivalent: (i) G is solvable (resp. nilpotent); (ii) overlineDⁱG = e (resp. overlineCⁱG = e) for sufficiently large i. DⁱG ⊂ overli… 0AIJ
Corollary 2 Let G be a Hausdorff topological group and A a subgroup of G. For A to be solvable (resp. nilpotent, commutative, it is necessary and sufficient that overlineA be so. ↩ 3 0AIK
Proposition 2 Let G be a topological group and A and B subgroups of G. If A is connected, (A, B) is connected. ↩ 1 0AIL
Proposition 3 Let G be a finite-dimensional Lie group and H1 and H2 subgroups of G. Let h1, h2 and h be the Lie subalgebras tangent at e to H1 H2 and (H1, H2) respectively. Then [h1, h2] ⊂ h. ↩ 3 0AIM
Remark Even if H1 and H2 are connected Lie subgroups of G, the Lie subalgebra of L(G) generated by [h1, h2] is in general distinct from h. 0AIN
Proposition 4 Let G be a finite-dimensional real or complex Lie group. Let A, B, C be integral subgroups of G such that [L(A), L(C)] ⊂ L(C) and [L(B), L(C)] ⊂ L(C). If [L(A), L(B)] ⊂ L(C), then… ↩ 3 0AIO
Corollary Let G be a finite-dimensional connected real or complex Lie group with Lie algebra g. The subgroups DⁱG (resp. CⁱG) are integral subgroups with Lie algebras Dⁱ𝔤 (resp. Cⁱ𝔤). If G i… ↩ 4 0AIP
Proposition 5 Let G be a finite-dimensional real or complex Lie group and A an integral subgroup of G. Then DA = DA. In particular, A is a normal subgroup of overlineA and overlineA/A is commuta… ↩ 1 0AIQ
Proposition 6 Suppose that K is ultrametric. Let G be a finite-dimensional Lie group. Let A, B, C be Lie subgroups of G such that [L(A), L(C)] ⊂ L(C), [L(B), L(C)] ⊂ L(C). If [L(A), L(B)] ⊂ L(C)… ↩ 1 0AIR
Corollary Suppose that K is ultrametric. Let G be a finite-dimensional Lie group with Lie algebra g. There exists an open subgroup G₀ of G such that, for all i, DⁱG₀ (resp. CⁱG₀) is a Lie su… ↩ 1 0AIS
Proposition 7 Let G be a finite-dimensional Lie group, g its Lie algebra and a a subset of g. Then Z_G(a) is a Lie subgroup of G with Lie algebra 𝔷_g(a). ↩ 2 0AIT
Proposition 8 Let G be a finite-dimensional real or complex Lie group, g its Lie algebra and A a subset of G. Then Z_G(A) is a Lie subgroup of G with Lie algebra 𝔷_g(A). ↩ 4 0AIU
Proposition 9 Let G be a finite-dimensional real or complex Lie group, g its Lie algebra, A an integral subgroup of G and a = L(A). Then Z_G(A) = Z_G(a), 𝔷_g(A) = 𝔷_g(a) and Z_G(A) is a Lie subg… ↩ 2 0AIV
Proposition 10 Let G be a finite-dimensional Lie group, g its Lie algebra and a a vector subspace of g. Then N_G(a) is a Lie subgroup of G with Lie algebra n_g(a). This follows from § 3, Proposit… ↩ 2 0AIW
Proposition 11 Let G be a finite-dimensional real or complex Lie group, g its Lie algebra, A an integral subgroup of G and a = L(A). Then N_G(A) = N_G(a) and N_G(A) is a Lie subgroup of G contain… ↩ 5 0AIX
Corollary If a = n_g(a), A is a Lie subgroup of G and is the identity component of N_G(A). This identity component is a Lie subgroup with Lie algebra n_g(a) (Proposition 11) and hence is equ… ↩ 1 0AIY
Proposition 12 Let G be a finite-dimensional Lie group. For L(G) to be nilpotent, it is necessary and sufficient that G have a nilpotent open subgroup. Suppose that G has a nilpotent open subgrou… ↩ 3 0AIZ
Proposition 13 Let G be a finite-dimensional simply connected nilpotent Lie group over R or C. ↩ 2 0AJ0
Proposition 14 Let G be a finite-dimensional connected Lie group over R or C. ↩ 3 0AJ1
Proposition 15 Let G be a finite-dimensional connected nilpotent Lie group over R or C and A an integral subgroup of G. Then Z_G(A) is the connected Lie subgroup of G with Lie algebra 𝔷_G(L(A)). 0AJ2
Proposition 16 Let G be a finite-dimensional nilpotent Lie group over R or C and A an integral subgroup of G distinct from G. Then N_G(A) is a connected Lie subgroup of G distinct from A. ↩ 1 0AJ3
Proposition 17 Let g be a finite-dimensional nilpotent Lie algebra over K and (g₀, g₁, …, gₙ) a decreasing sequence of ideals of g such that g₀ = g, gₙ = 0 and [g, gᵢ] ⊂ gᵢ₊₁ for 0 ≤ i < n. Let a… 0AJ4
Proposition 18 Let k be a commutative field. V a vector space of finite dimension 0 over k and G a subgroup of GL(V) whose elements are unipotent. (i) There exists a non-zero element v of V such… ↩ 1 0AJ5
Corollary 1 Let G be a finite-dimensional connected real or complex Lie group. For G to be nilpotent, it is necessary and sufficient that every element of Ad G be nilpotent. 0AJ6
Corollary 2 Every analytic subgroup of GL(n, K) consisting of unipotent elements is a simply connected Lie subgroup. 0AJ7
Proposition 19 Let G be a finite-dimensional Lie group. For L(G) to be solvable, it is necessary and sufficient that G possess a solvable open subgroup. ↩ 2 0AJ8
Proposition 20 Let G be a simply connected solvable Lie group of finite-dimension n over R or C and g = L(G). Let (gₙ, gₙ₋₁, …, g₀) be a sequence of subalgebras of g of dimensions n, n-1, …, 0, s… ↩ 3 0AJ9
Proposition 21 Let G be a simply connected finite-dimensional solvable Lie group over R or C and M an integral subgroup of G. Then M is a Lie subgroup of G and is simply connected. ↩ 1 0AJA
Proposition 22 Suppose that K = R or C. Let V be a finite-dimensional vector space and G a connected solvable subgroup of GL(V). Suppose that the identity representation of G is simple. (i) If K… ↩ 1 0AJB
Corollary Let V be a complex vector space of finite dimension > 0 and G a connected solvable subgroup of GL(V). (i) There exists a non-zero element v of V such that gv ∈ Cv for all g ∈ G. (i… 0AJC
Proposition 23 Let G be a finite-dimensional real or complex Lie group, r the radical of L(G) (Chapter I, § 5, Definition 2) and n the largest nilpotent ideal of L(G) (Chapter I, § 4, no. 4). Let… ↩ 1 0AJD
Definition 1 Let G be a finite-dimensional real or complex Lie group. The radical of G is the largest connected solvable normal subgroup of G. 0AJE
Remark Even if G is connected there may be solvable normal subgroups of G not contained in the radical of G. 0AJF
Proposition 24 Suppose that K = R or C. Let G₁, G₂ be two finite-dimensional connected Lie groups, R₁ and R₂ their radicals and φ a surjective morphism of G₁ into G₂. Then φ(R₁) = R₂. 0AJG
Proposition 25 Suppose that K = R or C. Let G₁, G₂ be finite-dimensional connected Lie groups and R₁ and R₂ their radicals. The radical of G₁ × G₂ is R₁ × R₂. 0AJH
Proposition 26 Let G be a finite-dimensional connected real or complex Lie group. The following conditions are equivalent: (i) L(G) is semi-simple; (ii) the radical of G is e; (iii) every normal… ↩ 1 0AJI
Definition 2 A connected real or complex Lie group is called semi-simple if it is finite-dimensional and satisfies the conditions of Proposition 26. 0AJJ
Remark 1 Let G be a finite-dimensional connected real or complex Lie group. If G is not semi-simple, G has a commutative connected Lie subgroup G' which is invariant under every continuous… 0AJK
Proposition 27 Let G be a finite-dimensional connected real or complex Lie group. The following conditions are equivalent: (i) L(G) is simple; (ii) the only normal integral subgroups of G are e a… ↩ 1 0AJL
Definition 3 A connected real or complex Lie group is called almost simple if it is finite-dimensional and satisfies the conditions of Proposition 27. ↩ 2 0AJM
Proposition 28 Let G be a simply connected real or complex Lie group. The following conditions are equivalent: (i) G is semi-simple; (ii) G is isomorphic to the product of a finite number of almo… ↩ 2 0AJN
Lemma 1 Let G be a connected topological group, Z its centre and Z' a discrete subgroup of Z. Then the centre of G/Z' is Z/Z'. Let y be an element of G whose class modulo Z' is a central e… ↩ 3 0AJO
Proposition 29 Let G be a semi-simple connected real or complex Lie group. (i) G = (G, G). (ii) The centre Z of G is discrete. (iii) The centre of G/Z is e. Assertion (i) follows from the Corolla… ↩ 1 0AJP
Proposition 30 (i) Let 𝔤 be a semi-simple real or complex Lie algebra. Then Int 𝔤 is the identity component of Aut 𝔤.. (ii) Let G be a semi-simple connected real or complex Lie group. The adjoint… ↩ 1 0AJQ
Remark 2 Let 𝔤 be a complex semi-simple Lie algebra and 𝔤₀ its underlying real Lie algebra. Then Aut(𝔤) is open in Aut(𝔤₀), for Int(𝔤₀) ⊂ Aut(𝔤). ↩ 1 0AJR
Proposition 31 Let G be a simply connected finite-dimensional real or complex Lie group and R its radical. There exists a semi-simple simply connected Lie subgroup S of G such that G, as a Lie gr… 0AJS
Lemma 2 Let G be a group (resp. a topological group), G' a normal subgroup of G, V a finite-dimensional vector space over a commutative field k (resp. over K), ρ a linear representation (r… ↩ 2 0AJT
Theorem 1 Let G be a finite-dimensional real or complex Lie group, G₀ its identity component, R its radical and r the radical of L(G); suppose that G/G₀ is finite. Let ρ be a finite-dimensio… ↩ 4 0AJU
Corollary 1 Let ρ, ρ₁, ρ₂ be finite-dimensional semi-simple analytic linear repre- ↩ 1 0AJV
Corollary 2 Let ρ be a finite-dimensional semi-simple analytic linear representation of G on a vector space V, S the symmetric algebra of V and Sᴳ the subalgebra of S consisting of the element… ↩ 1 0AJW
Corollary 3 Let G be a real or complex Lie group and G₀ its identity component. Suppose that G₀ is semi-simple and that G/G₀ is finite. Then every finite-dimensional analytic linear representa… ↩ 1 0AJX
Proposition 32 Let G be a finite-dimensional connected real Lie group. Suppose that L(G) is reductive. The following conditions are equivalent: ↩ 1 0AJY
Proposition 33 Let G be a finite-dimensional complex Lie group whose number of connected components is finite, φ a finite-dimensional analytic linear representation of G and G' an integral subgro… 0AJZ

§ 10. The automorphism group of a Lie group

Lemma 1 Let G be a Lie group and α a vector field on G. For all g ∈ G, let β(g) = α(g)g⁻¹ ∈ L(G). The following conditions are equivalent: (i) α is a homomorphism of the group G into the g… ↩ 2 0AL6
Definition 1 Let G be a Lie group. An infinitesimal automorphism of G is any analytic vector field on G satisfying the conditions of Lemma 1. 0AL7
Lemma 2 Let K' be a non-discrete closed subfield of K, A a K'-manifold, B and C K-manifolds and f a K'-analytic mapping of A × B into C. Suppose that, for all ↩ 1 0AL8
Proposition 1 Let K' be a non-discrete closed subfield of K, G a Lie group over K, V a manifold over K' and (v, g) ↦ vg a K'-analytic mapping of V × G into G. Suppose that, for all v ∈ V, the ma… 0AL9
Proposition 2 Let G be a real or complex Lie group and α an infinitesimal automorphism of G. There exists a law of analytic operation (λ, g) ↦ φ_λ(g) of K on G with the following properties: (1)… ↩ 1 0ALA
Proposition 3 Suppose that K is ultrametric. Let G be a compact Lie group and α an infinitesimal automorphism of G. There exist an open subgroup I of K and a law of analytic operation (λ, g) ↦ φ… ↩ 2 0ALB
Lemma 3 Let G and G' be Lie groups and φ a homomorphism of G into Aut(G'). Let f(g, g') = (φ(g))(g') for g ∈ G, g' ∈ G'. Consider the following conditions: (i) f is analytic; (ii) f is ana… ↩ 1 0ALC
Lemma 4 Let H be a finite-dimensional simply connected Lie group. (i) For all u ∈ Aut L(H), let θ(u) be the unique automorphism of H such that L(θ(u)) = u. Then the mapping (u, g) ↦ θ(u)g… ↩ 1 0ALD
Theorem 1 Let G be a finite-dimensional Lie group and G₀ its identity component. Suppose that G is generated by G₀ and a finite number of elements. ↩ 4 0ALE
Corollary 1 Let G be a finite-dimensional real Lie group and G₀ its identity component. Suppose that G is generated by G₀ and a finite number of elements. Then Aut G has the topology T_β and i… 0ALF
Corollary 2 Let G be a semi-simple connected real or complex Lie group. The group Int G is the identity component of Aut G. ↩ 2 0ALG
Theorem 2 When K is ultrametric and locally compact and G is a compact Lie group, assertions (i), (ii), (iii), (v) and (vi) of Theorem 1 are true. (a) The uniqueness of the analytic structur… ↩ 1 0ALH
Remark If K = Qp, and the Lie group G is generated by a compact subset (cf. Exercise 2), assertions (i), (ii), (iii) and (vi) of Theorem 1 are still true, but not (v) (Exercise 3). 0ALI

IV. Coxeter groups and Tits systems

§ 1. Coxeter Groups

Definition 1 Let w \in W. The length of w (with respect to S), denoted by lS(w) or simply by l(w), is the smallest integer q \geq 0 such that w is the product of a sequence of q elements of S… 0ALN
Proposition 1 Let w and w' be in W. We have the formulas: 0ALO
Corollary Let s = (s₁, …, sₚ) and s' = (s'₁, …, s'_q) be two sequences of elements of S such that w = s₁ … sₚ and w' = s'₁ … s'_q. If the sequence (s₁, …, sₚ, s'₁, …, s'_q) is a reduced deco… 0ALP
Remark In view of formulas (1) and (2), the formula d(w, w') = l(ww'⁻¹) defines a distance on W, invariant under right translations. 0ALQ
Definition 2 A dihedral group is a group generated by two distinct elements of order two. 0ALR
Example Let M be the multiplicative group 1, -1, and let m be an integer ≥ 2 (resp. m = ∞). Then M acts on the group Z/mZ (resp. on Z) by (-1).x = -x, and the corresponding semi-direct pro… 0ALS
Proposition 2 Assume that S consists of two distinct elements s and s' of order 2. ↩ 3 0ALT
Remark Consider a dihedral group W of order 2m generated by two distinct elements s and s' of order 2. Denote by s_q (resp. s'_q) the sequence of length q whose odd (resp. even) numbered… 0ALU
Definition 3 (W, S) is said to be a Coxeter system if it satisfies the following condition: 0ALV
Example 1 Let m be an integer ≥ 2 or ∞ and let W be a group defined by a set of generators S = {s, s'} and the relations s² = s'² = 1 when m = ∞, s² = s'² = (ss')ᵐ = 1 when m is finite. Cons… 0ALW
Example 2 Let 𝔖ₙ be the symmetric group of degree n, with n ≥ 2. Let sᵢ be the transposition of i and i+1 for 1 ≤ i < n, and let S = {s1, ..., s{n-1}}. One can show (§ 2, no.4, Example and §… 0ALX
Example 3 For the classification of finite Coxeter groups, cf. Chap. 4, § 4. 0ALY
Remark Suppose that (W, S) is a Coxeter system. There exists a homomorphism ε from W to the group {1, -1} characterized by ε(s) = -1 for all s ∈ S. We call ε(w) the signature of w; it is… 0ALZ
Proposition 3 Assume that (W, S) is a Coxeter system. Then, two elements s and s' of S are conjugate² in W if and only if the following condition is satisfied: ↩ 2 0AM0
Lemma 1 (i) Let w ∈ W and t ∈ T. The number (-1)^n(s, t) has the same value η(w, t) for all sequences s = (s₁, …, s_q) of elements of S such that w = s₁ … s_q. ↩ 3 0AM1
Lemma 2 Let s = (s₁, …, s_q), Φ(s) = (t₁, …, t_q) and w = s₁ … s_q. Let T_w be the set of elements of T such that η(w, t) = -1. Then s is a reduced decomposition of w if and only if the tᵢ… ↩ 4 0AM2
Lemma 3 Let w ∈ W and s ∈ S be such that l(sw) ≤ l(w). For any sequence s = (s₁, …, s_q) of elements of S with w = s₁ … s_q, there exists an integer j such that 1 ≤ j ≤ q and ↩ 2 0AM3
Remark The set T_w defined in Lemma 2 consists of the elements of the form w''sw''⁻¹ corresponding to the triples (w', w'', s) ∈ W × W × S such that w = w''sw' and l(w') + l(w'') + 1 = l(… 0AM4
Proposition 4 Let s ∈ S, w ∈ W and s = (s₁, …, s_q) be a reduced decomposition of w. Then one of the following must hold: ↩ 4 0AM5
Lemma 4 Let w ∈ W have length q ≥ 1, let D be the set of reduced decompositions of w, and let F be a map from D to a set E. Assume that F(s) = F(s') if the elements s = (s₁, …, s_q); s' =… ↩ 1 0AM6
Proposition 5 Let M be a monoid (with unit element 1) and f a map from S to M. For s, s' \in S, let m(s, s') be the order of ss' and put ↩ 3 0AM7
Theorem 1 (W, S) is a Coxeter system if and only if it satisfies the exchange condition (E) of no. 5. ↩ 2 0AM8
Proposition 6 Let (Pₛ)_s ∈ S be a family of subsets of W satisfying (C) and the following conditions: ↩ 5 0AM9
Proposition 7 Let w be in W. There exists a subset S_w of S such that s₁, …, s_q = S_w for any reduced decomposition (s₁, …, s_q) of w. 0AMA
Corollary 1 For any subset X of S, the subgroup W_X of W consists of the elements w of W such that S_w ⊂ X. ↩ 5 0AMB
Corollary 2 For any subset X of S, we have W_X ∩ S = X. ↩ 2 0AMC
Corollary 3 The set S is a minimal generating set of W. 0AMD
Corollary 4 For any subset X of S and any w in W_X, the length of w with respect to the generating set X of W_X is equal to l_S(w). ↩ 1 0AME
Theorem 2 (i) For any subset X of S, the pair (W_X, X) is a Coxeter system. ↩ 3 0AMF
Definition 4 Let I be a set. A Coxeter matrix of type I is a symmetric square matrix M = (mᵢⱼ)_i,j ∈ I whose entries are integers or +∞ satisfying the relations mᵢᵢ = 1 for all i ∈ I; tag25 mᵢⱼ… 0AMG
Proposition 8 Let (Sᵢ)_i ∈ I be a partition of S such that every element of Sᵢ commutes with every element of Sⱼ if i ≠ j. For all i ∈ I, let Wᵢ be the subgroup generated by Sᵢ. Then W is the re… ↩ 1 0AMH

§ 2. Tits Systems

Definition 1 A Tits system is a quadruple (G, B, N, S), where G is a group, B and N are two subgroups of G and S is a subset of N/(B ∩ N), satisfying the following axioms: ↩ 1 0AN8
Remark 1 We shall see in no. 5 (Cor. of Th. 3) that, if (G, B, N) is given, there exists at most one subset S of N/(B ∩ N) such that (G, B, N, S) is a Tits system. 0AN9
Remark 2 Let (G, B, N, S) be a Tits system, and let Z be a normal subgroup of G contained in B. Let G' = G/Z, B' = B/Z, N' = N/(Z ∩ N), and let S' be the image of S in N'/(B' ∩ N'). Then on… 0ANA
Lemma 1 Let s₁, …, s_q ∈ S and let w ∈ W. We have C(s₁ … s_q).C(w) ⊂ ⋃_(i₁, …, iₚ) C(s_i₁ … s_iₚ w), where (i₁, …, iₚ) denotes the set of strictly increasing sequences of integers in the i… ↩ 4 0ANB
Theorem 1 We have G = BWB. The map w ↦ C(w) is a bijection from W to the set B backslash G / B of double cosets of G with respect to B. ↩ 5 0ANC
Remark Axiom (T4) was not used in the preceding proof. 0AND
Theorem 2 The pair (W, S) is a Coxeter system. Moreover, for s ∈ S and w ∈ W, the relations C(sw) = C(s).C(w) and l_S(sw) > l_S(w) are equivalent. ↩ 1 0ANE
Corollary 1 Let w₁, …, w_q ∈ W and let w = w₁ … w_q. If l_S(w) = l_S(w₁) + ⋯ + l_S(w_q), then C(w) = C(w₁) … C(w_q). ↩ 5 0ANF
Corollary 2 Let w ∈ W and let T_w be the subset of W associated to w by the procedure of Lemma 2 of § 1, no. 4. If t ∈ T_w, then ↩ 1 0ANG
Corollary 3 Let w ∈ W and let H_w be the subgroup of G generated by C(w).C(w⁻¹). Then: ↩ 2 0ANH
Example Th. 2, applied to the Tits system described in no. 2, shows that the symmetric group 𝔖ₙ, with the set of transpositions of consecutive elements, is a Coxeter group. 0ANI
Theorem 3 a) For any subset X of S, the set GX is a subgroup of G, generated by ⋃_s ∈ X C(s). ↩ 4 0ANJ
Corollary The set S consists of the elements w ∈ W such that w ≠ 1 and B ∪ C(w) is a subgroup of G. ↩ 1 0ANK
Remark 1 The above corollary shows that S is determined by (G, B, N); for this reason, we sometimes allow ourselves to say that (G, B, N) is a Tits system, or that (B, N) is a Tits system i… 0ANL
Proposition 1 Let X be a subset of S and N' a subgroup of N whose image in W is equal to WX. Then, (GX, B, N', X) is a Tits system. 0ANM
Proposition 2 Let X, Y ⊂ S and w ∈ W. We have ↩ 2 0ANN
Remark 2 Denote by G_X backslash G / G_Y the set of subsets of G of the form G_X g G_Y, g ∈ G; and define W_X backslash W / W_Y analogously. The preceding proposition shows that the canonic… 0ANO
Proposition 3 Let X ⊂ S and g ∈ G. The relation g B g⁻¹ ⊂ G_X implies that g ∈ G_X. ↩ 1 0ANP
Definition 2 A subgroup of G is said to be parabolic if it contains a conjugate of B. ↩ 1 0ANQ
Proposition 4 Let P be a subgroup of G. ↩ 1 0ANR
Theorem 4 (i) Let P₁ and P₂ be two parabolic subgroups of G whose intersection is parabolic and let g ∈ G be such that gP₁g⁻¹ ⊂ P₂. Then g ∈ P₂ and P₁ ⊂ P₂. ↩ 2 0ANS
Proposition 5 Let P₁ and P₂ be two parabolic subgroups of G. Then P₁ ∩ P₂ contains a conjugate of T. 0ANT
Lemma 2 Let H be a normal subgroup of G. There exists a subset X of S such that BH = G_X and such that every element of X commutes with every element of S - X. ↩ 1 0ANU
Theorem 5 Let Z be the intersection of the conjugates of B, let U be a subgroup of B and let G1 be the subgroup generated by the conjugates of U in G. We make the following assumptions: ↩ 4 0ANV
Corollary Under the assumptions of Th. 5, the group G1/(G1 \cap Z) is either simple non-abelian or reduces to the identity element. 0ANW
Remark 1 Assumptions (2), (3), (4) were not used in the proof that (G', B', N', S') is a Tits system. 0ANX
Remark 2 Suppose that Z \cap U = \{1\}. Since Z and U are normal in B, it follows that every element of Z commutes with every element of U, and hence with every element of G1. In view of th… 0ANY
Remark 3 Assumption (3) is implied by the following condition: 0ANZ
Example 1 Let k be a field, n an integer \geqslant 0, G = \mathbf{GL}(n, k), and let (G, B, N, S) be the Tits system described in no. 2. Let U be the strictly upper triangular subgroup of G… 0AO0
Example 2 Let g be a simple Lie algebra over C, and let G be the group of inner automorphisms of g (cf. Chap. III, § 6, no. 2, Prop. 2). By using Th. 5, one can show that G is simple non-abe… 0AO1

App. 0. Graphs

Definition 1 A combinatorial graph (or simply a graph, if there is no risk of confusion) is a pair (A, S), where S is a set and A is a subset of 𝔓(S) consisting of sets with two elements. 0AOV
Proposition 1 Let Γ = (A, S) be a graph and (S_α)_α ∈ L its family of connected components. Denote by Γ_α the full subgraph of Γ having S_α as its set of vertices. ↩ 2 0AOW
Corollary 1 A graph Γ = (A, S) is connected if and only if there does not exist a partition (S', S'') of S into two non-empty subsets such that no element of S' is joined in Γ to any element o… ↩ 1 0AOX
Corollary 2 A subset S' of S is a union of connected components if and only if no vertex belonging to S' is joined to any vertex belonging to S - S'. 0AOY
Proposition 2 Let Γ = (A, S) be a forest having only a finite number of vertices. ↩ 1 0AOZ
Proposition 3 A graph is a chain if and only if its number of vertices is finite and non-zero and it is a tree with no ramification point. ↩ 1 0AP0

V. Groups generated by reflections

§ 1. Hyperplanes, chambers and facets

Proposition 1 Let a be a point of E. There exists a connected open neighbourhood of a that does not meet any hyperplane H that belongs to H and does not pass through a. Moreover, there exist onl… ↩ 1 0AP1
Definition 1 A facet of E relative to H is an equivalence class of the equivalence relation defined above. ↩ 1 0AP2
Proposition 2 The set of facets is locally finite. 0AP3
Proposition 3 Let F be a facet and L its affine support. ↩ 3 0AP4
Corollary Let F and F' be two facets. If overlineF = overlineF', the facets F and F' are equal. ↩ 1 0AP5
Proposition 4 Let F be a facet, and let L be an affine subspace of E that is the intersection of hyperplanes belonging to ℌ; denote by 𝔑 the set of hyperplanes H ∈ ℌ that do not contain L. The f… ↩ 5 0AP6
Definition 2 A chamber of E relative to ℌ (or simply a chamber if there is no ambiguity regarding ℌ) is a facet of E relative to ℌ that is not contained in any hyperplane belonging to ℌ. 0AP7
Proposition 5 Let C be a non-empty subset of E. Assume that there exists a subset ℌ' of ℌ with the following properties: ↩ 1 0AP8
Proposition 6 Every point of E is in the closure of at least one chamber. ↩ 2 0AP9
Proposition 7 Let L be an affine subspace of E and Ω a non-empty open subset of L. ↩ 3 0APA
Definition 3 Let C be a chamber of E. A face of C is a facet contained in the closure of C whose support is a hyperplane. A wall of C is a hyperplane that is the support of a face of C. 0APB
Proposition 8 Every hyperplane H belonging to ℌ is the wall of at least one chamber. ↩ 1 0APC
Proposition 9 Let C be a chamber and M the set of walls of C. Then C = D_M(C) and every subset L of ℌ such that C = D_L(C) contains M. A subset F of overlineC is a facet if and only if it is a f… ↩ 3 0APD
Remark 1 It follows from formula (6) and Prop. 9 that the closure of a chamber C is the intersection of the closed half-spaces that are bounded by a wall of C and contain C. 0APE
Remark 2 Let F be a facet whose support is a hyperplane L; we shall show that F is a face of two chambers. Let 𝔑 be the set of hyperplanes H ≠ L belonging to ℌ; put A = D_𝔑(F) and denote by… 0APF
Lemma 1 Any two non-parallel hyperplanes have non-empty intersection. ↩ 2 0APG
Lemma 2 Let H and H' be two distinct hyperplanes of E, and f, f' two affine functions on E such that H (resp. H') consists of the points a in E such that f(a) = 0 (resp. f'(a) = 0). Finall… ↩ 1 0APH
Proposition 10 Let C be a chamber, let H and H' be two walls of C, and let L be a hyperplane meeting D_H(C) ∩ D_H'(C). Assume that H is distinct from H' and that one of the following conditions i… 0API

§ 2. Reflections

Definition 1 An endomorphism s of the vector space V is said to be a pseudo-reflection if 1 - s is of rank 1. 0APJ
Proposition 1 Let G be a group and ρ an irreducible linear representation of G on a vector space V; assume that there exists an element g of G such that ρ(g) is a pseudo-reflection. ↩ 5 0APK
Proposition 2 Let s be an endomorphism of V. ↩ 1 0APL
Corollary If V is finite dimensional, every reflection is of determinant −1. 0APM
Proposition 3 Let s be a reflection in V. ↩ 5 0APN
Corollary Two distinct reflections s and u commute if and only if Vₛ⁻ ⊂ Vᵤ⁺ and Vᵤ⁻ ⊂ Vₛ⁺. 0APO
Remark Let a ≠ 0 be in V and let a^* be a non-zero linear form on V: it follows from formula (1) that 0APP
Proposition 4 Assume that V is finite dimensional. Let B be a non-degenerate symmetric bilinear form on V, X a subspace of V and X⁰ the orthogonal complement of X with respect to B; finally, let… 0APQ
Proposition 5 Let H and H' be two parallel, non-isotropic hyperplanes of E. There exists a unique vector v ∈ V orthogonal to H and such that H' = H + v. The displacement s_H' s_H is the translat… ↩ 4 0APR
Corollary Let H and H' be two distinct, parallel, non-isotropic hyperplanes. If K is of characteristic zero (resp. p > 0, with p ≠ 2), the group of displacements of E generated by s_H and s_… 0APS
Remark We retain the notation of Prop. 5 and assume in addition that K = R. Put s = s_H and s' = s_H'. Let Hₙ be the hyperplane H + n.v and let Cₙ be the set of points of E of the form a… 0APT
Proposition 6 Let s be the orthogonal reflection with respect to a line D of V. If Δ and Δ' are two half-lines starting at the origin (resp. two lines passing through the origin) of V, we have 0APU
Corollary Let D and D' be two lines of V and let θ be a measure of the angle (widehatD, widehatD'). Then s_D' s_D = ρ_2θ. ↩ 4 0APV
Proposition 7 The group W is dihedral (Chap. IV, § 1, no. 2) of order 2m. It consists of the rotations ρ_2nθ and the products ρ_2nθ s for n ∈ Z. The transforms of D and D' by the elements of W a… 0APW
Corollary Let C be the open angular sector formed by the union of the open half-lines Δ₁ starting at the origin and such that 0 < (widehatΔ, widehatΔ₁) < θ. Then no transform of D or D' by a… ↩ 1 0APX
Remark Assume that m is finite and that θ = π / m. If n ∈ Z, let Cₙ be the union of the open half-lines Δ₁ starting at 0 such that 0APY

§ 3. Groups of displacements generated by reflections

Lemma 1 The set of hyperplanes H is locally finite. 0AQ2
Lemma 2 Let C be a chamber. (i) For any x ∈ E, there exists an element w ∈ W such that w(x) ∈ overlineC. (ii) For any chamber C', there is an element w ∈ W such that w(C') = C. (iii) The g… ↩ 2 0AQ3
Theorem 1 Let C be a chamber and let S be the set of reflections with respect to the walls of C. ↩ 7 0AQ4
Corollary Let Σ be a set of reflections generating W. Then every reflection belonging to W is conjugate to an element of Σ. ↩ 5 0AQ5
Theorem 2 For any chamber C, the closure overlineC of C is a fundamental domain for the action of W on E. ↩ 7 0AQ6
Proposition 1 Let F be a facet and C a chamber such that F ⊂ overlineC. Let w ∈ W. The following conditions are equivalent: (i) w(F) meets F; (ii) w(F) = F; (iii) w(overlineF) = overlineF; ↩ 5 0AQ7
Proposition 2 Let A be a non-empty subset of E, let ℌ_A be the set of hyperplanes H ∈ ℌ containing A, let A' be the intersection of the H ∈ ℌ_A, and let F be a facet of E open in A' (§ 1, no. 3… ↩ 3 0AQ8
Remark 1 It follows from Th. 2 that, if C is a chamber and F a facet, there exists a unique facet F' contained in overlineC that is transformed into F by an element of W. 0AQ9
Remark 2 It follows from Props. 1 and 2 that, for any non-empty subset A of E, there exists a point a ∈ E such that W(A) = W(a); moreover, the group W(A) is a Coxeter group (Th. 1). 0AQA
Remark 3 Let C be a chamber of E and S the set of reflections with respect to the walls of C. Let w ∈ W and let (s₁, …, s_q) be a reduced decomposition of w with respect to S. If x ∈ overli… 0AQB
Proposition 3 Let C be a chamber and let i, j ∈ I with i ≠ j. Put sᵢ = sᵢ(C), Hᵢ = Hᵢ(C), eᵢ = eᵢ(C) and define sⱼ, Hⱼ and eⱼ similarly. ↩ 7 0AQC
Remark Formula (4) is actually valid for all i, j ∈ I: in fact, π / mᵢⱼ = 0 if mᵢⱼ = ∞, and if i = j then mᵢⱼ = 1 and (eᵢ|eⱼ) = 1. 0AQD
Lemma 3 Let q be a positive quadratic form on a real vector space V and let B be the associated symmetric bilinear form. Let a₁, …, aₙ be elements of V such that B(aᵢ, aⱼ) ≤ 0 for i ≠ j. ↩ 3 0AQE
Lemma 4 Let Q = (qᵢⱼ) be a real, symmetric, square matrix of order n such that: a) qᵢⱼ ≤ 0 for i ≠ j; b) there does not exist a partition of 1, 2, …, n into two non-empty subsets I and J s… ↩ 3 0AQF
Lemma 5 Let e₁, …, eₙ be vectors generating T such that: a) (eᵢ|eⱼ) ≤ 0 for i ≠ j; b) there does not exist a partition of 1, …, n into two non-empty subsets I and J such that (eᵢ|eⱼ) = 0 f… ↩ 1 0AQG
Lemma 6 Let (e₁, …, eₙ) be a basis of T such that (eᵢ|eⱼ) ≤ 0 for i ≠ j. ↩ 5 0AQH
Lemma 7 Let A be a set of unit vectors in T. If there exists a real number λ < 1 such that (a|a') ≤ λ for a, a' ∈ A and a ≠ a', then the set A is finite. ↩ 1 0AQI
Theorem 3 (i) The set of walls of a chamber is finite. (ii) The set of directions of hyperplanes belonging to ℌ is finite. (iii) The group U(W) is finite. ↩ 1 0AQJ
Proposition 4 Let C be a chamber and 𝔑 a set of walls of C. Let W_𝔑 be the subgroup of W generated by the orthogonal reflections with respect to the elements of 𝔑. For H ∈ 𝔑, denote by e_H the u… ↩ 2 0AQK
Remark 1 Since W is generated by reflections with respect to the walls of the chamber C, the preceding proposition gives a criterion for W to be finite. We shall return to this question in… 0AQL
Remark 2 Let F be a finite-dimensional real affine space and G a group of automorphisms of F. For all g ∈ G, denote by U(g) the automorphism of the space of translations V of F associated t… 0AQM
Proposition 5 Let J₁, …, Jₛ be the sets of vertices of the connected components of the Coxeter graph of W. For 1 ≤ p ≤ s, put ↩ 4 0AQN
Corollary Assume that W ≠ 1. Then W is irreducible if and only if it is essential and its Coxeter graph is connected. 0AQO
Remark Under the hypotheses of Prop. 5, the subspaces Tp for 0 ≤ p ≤ s are the isotypical components of the linear representation U of W on T (Algebra, Chap. VIII, §3, no. 4). It follows… 0AQP
Proposition 6 (i) The group Wₚ is a group of displacements of the euclidean affine space Eₚ; it is generated by reflections; provided with the discrete topology, it acts properly on Eₚ; it is ir… ↩ 3 0AQQ
Proposition 7 Assume that the group W is essential and finite. Then: (i) There exists a unique point a of E invariant under W. (ii) The family (cH){H \in \mathscr{M}} is a basis of T. (iii) The… ↩ 5 0AQR
Proposition 8 Assume that the group W is essential, irreducible and infinite. Then: ↩ 3 0AQS
Remark 1 Identify E with E₀ × E₁ × ⋯ × Eₛ and W with W₁ × ⋯ × Wₛ as in no. 8. By Prop. 6, the chamber C is then identified with 0AQT
Remark 2 Assume that W is irreducible and essential. If H and H' are two walls of C, then m_HH' = +∞ if and only if e_H = -e_H' (Prop. 3). By Props. 7 and 8, this can happen only if H and H… 0AQU
Proposition 9 Let a ∈ E. The following conditions are equivalent: (i) We have W = Wₐ.L; (ii) The restriction of the homomorphism U to Wₐ is an isomorphism from Wₐ to U(W); (iii) For every hyperp… ↩ 3 0AQV
Definition 1 A point a of E is special for W if it satisfies the equivalent conditions in Prop. 9. ↩ 1 0AQW
Proposition 10 There exists a special point for W. ↩ 1 0AQX
Proposition 11 Assume that W is essential. Let a be a special point for W. The chambers relative to Wₐ are the open simplicial cones with vertex a. For every chamber C' relative to Wₐ, there exis… ↩ 3 0AQY
Corollary Assume that W is essential. ↩ 2 0AQZ
Remark 1 Assume that W is essential, irreducible and infinite and retain the notation of Prop. 11. Since U is an isomorphism from Wₐ to U(W), it follows that the Coxeter graph of the group… 0AR0
Proposition 12 Assume that W is essential. Let a be a special point, let L(a) be the set of its transforms under the group of translations L, and let C be a chamber. Then overlineC meets L(a) in… ↩ 1 0AR1
Remark 2 The set L(a) is contained in the set of special points, but in general is distinct from it (cf. Chap. VI, § 2, no. 2 and Plates I to VI). 0AR2

§ 4. The geometric representation of a Coxeter group

Proposition 1 The restriction of B_M to E_s,s' is positive, and it is non-degenerate if and only if m is finite. ↩ 3 0ARB
Proposition 2 The subgroup of GL(E) generated by σₛ and σ_s' is a dihedral group of order 2m(s, s'). 0ARC
Proposition 3 There exists a unique homomorphism 0ARD
Remark 1 We shall see in no. 4 that σ is injective; the group W can thus be identified with the subgroup of GL(E) generated by the σₛ. 0ARE
Proposition 4 a) The map s ↦ gₛ from S to W is injective. ↩ 1 0ARF
Corollary The pair (W, S) is a Coxeter system with matrix M. ↩ 1 0ARG
Remark 2 We have thus shown that every Coxeter matrix corresponds to a Coxeter group. 0ARH
Theorem 1 (Tits) If w ∈ W and C ∩ w(C) ≠ ∅, then w = 1. ↩ 3 0ARI
Corollary 1 The group W acts simply-transitively on the set of w(C) for w ∈ W. 0ARJ
Corollary 2 The representations σ and σ are injective. 0ARK
Corollary 3 If S is finite, σ(W) is a discrete subgroup of \mathbf{GL}(E) provided with its canonical Lie group structure; similarly, σ(W) is a discrete subgroup of \mathbf{GL}(E^) . ↩ 2 0ARL
Lemma 1 Let s, s' ∈ S, with s ≠ s', and let v ∈ W_s,s'. Then v(Aₛ ∩ A_s') is contained in either Aₛ or in s(Aₛ), and in the second case l(sv) = l(v) - 1. ↩ 1 0ARM
Proposition 5 Let X, X' ⊂ S and w, w' ∈ W. If w(C_X) ∩ w'(C_X') ≠ ∅, then X = X', wW_X = w'W_X' and w(C_X) = w'(C_X'). ↩ 1 0ARN
Corollary Let X be a subset of S and x^* an element of C_X. The stabiliser of x^* in W is W_X. ↩ 1 0ARO
Proposition 6 (i) The cone U is convex. 0ARP
Proposition 7 Assume that (W, S) is irreducible (Chap. IV, §1, no. 9). Let E⁰ be the subspace of E orthogonal to E with respect to B_M. The group W acts trivially on E⁰, and every subspace of E… ↩ 3 0ARQ
Corollary Assume that (W, S) is irreducible. Then: a) If B_M is non-degenerate, the W-module E is absolutely simple. b) If B_M is degenerate, the W-module E is not semi-simple. 0ARR
Theorem 2 The following properties are equivalent: (1) W is finite. (2) The form B_M is positive and non-degenerate. ↩ 5 0ARS
Corollary If (W, S) is irreducible and finite, E is an absolutely simple W-module. 0ART
Proposition 8 If W is finite, the graph of (W, S) is a forest (Chap. IV, Appendix). ↩ 3 0ARU
Corollary If (W, S) is irreducible and finite, its graph is a tree. 0ARV
Proposition 9 With the preceding notations, W is the subgroup of O(E) generated by the reflections with respect to the hyperplanes of ℌ. It is an essential group (§ 3, no. 7) and C₀ is a chamber… 0ARW
Lemma 2 The orthogonal complement E⁰ of E for B_M is of dimension 1; it is generated by an element v = ∑_s ∈ S vₛ eₛ with vₛ > 0 for all s. 0ARX
Proposition 10 The group W provided with the discrete topology acts properly on A; it is generated by orthogonal reflections; it is infinite, irreducible and essential (§ 3, no. 7). The intersect… 0ARY
Proposition 11 Let W be a group of displacements of a euclidean affine space A, satisfying the assumptions of § 3. Assume that W is infinite, essential and irreducible. Then the form B_M attached… ↩ 1 0ARZ

§ 5. Invariants in the symmetric algebra

Definition 1 If there exists n₀ ∈ Z such that Mₙ = 0 for n ≤ n₀, the formal series ∑_n ≥ n₀ rk_K(Mₙ) Tⁿ, which is an element of Q((T)), is called the Poincaré series of M and denoted by P_M(T). 0ASK
Proposition 1 Let S = ⨁_n ≥ 0 Sₙ be a commutative graded K-algebra with a system of generators (x₁, x₂, …, xₘ) consisting of homogeneous and algebraically independent elements. Let dᵢ be the deg… ↩ 1 0ASL
Corollary The degrees dᵢ are determined up to order by S. 0ASM
Proposition 2 Retain the preceding notation and assume that S and R are graded polynomial algebras. ↩ 3 0ASN
Remark Let S = K[X₁, …, Xₙ] be a graded polynomial algebra over K, dᵢ the degree of Xᵢ and F(X₁, …, Xₙ) a homogeneous element of degree m of S. Then 0ASO
Theorem 1 Let K be a commutative field, V a finite dimensional vector space over K, S = S(V) the symmetric algebra of V, G a finite group of automorphisms of V, and R the graded subalgebra o… ↩ 4 0ASP
Remark 1 It follows from Algebra, Chap. II, §11, no. 4, Prop. 7 that, if (z₁, …, z_q) is a family of homogeneous elements of S whose canonical images in S/(R₊ S) form a basis of S/(R₊ S) ov… ↩ 1 0ASQ
Remark 2 Let g be a pseudo-reflection of V, whose order n ≥ 2 is finite and coprime to the characteristic of K. By Maschke’s theorem (Appendix, Prop. 2), V can be decomposed as D ⊕ H, where… 0ASR
Theorem 2 Retain the assumptions and notation of Theorem 1. 0ASS
Theorem 3 We retain the assumptions and notation of Th. 1. In the set of systems of generators of the ideal R₊ of R consisting of homogeneous elements, choose a minimal element (α₁, …, αₗ)… ↩ 7 0AST
Lemma 1 Let K be a commutative field, S a graded K-algebra of polynomials, and R a graded subalgebra of S of finite type such that the R-module S admits a basis (z_λ)_λ ∈ Λ consisting of h… 0ASU
Corollary With the assumptions and notation of Th. 3, the product of the characteristic degrees of R is Card(G). ↩ 2 0ASV
Lemma 2 Let K be a commutative field, V a finite dimensional vector space over K, S = ⨁_n ≥ 0 Sₙ the symmetric algebra of V, s an endomorphism of V, and s⁽ⁿ⁾ the canonical extension of s t… ↩ 1 0ASW
Lemma 3 Let K, V and S be as in Lemma 2, G a finite group of automorphisms of V, q the order of G, and R the graded subalgebra of S consisting of the elements invariant under G. Assume tha… ↩ 1 0ASX
Proposition 3 With the assumptions and notation of Th. 3, let H be the set of pseudo-reflections belonging to G and distinct from 1. Assume that K is of characteristic 0. Then Card(H) = ∑ᵢ₌₁ˡ (k… ↩ 2 0ASY
Remark When K = R, G is a Coxeter group and H is the set of reflections belonging to G, we know (§ 3) that the elements of H are in one-to-one correspondence with the walls of V. 0ASZ
Proposition 4 With the assumptions and notation of Th. 3, assume that K is of characteristic ≠ 2. Then -1 ∈ G if and only if the characteristic degrees k₁, …, kₗ of R are all even. ↩ 1 0AT0
Proposition 5 (i) Let D be the element ∏_g ∈ H e_g of S. The elements of S which are anti-invariant under G are the elements of RD. ↩ 6 0AT1
Lemma 4 Let K be a commutative field, V a finite dimensional vector space over K, G a finite group of automorphisms of V whose order q is invertible in K, S the symmetric algebra of V, and… ↩ 2 0AT2
Lemma 5 Let K be a commutative field, S a graded polynomial algebra over K, and R a graded subalgebra of S. Then S is a free graded R-module (Algebra, Chap. II, § 11, no. 2) if and only if… ↩ 3 0AT3
Theorem 4 Let K be a commutative field, V a finite dimensional vector space over K, S the symmetric algebra of V, G a finite group of automorphisms of V, and R the subalgebra of S consisting… ↩ 4 0AT4
Remark 1 Under the assumptions of Th. 4, the product of the characteristic degrees of R is q (formula (6) and Prop. 2 (iii)), so they are coprime to the characteristic of K. This was assert… 0AT5
Remark 2 If we do not assume that Card(G) is invertible in K, we still have the implications (ii) Longleftrightarrow (iii) (cf. Lemma 5) and (ii) Longrightarrow (i) (cf. Exerc. 8); but the… 0AT6
Proposition 6 The assumptions and notation are those of Th. 4. Let H be the set of pseudo-reflections belonging to G and distinct from 1. Assume that H generates G. For all g ∈ G, put g(x) = x +… 0AT7

§ 6. The Coxeter transformation

Definition 1 The Coxeter transformation determined by an ordered chamber (C, (Hᵢ)_1 ≤ i ≤ l) is the element c = s_H₁ s_H₂ ⋯ s_Hₗ of W. 0ATH
Proposition 1 All Coxeter transformations are conjugate in W. ↩ 2 0ATI
Lemma 1 Let X be a finite forest, and x ↦ gₓ a map from X to a group Γ such that gₓ and g_y are conjugate whenever x and y are not linked in X. Let T be the set of total orderings on X. Fo… 0ATJ
Remark Let W₁, …, Wₘ be essential finite groups acting in the spaces V₁, …, Vₘ and generated by reflections. Let Cⱼ be a chamber relative to Wⱼ. Let W be the group W₁ × ⋯ × Wₘ acting in t… 0ATK
Definition 2 The integers m₁, m₂, …, mₗ are called the exponents of W. ↩ 1 0ATL
Lemma 2 Assume that W is irreducible and that dim V ≥ 2. With the preceding notation, there exist two linearly independent vectors z', z'' such that (i) the plane P generated by z', z'' is… 0ATM
Theorem 1 Assume that W is irreducible. Then: (i) m₁ = 1, mₗ = h-1. (ii) Card(ℌ) = 1/2 lh. ↩ 4 0ATN
Remark Retain the notation of the preceding proof. Let c' be the C-linear extension of c to V ⊗_R C, and c'' the restriction of c' to P ⊗_R C. From our study of c|P, c'' has an eigenvecto… 0ATO
Corollary Let R₀ be the set of unit vectors of V orthogonal to an element of ℌ. If W is irreducible, ↩ 3 0ATP
Proposition 2 If W is irreducible and h is even, the unique element of W that transforms C to −C is c^h/2. ↩ 1 0ATQ
Proposition 3 Assume that W is irreducible. Let u₁, …, uₗ be homogeneous elements of the symmetric algebra S = S(V), algebraically independent over R and generating the algebra of elements of S… ↩ 1 0ATR
Corollary 1 If (mᵢ)_1 ≤ i ≤ l is the increasing sequence of exponents of W, the order of W is equal to (m₁ + 1)(m₂ + 1) … (mₗ + 1). ↩ 1 0ATS
Corollary 2 If c is a Coxeter transformation of W, ↩ 1 0ATT
Corollary 3 The homothety with ratio −1 of V belongs to W if and only if all the exponents of W are odd. In that case, h is even and c^h/2 = -1 for any Coxeter transformation c of W. ↩ 1 0ATU

App. 0. Complements on linear representations

Proposition 1 Let K be a commutative field, A a K-algebra, and V and W two left A-modules that are finite dimensional vector spaces over K. If there exists an extension L of K such that the (A \… 0ATZ
Proposition 2 (Maschke) Let A be a ring with unit element, E a left A-module, F a direct summand of E, G a group of finite order q, and ρ a linear representation of G on E. Assume that q.1 is invertible i… ↩ 3 0AU0
Corollary Let G be a finite group of order q, and K a commutative field whose characteristic does not divide q. Then the group algebra of G over K is semi-simple. 0AU1
Proposition 3 Let A be a commutative ring, M an A-module, G a finite group acting on M, and A' an A-module. Assume that the order q of G is invertible in A. Let Mᴳ be the set of elements of M in… 0AU2
Remark The preceding proposition applies in particular when A' is an A-algebra. In that case, Mᴳ ⊗_A A' is an A'-submodule of M ⊗_A A'. 0AU3

VI. Root systems

§ 1. Root systems

Lemma 1 Let V be a vector space over k, R a finite subset of V generating V. For any α ∈ R such that α ≠ 0, there exists at most one reflection s of V such that s(α) = -α and s(R) = R. ↩ 1 0AU4
Definition 1 Let V be a vector space over k, and R a subset of V. Then R is said to be a root system in V if the following conditions are satisfied: 0AU5
Remark 1 Let k' be an extension of k. Identify V canonically with a subset of V ⊗ k' and V^* with a subset of V^* ⊗ k' = (V ⊗ k')^*. Then, R is a root system in V ⊗ k', and the α^∼ are the… ↩ 2 0AU6
Lemma 2 Let R be a root system in V. Let (x|y) be a symmetric bilinear form on V, non-degenerate and invariant under W(R). Identify V with V^* by means of this form. If α ∈ R, then α is no… ↩ 5 0AU7
Proposition 1 Let V_Q (resp. V_Q^*) be the Q-vector subspace of V (resp. V^*) generated by the α (resp. the α^∼). Then V_Q (resp. V_Q^*) is a Q-structure on V (resp. V^*) (Algebra, Chap. II, § 8… ↩ 5 0AU8
Remark 2 Proposition 1 reduces the study of root systems to the case k = Q. Remark 1 reduces it further to the study of root systems in the real vector space V_R = V_Q ⊗_Q R. The Weyl group… ↩ 2 0AU9
Remark 3 Since the α tilde generate V^*, the group W(R), considered as a subgroup of GL(V_R), is essential (Chap. V, § 3, no. 7). Moreover, the Cor. of Th. 1 of Chap. V, § 3, no. 2 shows th… ↩ 1 0AUA
Proposition 2 The α tilde form a root system in V^*, and α tilde = α for all α ∈ R. ↩ 1 0AUB
Proposition 3 For x, y ∈ V, put (x|y) = ∑_α ∈ R ⟨ α tilde, x ⟩ ⟨ α tilde, y ⟩. Then (x|y) is a non-degenerate symmetric bilinear form on V, invariant under A(R). For x, y ∈ V_Q we have (x|y) ∈ Q… ↩ 5 0AUC
Proposition 4 (i) Let X be a subset of R, let V_X be the vector subspace of V generated by X, and let V'_X be the vector subspace of V^* generated by the α⁻, where α ∈ X. Then V is the direct su… ↩ 1 0AUD
Corollary Let V₁ be a vector subspace of V, and let V₂ be the vector subspace generated by R ∩ V₁. Then R ∩ V₁ is a root system in V₂. ↩ 2 0AUE
Proposition 5 Let V be a vector space over k that is the direct sum of vector spaces V₁, …, Vᵣ. Let R be a root system in V. Put Rᵢ = R ∩ Vᵢ. The following three conditions are equivalent: ↩ 3 0AUF
Corollary Let R be a root system in V. The following conditions are equivalent: (i) R is irreducible; (ii) the W(R)-module V is simple; (iii) the W(R)-module V is absolutely simple. (ii) Lon… ↩ 4 0AUG
Proposition 6 Every root system R in V is the direct sum of a family (Rᵢ)_i ∈ I of irreducible root systems that is unique up to a bijection of the index set. 0AUH
Proposition 7 Let R be a root system in V, (Rᵢ) the family of its irreducible components, Vᵢ the vector subspace of V generated by Rᵢ, B the invariant symmetric bilinear form on V defined in Pro… ↩ 2 0AUI
Remark Choose a scalar product on V_R invariant under W(R). It is then possible to speak of the length of a root and the angle between two roots. Prop. 7 shows that this angle is independ… 0AUJ
Proposition 8 (i) If two roots are proportional, the factor of proportionality can only be ± 1, ± 1/2, ± 2. ↩ 1 0AUK
Theorem 1 Let α, β be two roots. ↩ 2 0AUL
Corollary Let α and β be two roots. ↩ 6 0AUM
Proposition 9 Let α and β be two non-proportional roots. ↩ 8 0AUN
Corollary Let S be an α-chain of roots, and γ the origin of S. The length of S is -n(γ, α); it is equal to 0, 1, 2 or 3. ↩ 1 0AUO
Remark We retain the notation above. Then: 0AUP
Proposition 10 Let α, β be two non-proportional roots such that β + α is a root. Let p, q be the integers in Prop. 9. Then ↩ 1 0AUQ
Proposition 11 Assume that R is irreducible. Let α and β be two roots such that | α | = | β |. There exists g ∈ W(R) such that g(α) = β. ↩ 2 0AUR
Proposition 12 Assume that R is irreducible and reduced. ↩ 1 0AUS
Proposition 13 Assume that R is irreducible, non-reduced and of rank ≥ 2. ↩ 1 0AUT
Proposition 14 Assume that R is irreducible and reduced, and that (α|α) takes the values λ and 2λ for α ∈ R. Let A be the set of roots α such that (α|α) = λ. Assume that any two non-proportional… 0AUU
Theorem 2 (i) The group W(R) acts simply-transitively on the set of chambers. ↩ 14 0AUV
Remark 1 Assertion (vii) shows in particular that W(R) is generated by the reflections s_αᵢ. ↩ 1 0AUW
Remark 2 If x, y ∈ C, then (x|y) > 0 (Chap. V, § 3, no. 5, Lemma 6), in other words the angle (x, y) is acute. 0AUX
Remark 3 Let m(α, β) be the order of s_α s_β (α, β ∈ B(C)). The matrix (m(α, β)) is identified with the Coxeter matrix (Chap. IV, § 1, no. 9) of (W, S). If α ≠ β, Prop. 3 of Chap. V, § 3, n… 0AUY
Definition 2 A subset B of R is called a basis of R if there exists a chamber C of R such that B = B(C). If C is a chamber, B(C) is called the basis of R defined by C. ↩ 3 0AUZ
Remark 4 Assertion (vi) of Th. 2 shows that the map C ↦ B(C) is a bijection from the set of chambers to the set of bases. Consequently, W(R) acts simply-transitively on the set of bases. ↩ 2 0AV0
Remark 5 Let C be a chamber of R, and let B be the corresponding basis. If α ∈ B, put φ(α) = α⁻ if 2α ∉ R and φ(α) = 1/2α⁻ if 2α ∈ R. Then φ(B) is the basis of R⁻ defined by C⁻; this follow… ↩ 1 0AV1
Definition 3 Let B be a basis of R. The Cartan matrix of R (relative to B) is the matrix (n(α, β))_α, β ∈ B. 0AV2
Remark 6 The Cartan matrix (n(α, β)) should not be confused with the Coxeter matrix (m(α, β)). Note in particular that the Cartan matrix is not necessarily symmetric. 0AV3
Remark 7 Canonical indexing. If B and B' are two bases of R, there exists a unique element w ∈ W such that w(B) = B'. We have 0AV4
Proposition 15 Let B be a basis of R and α an indivisible root. There exist β ∈ B and w ∈ W(R) such that α = w(β). ↩ 6 0AV5
Corollary Let R₁ and R₂ be two reduced root systems in vector spaces V₁ and V₂, and let B₁ and B₂ be bases of R₁ and R₂. Let f : B₁ → B₂ be a bijection that transforms the Cartan matrix of R… ↩ 1 0AV6
Proposition 16 Let B be a basis of R, and G the subgroup of A(R) consisting of the elements leaving B stable. Then W(R) is a normal subgroup of A(R) and A(R) is the semi-direct product of G and W… ↩ 3 0AV7
Remark 8 Let R₁, …, Rₚ be root systems in vector spaces V₁, …, Vₚ, R the direct sum of the Rᵢ in V = ∏ᵢ Vᵢ, Cᵢ a chamber of Rᵢ, and Bᵢ = B(Cᵢ). It is immediate that C = ∏ᵢ Cᵢ is a chamber o… 0AV8
Theorem 3 Every root is a linear combination with integer coefficients of the same sign of elements of B(C). In particular, every root is either positive or negative for C. ↩ 8 0AV9
Corollary Let γ be a linear combination of roots with integer coefficients, and α an indivisible root. If γ is proportional to α, then γ ∈ Zα. 0AVA
Proposition 17 Assume that R is reduced and let w ∈ W and α ∈ R. ↩ 4 0AVB
Corollary 1 Let β ∈ B(C). The reflection s_β permutes the positive roots not proportional to β. ↩ 2 0AVC
Corollary 2 Assume that R is reduced. Let w ∈ W, let q be the length of w with respect to S (Chap. IV, § 1, no. 1), and let w = s₁ … s_q be a reduced decomposition of w. Let α₁, …, α_q be the… ↩ 4 0AVD
Corollary 3 Assume that R is reduced. There exists a unique longest element w₀ in W. Its length is equal to the number of positive roots and w₀ transforms the chamber C to -C. We have w₀² = 1… ↩ 2 0AVE
Proposition 18 Let x ∈ V. The following three properties are equivalent: ↩ 3 0AVF
Corollary An element x ∈ C if and only if x > w(x) for all w ∈ W such that w ≠ 1. ↩ 1 0AVG
Proposition 19 Let (βᵢ)_1 ≤ i ≤ n be a sequence of positive roots for the chamber C such that β₁ + β₂ + ⋯ + βₙ is a root. Then there exists a permutation π ∈ Sₙ such that, for all i ∈ 1, 2, …, n… ↩ 8 0AVH
Corollary 1 Let α ∈ R₊(C). Then α ∈ B(C) if and only if α is the sum of two positive roots. ↩ 1 0AVI
Corollary 2 Let φ be a map from R to a abelian group Γ having the following properties: ↩ 2 0AVJ
Corollary 3 a) Let α ∈ R. Then Y(α) is a connected subset of B(C) (Chap. IV, Appendix). 0AVK
Definition 4 Let P be a subset of R. ↩ 2 0AVL
Lemma 3 Let C be a chamber of R and P a closed subset of R containing R₊(C) (in the notation of no. 6). Let Σ = B(C) ∩ (-P), and let Q be the set of roots that are linear combinations of e… ↩ 3 0AVM
Proposition 20 Let P be a subset of R. The following conditions are equivalent: ↩ 4 0AVN
Corollary 1 Let P be a subset of R. The following conditions are equivalent: ↩ 5 0AVO
Corollary 2 Assume that V is equipped with the structure of an ordered vector space such that, for this structure, every root of R is either positive or negative. Let P be the set of positive… ↩ 2 0AVP
Corollary 3 A subset B of R is a basis of R if and only if the following conditions are satisfied: ↩ 2 0AVQ
Corollary 4 Let B be a basis of R, B' a subset of B, V' the vector subspace of V generated by B', and R' = R ∩ V'. Then B' is a basis of the root system B'. ↩ 4 0AVR
Corollary 5 Let B be a basis of R, A₁, A₂, …, Aᵣ pairwise orthogonal subsets of B, and A = A₁ ∪ A₂ ∪ ⋯ ∪ Aᵣ. Then every root α that is a linear combination of elements of A is actually a linea… ↩ 1 0AVS
Corollary 6 We adopt the hypotheses and notation of Prop. 20. Let V₁ be the vector subspace of V generated by Σ. Then P ∩ (-P) = Q ∪ (-Q) = V₁ ∩ R is a root system in V₁ with basis Σ. ↩ 1 0AVT
Proposition 21 Let C (resp. C') be a chamber of R, Σ (resp. Σ') a subset of B(C) (resp. B(C')), Q (resp. Q') the set of linear combinations of elements of Σ (resp. Σ') with negative integer coeff… ↩ 1 0AVU
Corollary Let P, P' be two parabolic subsets of R transformed into each other by an element of the Weyl group. If there exists a chamber C of R such that R₊(C) ⊆ P and R₊(C) ⊆ P', then P = P… 0AVV
Proposition 22 Let P be a closed subset of R such that P ∩ (-P) = ∅. Then there exists a chamber C of R such that P ⊆ R₊(C). ↩ 2 0AVW
Proposition 23 Let P be a subset of R and V₁ (resp. Γ) the vector subspace (resp. the subgroup) of V generated by P. The following conditions are equivalent: ↩ 2 0AVX
Proposition 24 Let R' be the intersection of R with a vector subspace of V, so that R' is a root system in the vector subspace V' that it generates (cf. Cor. of Prop. 4, no. 1). Let B' be a basis… 0AVY
Proposition 25 Assume that R is irreducible. Let C be a chamber of R, and let B(C) = α₁, …, αₗ be the corresponding basis. ↩ 3 0AVZ
Remark The root 0AW0
Proposition 26 The set of x ∈ V such that ⟨ x, y^* ⟩ ∈ Z for all y^* ∈ Q(R^*) (or, equivalently, for all y^* ∈ R^*) is a discrete subgroup G of V containing Q(R). If B' is a basis of R^*, the bas… 0AW1
Proposition 27 Let R₁ be a subset of R, Q₁ the subgroup of Q(R) generated by R₁, and W₁ the subgroup of W(R) generated by the s_α (α ∈ R₁). If p ∈ P(R) and w ∈ W₁, then p - w(p) ∈ Q₁. ↩ 1 0AW2
Proposition 28 Let B be a basis of R, B' a subset of B, V' the vector subspace of V generated by B', R' = R ∩ V' (which is a root system in V'), R'tilde the inverse root system (which is identifi… ↩ 1 0AW3
Proposition 29 Let ρ be half the sum of the roots > 0. ↩ 5 0AW4
Corollary Let σ be half the sum of the elements > 0 of R⁻ (for B⁻). For all α ∈ V, the sum of the coordinates of α with respect to the basis B is ⟨ α, σ ⟩. If α ∈ R, this sum is equal to 1/2… ↩ 3 0AW5
Proposition 30 Assume that R is irreducible. Let m be an integer between 1 and h - 1 and prime to h. Then exp((2iπ m)/h) is an eigenvalue of c of multiplicity 1. ↩ 1 0AW6
Lemma 4 For all w ∈ W, the characteristic polynomial of w has integer coefficients. 0AW7
Proposition 31 Assume that R is irreducible and reduced, and let β = n₁ α₁ + ⋯ + nₗ αₗ be the highest root of R (cf. no. 8). Then n₁ + ⋯ + nₗ = h - 1. ↩ 4 0AW8
Proposition 32 Assume that R is irreducible, and that all roots have the same length. Let α ∈ R. The number of elements of R not orthogonal to α is 4h - 6. 0AW9
Proposition 33 Assume that R is irreducible and reduced. Put s_αᵢ = sᵢ, and let Γ be the subgroup of W generated by c = s₁ … sₗ. ↩ 4 0AWA

§ 2. Affine Weyl group

Definition 1 The group of affine transformations of E generated by the reflections s_α,k for α ∈ R and k ∈ Z is called the affine Weyl group of the root system R and denoted by Wₐ(R) (or simply… 0AWZ
Proposition 1 The group Wₐ is the semi-direct product of W by Q. 0AX0
Proposition 2 The group Wₐ, with the discrete topology, acts properly on E and permutes the hyperplanes L_α,k (for α ∈ R and k ∈ Z). 0AX1
Proposition 3 The special points (Chap. V, § 3, no. 10, Def. 1) of Wa are the weights of R'. ↩ 1 0AX2
Corollary (i) If \overline{\omega} \in P(R'), there exists an alcove C such that \overline{\omega} is an extremal point of \overline{C}. (ii) If C is an alcove, \overline{C} \cap Q(R') reduc… 0AX3
Proposition 4 Let C' be a chamber of R'. ↩ 1 0AX4
Proposition 5 Let C be the alcove contained in C' and containing 0 in its closure (Prop. 4). ↩ 5 0AX5
Corollary The alcove C is an open simplex with vertices 0 and the overlineωᵢ / nᵢ, i ∈ I. ↩ 1 0AX6
Proposition 6 For all i ∈ J, the element γᵢ = t(overlineωᵢ)wᵢw₀ belongs to Γ_C and the map i ↦ γᵢ is a bijection from J to Γ_C - 1. ↩ 1 0AX7
Corollary The (overlineωᵢ)_i ∈ J form a system of representatives in P(R⁻) of the non-zero elements of P(R⁻)/Q(R⁻). ↩ 2 0AX8
Remark 1 The map γ ↦ γ(0) is a bijection from Γ_C to overlineC ∩ P(R⁻). 0AX9
Remark 2 The group G is also the normaliser of Wₐ in the group of automorphisms of E provided only with its affine structure (cf. Exerc. 3). 0AXA
Lemma 1 Let X be a locally compact space countable at infinity, G a group acting continuously and properly on X, μ a measure on X invariant under G, G' a subgroup of G, U and U' two open s… ↩ 1 0AXB
Proposition 7 Assume that R is irreducible. Let B = α₁, …, αₗ be a basis of R, f the connection index of R (§ 1, no. 9) and tildeα = n₁ α₁ + ⋯ + nₗ αₗ the highest root of R (for the order define… ↩ 2 0AXC
Proposition 8 Let F be a real Hilbert space of finite dimension l. Let ℌ be the set of affine hyperplanes of F and G the group generated by the orthogonal reflections s_H with respect to the hyp… 0AXD
Remark 1 The assumption that T is of rank l is satisfied in particular when G is irreducible and infinite. Indeed, the vector space generated by vectors corresponding to the translations in… 0AXE
Proposition 9 Let V be a real vector space of finite dimension l, and let G be a finite subgroup of GL(V), generated by reflections and essential. Give V a scalar product invariant under G. The… ↩ 1 0AXF
Definition 2 A group G satisfying the equivalent conditions of Prop. 9 is called a crystallographic group. 0AXG
Remark 2 Let G be a finite group generated by reflections and essential. Then G is crystallographic if and only if every element of its Coxeter matrix is one of the integers 1, 2, 3, 4, 6… 0AXH

§ 3. Exponential invariants

Lemma 1 Assume that A is factorial (Commutative Algebra, Chap. VII, § 3, no. 1, Def. 1). (i) The ring A[P] is factorial. (ii) If u, v are non-proportional elements of P, the elements 1 - e… ↩ 1 0AXN
Definition 1 Let x = ∑_p ∈ P xₚ eᵖ be an element of A[P]. The set S of p ∈ P such that xₚ ≠ 0 is called the support of x and the set X of maximal elements of S is called the maximal support of… ↩ 2 0AXO
Lemma 2 Let x ∈ A[P] and let (xₚ eᵖ)_p ∈ X be the family of maximal terms of x. Let q ∈ P and let y ∈ A[P] be such that e^q is the unique maximal term of y. Then, the family of maximal ter… ↩ 4 0AXP
Remark Since x ≠ 0 means that the maximal support of x is non-empty, Lemma 2 shows that x ≠ 0 implies xy ≠ 0 whenever y admits a unique maximal term of the form e^q. 0AXQ
Definition 2 An element x ∈ A[P] is said to be anti-invariant under W if w(x) = (-1)ˡ⁽ʷ⁾.x for all w ∈ W. 0AXR
Proposition 1 If 2 is not a zero divisor in A, the elements J(eᵖ) for p ∈ P ∩ C form a basis of the module of anti-invariant elements of A[P]. ↩ 3 0AXS
Proposition 2 (i) The element d defined by (3) is an anti-invariant element of A[P]; its unique maximal term (no. 2, Def. 1) is e^ρ and d = J(e^ρ). (ii) For any p ∈ P, the element J(eᵖ) is divis… ↩ 3 0AXT
Remark 1 If 2 is not a zero divisor in A, it is easy to check that d is the unique anti-invariant element of A[P] with e^ρ as its maximal term. 0AXU
Remark 2 Lemma 2 of no. 2 shows that the unique maximal term of the quotient J(eᵖ)/d (for p ∈ P ∩ C) is e^p-ρ. ↩ 1 0AXV
Lemma 3 The S(eᵖ) for p ∈ P ∩ overlineC form a basis of the A-module A[P]ᵂ. ↩ 1 0AXW
Theorem 1 Let overlineω₁, …, overlineωₗ be the fundamental weights corresponding to the chamber C, and, for 1 ≤ i ≤ l, let xᵢ be an element of A[P]ᵂ with e^overlineωᵢ as its unique maximal t… ↩ 2 0AXX
Example 1 We can take xᵢ = S(e^overlineωᵢ). 0AXY
Example 2 By Remark 2 of no. 3, we can take xᵢ = J(e^ρ + overlineωᵢ)/d (with the notation in no. 3). 0AXZ

§ 4. Classification of root systems

Lemma 1 For all i, ∑_j ≠ i qᵢⱼ² < 1. ↩ 4 0AY2
Lemma 2 Any vertex of X belongs to at most 3 edges. ↩ 3 0AY3
Lemma 3 If i belongs to 3 edges, these edges are of order 3. 0AY4
Lemma 4 If there exists an edge of order ≥ 6, then l = 2. ↩ 2 0AY5
Lemma 5 A vertex cannot belong to two distinct edges of order ≥ 4. ↩ 1 0AY6
Lemma 6 If i, j is of order 3, W(M') is a finite Coxeter group. ↩ 1 0AY7
Lemma 7 We have one of the following: ↩ 2 0AY8
Lemma 8 Let i₁, i₂, …, iₚ be vertices of X such that i₁, i₂, i₂, i₃, …, iₚ₋₁, iₚ are edges of order 3. Then q(∑ᵣ₌₁ᵖ r e_iᵣ) = 1/2 p(p+1). ↩ 2 0AY9
Lemma 9 Assume that X is a chain with vertices 1, 2, ..., l and edges {1, 2}, {2, 3}, ..., {l - 1, l}. ↩ 1 0AYA
Lemma 10 If X has a ramification point i, the full subgraph X - {i} is the union of three chains, and if p - 1, q - 1, r - 1 are the lengths of these chains, the triple {p, q, r} is equal… ↩ 1 0AYB
Theorem 1 The graph of any irreducible finite Coxeter system (W, S) is isomorphic to one of the following: ↩ 6 0AYC
Theorem 2 The Coxeter groups defined by the Coxeter graphs Al, Bl, ..., I2(p) of Th. 1 are finite. ↩ 2 0AYD
Proposition 1 Let R₁ and R₂ be two reduced root systems in vector spaces V₁ and V₂. Let B₁ = (αᵢ)_i ∈ I₁ and B₂ = (αᵢ)_i ∈ I₂ be bases of R₁ and R₂, indexed canonically. Let λ be an isomorphism… ↩ 2 0AYE
Corollary The homomorphism φ defines by passage to the quotient an isomorphism from the group A(R)/W(R) to the group of automorphisms of the Dynkin graph of R. ↩ 2 0AYF
Theorem 3 If R is an irreducible reduced root system, its Dynkin graph is isomorphic to one of the graphs represented by the following diagrams: ↩ 3 0AYG
Remark 1 The graph A1 reduces to a single node; we denote it also by B1 or C1. The graph B2 \circ-\circ-\circ-\circ is also denoted by C2. The graph A3 \circ-\cdots-\circ is also denoted by… 0AYH
Remark 2 If (X, f) is the Dynkin graph of a reduced root system R, the Dynkin graph of the inverse system can be identified with (X, f^{-1}). In other words, the diagram associated to the D… 0AYI
Remark It can be shown that the canonical map Aut(tildeX, tildef) → Aut(Σ) is an isomorphism. 0AYJ
Theorem 4 Let (W, S) be an irreducible Coxeter system with S finite. The associated quadratic form (Chap. V, § 4, no. 1) is positive and degenerate if and only if the Coxeter graph of (W, S)… 0AYK

VII. Cartan subalgebras and regular elements

§ 1. Primary decomposition of linear representations

Proposition 1 For all λ ∈ P, ↩ 5 00SJ
Proposition 2 Let V,V',W be vector spaces. Let r : S → End(V), r': S→ End(V') and q: S→ End(W) be maps. ↩ 9 00SK
Proposition 3 The sum ∑_λ∈ PV^λ(S) is direct. The sum ∑_λ∈ PV_λ(S) is direct. ↩ 2 00SL
Lemma 1 Let x, y∈ End(V). ↩ 4 00SM
Remark The argument proves that, if V is finite dimensional and if there exists an integer n such that (ad x)ⁿy= 0, then (ad x)^2 dim V-1y= 0. 00SN
Theorem 1 Assume that V is finite dimensional. The following conditions are equivalent: ↩ 5 00SO
Corollary 1 Assume that V is finite dimensional and that condition (AC) is satisfied. Let k' be an extension of k. Assume that, for all s∈ S, the endomorphism r(s)⊗ 1 of V⊗ₖk' is triangulariza… 00SP
Corollary 2 Assume that V is finite dimensional and that condition (AC) is satisfied. Denote by V⁺(S) the vector subspace ∑_s∈ S((⋂_i≥ 1r(s)ⁱV). Then: ↩ 1 00SQ
Corollary 3 Let V and V' be finite dimensional vector spaces, and let r: S→ End(V) and r': S→ End(V') be maps satisfying condition (AC). Let f: V→ V' be a surjective linear map such that f(r(s… ↩ 2 00SR
Proposition 4 Assume that k is perfect. Let V be a finite dimensional vector space, u an element of End(V),uₛ, uₙ the semi-simple and nilpotent components of u(Algebra, Chap. VII, §5, no. 8). ↩ 4 00SS
Proposition 5 Assume that condition (AC) is satisfied, and let λ : S→ k be such that V^λ(S)not= 0. If k is of characteristic 0, the map λ is linear. If k is of characteristic pnot= 0, there exis… ↩ 2 00ST
Proposition 6 Assume that k is infinite and that condition (AC) is satisfied. Let k' be an extension of k. Put V'= V⊗ₖk',S'= S⊗ₖk'. Let r': S'→ End(V') be the map obtained from r by extension of… ↩ 1 00SU
Proposition 7 Assume that k is infinite and that condition (AC) is satisfied. Let widetildeS be the set of s∈ S such that V⁰(s) = V⁰(S). If s∈ S, let P(s) be the determinant of the endomorphism… ↩ 3 00SV
Proposition 8 Let 𝔥 be a Lie algebra, and L,M,N𝔥-modules. Denote by P the set of maps from 𝔥 to k. ↩ 1 00SW
Proposition 9 Let 𝔥 be a nilpotent Lie algebra and M a finite dimensional 𝔥-module. Denote by P the set of maps from 𝔥 to k. ↩ 7 00SX
Remark Assume that k is perfect and of characteristic 2. Let 𝔥=𝔰𝔩(2, k), and 00SY
Corollary Let 𝔥 be a nilpotent Lie algebra, and M a finite dimensional 𝔥-module such that M⁰(𝔥) = 0. Let f:𝔥→ M be a linear map such that ↩ 2 00SZ
Proposition 10 Let 𝔤 be a Lie algebra, 𝔥 a nilpotent subalgebra of 𝔤. Denote by P the set of maps from 𝔥 to k. ↩ 9 00T0
Lemma 2 Assume that k is of characteristic 0. Let 𝔤 be a semi-simple Lie algebra over k, B the Killing form of 𝔤,𝔪 a subalgebra of 𝔤. Assume that the following conditions are satisfied: ↩ 4 00T1
Proposition 11 Assume that k is of characteristic 0. Let 𝔤 be a semi-simple Lie algebra, 𝔥 a nilpotent subalgebra of 𝔤. The algebra 𝔤⁰(𝔥) satisfies conditions (1) and (2) of Lemma 2 ; it is reduc… ↩ 2 00T2
Proposition 12 Let 𝔤 be a Lie algebra, a an automorphism of 𝔤. ↩ 1 00T3
Corollary Assume that k is of characteristic zero and that 𝔤 is semi-simple. Then the subalgebra 𝔤¹(a) satisfies conditions (1) and (2) of Lemma 2 ; it is reductive in 𝔤. 00T4
Proposition 13 Let 𝔤 be a semi-simple Lie algebra, 𝔞 a subalgebra of 𝔤 reductive in 𝔤, and 𝔪 the commutant of 𝔞 in 𝔤. The subalgebra 𝔪 satisfies conditions (1) and (2) of Lemma 2 of no. 3; it is… 00T5
Remark The commutant of 𝔪 in 𝔤 does not necessarily reduce to 𝔞, cf. Exerc. 4. 00T6
Proposition 14 Let 𝔤 be a semi-simple Lie algebra, A a group and r a homomorphism from A to Aut(𝔤). Let 𝔪 be the subalgebra of 𝔤 consisting of the elements invariant under r(A). Assume that the l… 00T7

§ 2. Cartan subalgebras and regular elements of a Lie algebra

Definition 1 Let 𝔤 be a Lie algebra. A Cartan subalgebra of 𝔤 is a nilpotent subalgebra of 𝔤 equal to its own normalizer. ↩ 2 00TJ
Example 1 If 𝔤 is nilpotent, the only Cartan subalgebra of 𝔤 is 𝔤 itself (Chap. I, §4, no. 1, Prop. 3). ↩ 2 00TK
Example 2 Let 𝔤=𝔤𝔩(n, k), and let 𝔥 be the set of diagonal matrices belonging to 𝔤. We show that 𝔥 is a Cartan subalgebra of 𝔤. First, 𝔥 is commutative, hence nilpotent. Let (Eᵢⱼ) be the can… ↩ 2 00TL
Example 3 Let 𝔥 be a Cartan subalgebra of 𝔤 and let 𝔤₁ be a subalgebra of 𝔤 containing 𝔥. Then 𝔥 is a Cartan subalgebra of 𝔤₁; this follows immediately from Def. 1. ↩ 1 00TM
Proposition 1 Let 𝔤 be a Lie algebra and let 𝔥 be a Cartan subalgebra of 𝔤. Then 𝔥 is a maximal nilpotent subalgebra of 𝔤. ↩ 1 00TN
Proposition 2 Let (𝔤ᵢ)_i∈ I be a finite family of Lie algebras and 𝔤=∏_i∈ I𝔤ᵢ. ↩ 3 00TO
Example 4 If k is of characteristic 0,𝔤𝔩(n, k) is the product of the ideals 𝔰𝔩(n, k) and k.1. It follows from Example 2 and Prop. 2 that the set of diagonal matrices of trace 0 in 𝔰𝔩(n, k) i… ↩ 1 00TP
Proposition 3 Let 𝔤 be a Lie algebra, 𝔥 a subalgebra of 𝔤, and k' an extension of k. Then 𝔥 is a Cartan subalgebra of 𝔤 if and only if 𝔥⊗ₖk' is a Cartan subalgebra of 𝔤⊗ₖk'. ↩ 2 00TQ
Proposition 4 Let 𝔤 be a Lie algebra, 𝔥 a nilpotent subalgebra of 𝔤. Then 𝔥 is a Cartan subalgebra of 𝔤 if and only if 𝔤⁰(𝔥) =𝔥. ↩ 5 00TR
Corollary 1 Let 𝔤 be a Lie algebra, 𝔥 a Cartan subalgebra of 𝔤. If k is infinite, there exists x∈ 𝔥 such that 𝔥=𝔤⁰(x). ↩ 1 00TS
Corollary 2 Let f:𝔤→ 𝔤' be a surjective homomorphism of Lie algebras. If 𝔥 is a Cartan subalgebra of 𝔤,f(𝔥) is a Cartan subalgebra of 𝔤'. ↩ 6 00TT
Corollary 3 Let 𝔥 be a Cartan subalgebra of a Lie algebra 𝔤, and let Cⁿ𝔤(n≥ 1) be a term of the descending central series of 𝔤 (Chap. I, §1, no. 5). Then 𝔤=𝔥+Cⁿ𝔤. ↩ 1 00TU
Corollary 4 Let 𝔤 be a Lie algebra, 𝔥 a Cartan subalgebra of 𝔤, and 𝔞 a subalgebra of 𝔤 containing 𝔥. ↩ 3 00TV
Corollary 5 Let 𝔤 be a Lie algebra, E a subset of 𝔤. Let E operate on 𝔤 by the adjoint representation. Then E is a Cartan subalgebra of 𝔤 if and only if E =𝔤⁰(E). ↩ 1 00TW
Corollary 6 Let 𝔤 be a Lie algebra, let k₀ be a subfield of k such that [k:k₀]<+∞, and let 𝔤₀ be the Lie algebra obtained from 𝔤 by restricting the field of scalars to k₀. Let 𝔥 be a subset of… 00TX
Proposition 5 Let 𝔤 be a Lie algebra, 𝔠 its centre, 𝔥 a vector subspace of 𝔤. Then 𝔥 is a Cartan subalgebra of 𝔤 if and only if 𝔥 contains 𝔠 and 𝔥/𝔠 is a Cartan subalgebra of 𝔤/𝔠. ↩ 3 00TY
Corollary Let C_∞𝔤 be the union of the ascending central series of the Lie algebra 𝔤 (Chap. I, §1, no. 6). The Cartan subalgebras of 𝔤 are the inverse images of the Cartan subalgebras of 𝔤/C… 00TZ
Remark C_∞𝔤 is the smallest ideal 𝔫 of 𝔤 such that the centre of 𝔤/𝔫 is zero; it is a characteristic and nilpotent ideal of 𝔤. 00U0
Remark 1 If 𝔤not=0,a₀= 0 since (ad x)(x) = 0 for all x∈ 𝔤. 00U1
Remark 2 Let k' be an extension of k. Write det(T - ad x') =∑ᵢ₌₀ⁿa'ᵢ(x')Tⁱ for x'∈ 𝔤⊗ₖk'. Then a'ᵢ|𝔤=aᵢ for all i. ↩ 1 00U2
Definition 2 The rank of 𝔤, denoted by rk(𝔤), is the smallest integer l such that aₗnot= 0. An element x of 𝔤 is called regular if aₗ(x)not= 0. ↩ 1 00U3
Example 1 If 𝔤 is nilpotent, rk(𝔤) = dim𝔤 and all elements of 𝔤 are regular. 00U4
Example 2 Let 𝔤=𝔰𝔩(2, k). If x=(γ -α )∈ 𝔤, an easy calculation gives 00U5
Example 3 Let V be a vector space of finite dimension n, and 𝔤=𝔤𝔩(V). Let overlinek be an algebraic closure of k. Let x∈ 𝔤, and let λ₁, . . . , λₙ be the roots in overlinek of the characteri… 00U6
Proposition 6 Let 𝔤 be a Lie algebra, k' an extension of k, and 𝔤'= 𝔤⊗ₖk'. ↩ 1 00U7
Proposition 7 Let (𝔤ᵢ)_i∈ I be a finite family of Lie algebras, and let 𝔤= ∏_i∈ I𝔤ᵢ. 01IY
Proposition 8 Let f:𝔤→ 𝔤' be a surjective homomorphism of Lie algebras. 00U8
Corollary Let Cₙ𝔤(n≥ 0) be a term of the ascending central series of 𝔤 (Chap. I, §1, no. 6). The regular elements of 𝔤 are those whose image in 𝔤/Cₙ𝔤 is regular. 00U9
Proposition 9 Let 𝔤 be a Lie algebra, 𝔤' a subalgebra of 𝔤. Every element of 𝔤' regular in 𝔤 is regular in 𝔤'. ↩ 2 00UA
Remark 3 Elements of 𝔤' regular in 𝔤 do not necessarily exist. If at least one does exist, the set of these elements is precisely the set denoted by S in the above proof. 00UB
Theorem 1 Let 𝔤 be a Lie algebra. ↩ 5 00UC
Corollary 1 Every Lie algebra 𝔤 has Cartan subalgebras, and the rank of 𝔤 is the minimum dimension of a Cartan subalgebra. ↩ 1 00UD
Corollary 2 Let f:𝔤→ 𝔤' be a surjective homomorphism of Lie algebras. If 𝔥' is a Cartan subalgebra of 𝔤', there exists a Cartan subalgebra 𝔥 of 𝔤 such that 𝔥'=f(𝔥). ↩ 1 00UE
Corollary 3 Every Lie algebra 𝔤 is the sum of its Cartan subalgebras. ↩ 1 00UF
Proposition 10 Let 𝔤 be a Lie algebra, 𝔞 a commutative subalgebra of 𝔤 and 𝔠 the commutant of 𝔞 in 𝔤. Assume that ad_𝔤x is semi-simple for all x∈ 𝔞. Then the Cartan subalgebras of 𝔠 are the Carta… ↩ 7 00UG
Proposition 11 Let 𝔫 be a nilpotent subalgebra of a Lie algebra 𝔤. There exists a Cartan subalgebra of 𝔤 contained in 𝔤⁰(𝔫). ↩ 1 00UH
Theorem 2 Assume that k is of characteristic 0. Let 𝔤 be a semi-simple Lie algebra, 𝔥 a Cartan subalgebra of 𝔤. Then 𝔥 is commutative, and all of its elements are semi-simple in 𝔤 (Chap. I… ↩ 3 00UI
Corollary 1 If x∈ 𝔥 and y∈ 𝔤^λ(𝔥), we have [x, y] =λ (x)y. ↩ 1 00UJ
Corollary 2 Every regular element of 𝔤 is semi-simple. 00UK
Corollary 3 Let 𝔥 be a Cartan subalgebra of a reductive Lie algebra 𝔤. ↩ 4 00UL

§ 3. Conjugacy theorems

Definition 1 A finite product of automorphisms of 𝔤 of the form eᵃᵈˣ with ad x nilpotent is called an elementary automorphism of 𝔤. The group of elementary automorphisms of 𝔤 is denoted by Autₑ… 00V0
Lemma 1 Let V be a finite dimensional vector space, 𝔫 a Lie subalgebra of 𝔞=𝔤𝔩(V) consisting of nilpotent elements. ↩ 6 00V1
Proposition 1 Let 𝔤 be a Lie algebra, 𝔫 a subalgebra of 𝔤 such that ad_𝔤x is nilpotent for all x∈ 𝔫. Then e^ad_𝔤𝔫 is a subgroup of Autₑ(𝔤). ↩ 1 00V2
Remark 1 Let V be a finite dimensional vector space, 𝔤 a Lie subalgebra of 𝔞=𝔤𝔩(V), x an element of 𝔤 such that ad_𝔤x is nilpotent. Then there exists a nilpotent element n of 𝔞 such that ad… ↩ 2 00V3
Remark 2 Let V be a finite dimensional vector space. For all g∈ SL(V), let φ (g) be the automorphism x → gxg⁻¹ of 𝔤𝔩(V). Then ↩ 3 00V4
Lemma 2 (i) Let 𝔥ᵣ be the set of x∈ 𝔥 such that 𝔤⁰(x) =𝔤⁰(𝔥); this is the set of x∈ 𝔥 such that λ (x)not= 0 for all λ ∈ R, and 𝔥ᵣ is open and dense in 𝔥 in the Zariski topology. ↩ 2 00V5
Proposition 2 Assume that k is algebraically closed. Let 𝔤 be a Lie algebra, 𝔥 and 𝔥' Cartan subalgebras of 𝔤. There exist u∈ E(𝔥) and u'∈ E(𝔥') such that u(𝔥) =u'(𝔥'). ↩ 1 00V6
Corollary E(𝔥) = E(𝔥'). 00V7
Theorem 1 Assume that k is algebraically closed. Let 𝔤 be a Lie algebra. The group E is normal in Aut(𝔤) and operates transitively on the set of Cartan subalgebras of 𝔤. ↩ 10 00V8
Theorem 2 Let 𝔤 be a Lie algebra. ↩ 7 00V9
Proposition 3 Let 𝔤 be a Lie algebra, 𝔤' a subalgebra of 𝔤. The following conditions are equivalent: ↩ 8 00VA
Corollary Let 𝔥 be a nilpotent subalgebra of 𝔤. The subalgebra 𝔤⁰(𝔥) has properties (i), (ii), (iii) in Prop. 3. ↩ 1 00VB
Proposition 4 Let 𝔤 be a Lie algebra, l the rank of 𝔤,c the nilpotency class of the Cartan subalgebras of 𝔤, and x∈ 𝔤. There exists an l-dimensional subalgebra of 𝔤 whose nilpotency class is ≤ c… ↩ 1 00VC
Theorem 3 Let 𝔤 be a solvable Lie algebra, and let 𝔥,𝔥' be Cartan subalgebras of 𝔤. There exists x∈ C^∞(𝔤) such that eᵃᵈˣ𝔥=𝔥'. ↩ 7 00VD
Lemma 3 Let 𝔤 be a Lie algebra, 𝔯 its radical, φ the canonical homomorphism from 𝔤 to 𝔤/𝔯,v an elementary automorphism of 𝔤/𝔯. There exists an elementary automorphism u of 𝔤 such that φ ∘… ↩ 1 00VE
Proposition 5 Let 𝔤 be a Lie algebra, 𝔯 its radical, 𝔥 and 𝔥' Cartan subalgebras of 𝔤, and φ the canonical homomorphism from 𝔤 to 𝔤/𝔯. The following conditions are equivalent: ↩ 1 00VF
Proposition 6 Assume that k is R,C or a non-discrete complete ultrametric field of characteristic 0. Let G be a finite dimensional Lie group over k,e its identity element, 𝔤 its Lie algebra, 𝔥 a… 00VG

§ 4. Regular elements of a Lie group

Lemma 1 Let M be an analytic manifold over k and a= (a₀, . . . , aₙ₋₁, aₙ= 1) a sequence of analytic functions on M. For all x∈ M, let rₐ(x) be the upper bound of those i∈ 0, n such that a… ↩ 4 00VU
Lemma 2 Let 0→ V'→ V→ V''→ 0 be an exact sequence of G-modules defined by analytic linear representations ρ ', ρ , ρ '' of G, respectively. Then: 00VV
Definition 1 An element g∈ G is called regular for the linear representation ρ if r_ρ(g) =r⁰_ρ(g). 00VW
Proposition 1 The regular points for an analytic linear representation ρ of G are the points of G in a neighbourhood of which r_ρ is constant. They constitute a dense open subset of G. If k=C an… ↩ 2 00VX
Remark Let G^* be an open subgroup of G. An element a∈ G^* is a regular element of G for the linear representation ρ of G if and only if it is a regular element of G^* for the linear repr… 00VY
Definition 2 An element of G is said to be regular if it is regular for the adjoint representation of G. ↩ 1 00VZ
Proposition 2 Let G' be a finite dimensional Lie group over k and f an open morphism from G to G'. The image under f of a regular element of G is a regular element of G'. If the kernel of f is c… ↩ 2 00W0
Proposition 3 Let G₁ and G₂ be two finite dimensional Lie groups over k. An element (g₁, g₂) of G₁× G₂ is regular if and only if g₁ and g₂ are regular elements of G₁ and G₂, respectively. 00W1
Lemma 3 Let a∈ G and let 𝔪 be a complement of 𝔤¹(a) in 𝔤. Let U be a neighbourhood of 0 in 𝔤 and exp an exponential map from U to G. The map ↩ 1 00W2
Proposition 4 Let a∈ G and H be a Lie subgroup germ of G with Lie algebra 𝔤¹(a). The map (b, c) → cabc⁻¹ from H× G to G is a submersion at (e, e). ↩ 2 00W3
Proposition 5 Let a∈ G and let W be a neighbourhood of e in G. There exists a neighbourhood V of a with the following property: for all a'∈ V, there exists an element g∈ W such that 𝔤¹(a')⊂ Ad(g… ↩ 1 00W4
Corollary Let G^* be an open subgroup of G. If a∈ G is regular, there exists a neighbourhood V of a such that, for all a'∈ V,𝔤¹(a') is conjugate to 𝔤¹(a) under Ad(G^*). ↩ 1 00W5
Proposition 6 Let V be an open subgroup of 𝔤 and let exp: V→ G be an exponential map defined on V. ↩ 2 00W6
Lemma 4 Let U be a neighbourhood of 0 in 𝔤 and exp an exponential map from U to G, étale at every point of U and such that 𝔤¹(exp x) =𝔤⁰(x) for all x∈ U. ↩ 2 00W7
Proposition 7 Let V be a neighbourhood of e in G. Every Cartan subalgebra of 𝔤 is of the form 𝔤¹(a) where a is a regular element of G belonging to V. 00W8
Proposition 8 Let l be the rank of 𝔤. There exists an open subgroup G^* of G such that: ↩ 1 00W9
Remark If k=C, the subalgebras 𝔤¹(a), for a regular and belonging to a connected component M of G, are conjugate under Int(𝔤). Indeed, let R be the set of regular elements of G. For all a… 00WA
Proposition 9 Let k be a field of characteristic 0 and 𝔤 a Lie algebra over k. If a∈ Autₑ(𝔤), the dimension of the nilspace of a-1 is greater than or equal to the rank of 𝔤. ↩ 2 00WB

§ 5. Decomposable linear Lie algebras

Definition 1 Let 𝔤 be a Lie subalgebra of 𝔤𝔩(V). Then 𝔤 is said to be decomposable if 𝔤 contains the semi-simple and nilpotent components of each of its elements (Algebra, Chap. VII, §5, no. 8)… ↩ 1 00WG
Example 1 Let V' and V'' be vector subspaces of V such that V''⊃ V'. The set of x∈ 𝔤𝔩(V) such that x(V'')⊂ V' is a decomposable Lie subalgebra of 𝔤𝔩(V); indeed, for all x∈ 𝔤𝔩(V), the semi-si… ↩ 1 00WH
Example 2 Assume that V has an algebra structure. The set of derivations of V is a decomposable Lie subalgebra of 𝔤𝔩(V) (§1, no. 1, Prop. 4 (ii)). 00WI
Example 3 More generally, it can be shown that the Lie algebra of any algebraic subgroup of GL(V) is decomposable.\ 01KQ
Proposition 1 Let 𝔤 be a decomposable Lie subalgebra of 𝔤𝔩(V),x∈ 𝔤,s and n the semi-simple and nilpotent components of x. 00WJ
Proposition 2 Let 𝔤 be a Lie subalgebra of 𝔤𝔩(V) reductive in 𝔤𝔩(V), E a finite dimensional vector space and π :𝔤→ 𝔤𝔩(E) a semi-simple linear representation of 𝔤 on E. Then: ↩ 5 00WK
Remark 1 The semi-simplicity assumption on π is equivalent to saying that π (x) is semi-simple for all x∈ 𝔠. Note that this assumption is satisfied when π is obtained from the identity repr… 00WL
Remark 2 Let 𝔤⊂ 𝔤𝔩(V), 𝔤'⊂ 𝔤𝔩(V') be decomposable Lie algebras, φ an isomorphism from 𝔤 to 𝔤'. Note that φ does not necessarily transform semi-simple (resp. nilpotent) elements of 𝔤 to semi… 00WM
Proposition 3 Let 𝔞 be a decomposable Lie subalgebra of 𝔤𝔩(V) and let 𝔟 and 𝔠 be vector subspaces of 𝔤𝔩(V) such that 𝔟⊂ 𝔠. Let 𝔞' be the set of x∈ 𝔞 such that [x,𝔠]⊂ 𝔟. Then 𝔞' is decomposable. ↩ 3 00WN
Corollary 1 If 𝔞 is a decomposable Lie subalgebra of 𝔤𝔩(V), and 𝔫 a Lie subalgebra of 𝔞, the normalizer (resp. centralizer) of 𝔫 in 𝔞 is decomposable. ↩ 2 00WO
Corollary 2 The Cartan subalgebras of a decomposable Lie subalgebra of 𝔤𝔩(V) are decomposable. ↩ 2 00WP
Remark We shall prove later (no. 5, Th. 2) a converse of Cor. 2. 00WQ
Proposition 4 Let 𝔤 be a Lie subalgebra of 𝔤𝔩(V) and 𝔫 an ideal of 𝔤. Then 𝔫 and e(𝔫) are ideals of e(𝔤), and [e(𝔤), e(𝔫)] = [𝔤,𝔫]. ↩ 6 00WR
Corollary 1 (i) Dⁱ𝔤=Dⁱe(𝔤) for i≥ 1, and Cⁱ𝔤=Cⁱe(𝔤) for i≥ 2. ↩ 2 00WS
Corollary 2 Let 𝔯 be the radical of 𝔤. If 𝔤 is decomposable, 𝔯 is decomposable. ↩ 3 00WT
Proposition 5 Let 𝔤 be a decomposable nilpotent Lie subalgebra of 𝔤𝔩(V). Let 𝔱 be the set of semi-simple elements of 𝔤. Then 𝔱 is a central subalgebra of 𝔤, and 𝔤 is the product of 𝔱 and 𝔫_V(𝔤)… ↩ 1 00WU
Proposition 6 Let 𝔤 be a decomposable Lie subalgebra of 𝔤𝔩(V). Let T be the set of commutative subalgebras of 𝔤 consisting of semi-simple elements, and T₁ the set of maximal elements of T. Let H… ↩ 1 00WV
Corollary 1 The Cartan subalgebras of 𝔤 are the centralizers of the regular semi-simple elements of 𝔤. 00WW
Corollary 2 Assume in addition that 𝔤 is solvable. Then: ↩ 2 00WX
Proposition 7 Let 𝔤 be a decomposable Lie subalgebra of 𝔤𝔩(V). ↩ 4 00WY
Lemma 1 Let 𝔫 be a Lie subalgebra of 𝔤𝔩(V) consisting of nilpotent endomorphisms, and N the subgroup exp𝔫 of GL(V) (§3, no. 1, Lemma 1). ↩ 4 00WZ
Lemma 2 ² (i) Let W be a vector subspace of V of dimension d, D the line bigwedgeᵈW⊂ bigwedgeᵈV,θ the canonical representation of 𝔤𝔩(V) on bigwedge V (Chap. III, App.). Let x∈ 𝔤𝔩(V). Then… ↩ 2 01JS
Proposition 8 Let 𝔫 be a Lie subalgebra of 𝔤𝔩(V) consisting of nilpotent elements, 𝔮 the normalizer of 𝔫 in 𝔤𝔩(V). There exists a finite dimensional vector space E, a representation ρ of 𝔤𝔩(V) o… ↩ 1 00X0
Theorem 1 Let 𝔤 be a Lie subalgebra of 𝔤𝔩(V) and let X be a subset of 𝔤 generating 𝔤 as a Lie algebra over k. If every element of X is either semi-simple or nilpotent, 𝔤 is decomposable. ↩ 1 00X1
Corollary 1 Every subalgebra of 𝔤𝔩(V) generated by its decomposable subalgebras is decomposable. ↩ 4 00X2
Corollary 2 Let 𝔤 be a Lie subalgebra of 𝔤𝔩(V). Then [𝔤,𝔤] is decomposable. 00X3
Corollary 3 Let 𝔤 be a Lie subalgebra of 𝔤𝔩(V), and let X be a subset of 𝔤 generating 𝔤(as a Lie algebra over k). ↩ 2 00X4
Corollary 4 Let 𝔤 be a decomposable Lie subalgebra of 𝔤𝔩(V). Let T be the set of commutative subalgebras of 𝔤 consisting of semi-simple elements (cf. Prop. 6). The maximal elements of T all ha… 00X5
Theorem 2 Let 𝔤 be a Lie subalgebra of 𝔤𝔩(V). The following conditions are equivalent: ↩ 1 00X6

App. 1. Polynomial maps and Zariski topology

Proposition 1 Let H be the set of algebra homomorphisms from A_V to k. For any x∈ V, let hₓ be the homomorphism f → f(x) from A_V to k. Then, the map x → hₓ is a bijection from V to H. 00XC
Corollary For any x∈ V, let 𝔪ₓ= Ker(hₓ). Then the map x → 𝔪ₓ is a bijection from V to the set of ideals 𝔪 of A_V such that A_V/𝔪=k. ↩ 1 00XD
Proposition 2 The vector space V, equipped with the Zariski topology, is an irreducible noetherian space. In particular, every non-empty open subset of V is dense. 00XE
Definition 1 A polynomial map f: V→ W is said to be dominant if the homomorphism from A_W to A_V associated to f is injective. 00XF
Proposition 3 Assume that k is algebraically closed. Let f: V→ W be a dominant polynomial map. The image under f of any dense open subset of V contains a dense open subset of W. 00XG
Proposition 4 Let f: V→ W be a polynomial map. Assume that there exists x₀∈ V such that (Df)(x₀) is surjective. Then f is dominant. ↩ 1 00XH
Corollary If k is algebraically closed and if f satisfies the assumptions of Prop. 4, the image under f of any dense open subset of V contains a dense open subset of W. 00XI

App. 2. A connectedness property

Lemma 1 Let X be a connected topological space and Ω a dense open subset of X. If, for any x∈ X, there exists a neighbourhood V of x such that V∩ Ω is connected, then Ω is connected. ↩ 1 00XN
Lemma 2 Let U be an open ball in Cⁿ and f: U→ C a holomorphic function, not identically zero. Let A be a subset of U such that f= 0 on A. Then U -- A is dense in U and connected. 00XO
Lemma 3 Let X be a finite dimensional connected complex-analytic manifold and let A be a subset of X satisfying the following condition: 00XP

VIII. Split semi-simple Lie algebras

§ 1. The Lie algebra sl(2, k) and its representations

Lemma 1 Let A be an associative algebra over k,H and X elements of A such that [H, X] = 2X. ↩ 5 00XS
Lemma 2 In the enveloping algebra of 𝔰𝔩(2, k), ↩ 1 00XT
Lemma 3 If x is an element of weight λ, then X₊x is an element of weight λ + 2 and X₋x is an element of weight λ -2. ↩ 2 00XU
Definition 1 Let E be an 𝔰𝔩(2, k)-module. An element of E is said to be primitive if it is a non-zero eigenvector of H_E and belongs to the kernel of X_+E. 00XV
Lemma 4 Let E be a non-zero finite dimensional 𝔰𝔩(2, k)-module. Then E has primitive elements. ↩ 2 00XW
Proposition 1 Let E be an 𝔰𝔩(2, k)-module, and e a primitive element of E of weight λ. Put eₙ=((-1)ⁿ)/nX₋ⁿe for n≥ 0, and e₋₁= 0. Then ↩ 4 00XX
Corollary The submodule of E generated by e is the vector subspace generated by the eₙ. ↩ 1 00XY
Proposition 2 If the submodule V of E generated by the primitive element e is finite dimensional, then: ↩ 6 00XZ
Corollary Let E be a finite dimensional 𝔰𝔩(2, k)-module. ↩ 9 00Y0
Proposition 3 For any integer m≥ 0, V(m) is an absolutely simple 𝔰𝔩(2, k)-module. In this module, e⁽ᵐ⁾₀=uᵐ is primitive of weight m. 00Y1
Theorem 1 Every simple 𝔰𝔩(2, k)-module of finite dimension n is isomorphic to V(n-1). Every finite dimensional 𝔰𝔩(2, k)-module is a direct sum of submodules isomorphic to the modules V(m). ↩ 4 00Y2
Remark 1 The adjoint representation of 𝔰𝔩(2, k) defines on 𝔰𝔩(2, k) the structure of a simple 𝔰𝔩(2, k)-module. This module is isomorphic to V(2) by an isomorphism that takes u² to X₊, 2uv t… 00Y3
Remark 2 For n≥ 0 and m > n, 00Y4
Remark 3 Let Φ be the bilinear form on V(m) such that ↩ 2 00Y5
Proposition 4 Let E be a finite dimensional 𝔰𝔩(2, k)-module, m an integer ≥ 0, Pₘ the set of primitive elements of weight m. Let L be the vector space of homomorphisms from the 𝔰𝔩(2, k)-module V… 00Y6
Corollary The isotypical component of E of type V(m) has length 00Y7
Definition 2 Let E be a finite dimensional vector space, and ρ (resp. π ) a linear representation of 𝔰𝔩(2, k) (resp. SL(2, k)) on E. Then ρ and π are said to be compatible if, for every nilpote… 00Y8
Theorem 2 Let ρ be a linear representation of 𝔰𝔩(2, k) on a finite dimensional vector space E. ↩ 3 00Y9
Remark When k is R or C or a complete non-discrete ultrametric field, 𝔰𝔩(2, k) is the Lie algebra of SL(2, k). Let ρ and π be as in Th. 2. The homomorphism π is a homomorphism of Lie grou… 00YA
Proposition 5 Let E, F be finite dimensional 𝔰𝔩(2, k)-modules, and let f∈ Homₖ(E,F). The following conditions are equivalent: 00YB
Definition 3 The adjoint representation of the group SL(2, k) is the linear representation Ad of SL(2, k) on 𝔰𝔩(2, k) defined by 00YC
Proposition 6 Let E be a finite dimensional 𝔰𝔩(2, k)-module, and t∈ k^*. Let Eₚ be the set of elements of E of weight p. ↩ 1 01L4
Corollary Let E = E'⊕ E'' be the decomposition of E defined in the Cor. of Prop. 2. The element (-10) of SL(2, k) operates by +1 on E' 00YD

§ 2. Root system of a split semi-simple Lie algebra

Definition 1 Let 𝔤 be a semi-simple Lie algebra. A Cartan subalgebra 𝔥 of 𝔤 is called splitting if, for all x∈ 𝔥, ad_𝔤x is triangularizable. A semi-simple Lie algebra is called splittable if it… 00YK
Remark 1 Let 𝔤 be a semi-simple Lie algebra, 𝔥 a Cartan subalgebra of 𝔤. For all x∈ 𝔥, ad_𝔤x is semi-simple (Chap. VII, §2, no. 4, Th. 2). Thus, to say that 𝔥 is splitting means that ad_𝔤x… 00YL
Remark 2 If k is algebraically closed, every semi-simple Lie algebra 𝔤 is splittable, and every Cartan subalgebra of 𝔤 is splitting. When k is not algebraically closed, there exist non-spli… ↩ 1 00YM
Remark 3 Let 𝔤 be a semi-simple Lie algebra, 𝔥 a Cartan subalgebra of 𝔤, and ρ a finite dimensional injective representation of 𝔤 such that ρ (𝔥) is diagonalizable. Then ad_𝔤x is diagonaliz… 00YN
Remark 4 We shall see (§3, no. 3, Cor. of Prop. 10) that if 𝔥,𝔥' are splitting Cartan subalgebras of 𝔤, there exists an elementary automorphism of 𝔤 transforming 𝔥 into 𝔥'. 00YO
Remark 5 Let 𝔤 be a reductive Lie algebra. Then 𝔤=𝔠× 𝔰 where 𝔠 is the centre of 𝔤 and 𝔰=D𝔤 is semi-simple. The Cartan subalgebras of 𝔤 are the subalgebras of the form 𝔥=𝔠× 𝔥' where 𝔥' is a… ↩ 1 00YP
Definition 2 A root of (𝔤,𝔥) is a non-zero weight of 𝔥 on 𝔤. 00YQ
Proposition 1 Let α ,β be roots of (𝔤,𝔥) and let ⟨ · ,· ⟩ be a non-degenerate invariant symmetric bilinear form on 𝔤(for example the Killing form of 𝔤). ↩ 5 00YR
Theorem 1 Let α be a root of (𝔤,𝔥). ↩ 3 00YS
Corollary Let Φ be the Killing form of 𝔤. For all a, b∈ 𝔥, ↩ 2 00YT
Proposition 2 Let α , β ∈ R. ↩ 1 00YU
Lemma 1 (i) For all h∈ 𝔥,θ_α(t).h=h-α (h)H_α. ↩ 3 00YV
Theorem 2 (i) The set R = R(𝔤,𝔥) is a reduced root system in 𝔥^*. ↩ 2 00YW
Corollary Every element of the Weyl group of (𝔤,𝔥), operating on 𝔥, is the restriction to 𝔥 of an elementary automorphism of 𝔤. ↩ 8 00YX
Remark 1 If 𝔥_Q (resp. 𝔥^*_Q) denotes the Q-vector subspace of 𝔥 (resp. 𝔥^*) generated by the H_α (resp. the α ), where α ∈ R, then 𝔥 (resp. 𝔥^*) can be identified canonically with 𝔥_Q⊗_Qk… 00YY
Remark 2 The root system R^∨ in 𝔥 defines a non-degenerate symmetric bilinear form β on 𝔥 (Chap. VI, §1, no. 1, Prop. 3), namely the form (a, b) → ∑_α∈ R⟨ α , a⟩ ⟨ α , b⟩. By the Cor. to Th… ↩ 1 00YZ
Proposition 3 Let 𝔤 be a semi-simple Lie algebra, 𝔥₁ and 𝔥₂ splitting Cartan subalgebras of 𝔤. There exists an isomorphism from 𝔥^*₁ to 𝔥^*₂ that takes R(𝔤,𝔥₁) to R(𝔤,𝔥₂). 00Z0
Proposition 4 Let α and β be linearly independent roots. Let p (resp. q) be the largest integer j such that β +jα (resp. β -jα ) is a root. ↩ 2 00Z1
Remark 3 Recall that, by Chap. VI, §1, no. 3, Cor. of Prop. 9, the integer p+q+ 1 can only take the values 1,2,3,4. 00Z2
Remark 4 Let (𝔤,𝔥) be a split reductive Lie algebra, 𝔠 the centre of 𝔤,𝔤'=D𝔤, 𝔥'=𝔥∩ 𝔤'. Then 𝔥=𝔠× 𝔥', and we identify 𝔥^'* with a vector subspace of 𝔥^*. For any λ ∈ 𝔥^* such that λ not= 0… ↩ 2 00Z3
Proposition 5 Let (𝔤,𝔥) be a split semi-simple Lie algebra, Φ an invariant symmetric bilinear form on 𝔤, and W the Weyl group of (𝔤,𝔥). Then the restriction Φ ' of Φ to 𝔥 is invariant under W. M… ↩ 2 00Z4
Proposition 6 Let (𝔤,𝔥) be a split semi-simple Lie algebra, Φ a non-degenerate invariant symmetric bilinear form on 𝔤. For all α ∈ R, let X_α be a non-zero element of 𝔤^α. Let (Hᵢ)_i∈ I be a bas… ↩ 2 00Z5
Lemma 2 There exists a family (X_α)_α∈ R such that, for all α ∈ R, ↩ 3 00Z6
Lemma 3 For all α ∈ R, ↩ 2 00Z7
Lemma 4 Let α , β ∈ R be such that α +β ∈ R. Let p (resp. q) be the largest integer j such that β +jα ∈ R (resp. β -jα ∈ R). Then, ↩ 3 00Z8
Definition 3 A Chevalley system for (𝔤,𝔥) is a family (X_α)_α∈ R such that ↩ 4 00Z9
Proposition 7 Let (X_α)_α∈ R be a Chevalley system for (𝔤,𝔥). We retain the notation of Lemma 4. Then, N_-α ,-β= N_α ,β and N_α ,β=± (q+ 1) for α , β , α +β ∈ R. ↩ 3 00ZA
Proposition 8 Let (X_α)_α∈ R be a Chevalley system for (𝔤,𝔥). Let M be a Z-submodule of 𝔥 containing the H_α and contained in the group of weights of R^∨. Let 𝔤_Z be the Z-submodule of 𝔤 generat… ↩ 1 00ZB

§ 3. Subalgebras of split semi-simple Lie algebras

Lemma 1 Let V be a vector subspace of 𝔤 and R(V) the set of α ∈ R such that 𝔤^α⊂ V. Then, (V∩ 𝔥) +∑_α∈ R(V)𝔤^α is the largest vector subspace of V stable under ad 𝔥. ↩ 1 00ZO
Lemma 2 Let 𝔥' be a vector subspace of 𝔥 and P a subset of R. Then 𝔥'+𝔤ᴾ is a subalgebra of 𝔤 if and only if P is a closed subset of R and ↩ 1 00ZP
Proposition 1 (i) The subalgebras of 𝔤 stable under ad𝔥 are the vector subspaces of the form 𝔥'+𝔤ᴾ, where P is a closed subset of R and 𝔥' is a vector subspace of 𝔥 containing 𝔥_P∩(-P). ↩ 3 00ZQ
Proposition 2 Let 𝔞 be a subalgebra of 𝔤 stable under ad𝔥, and let 𝔥'⊂ 𝔥, P⊂ R be such that 𝔞=𝔥'+𝔤ᴾ. ↩ 4 00ZR
Proposition 3 Let 𝔞 be a semi-simple subalgebra of 𝔤 stable under ad(𝔥) and let P be the subset of R such that 𝔞=𝔥_P+𝔤ᴾ. 00ZS
Proposition 4 Let 𝔞 be a subalgebra of 𝔤 containing 𝔥,x an element of 𝔞,s and n its semi-simple and nilpotent components. Then s∈ 𝔞 and n∈ 𝔞. 00ZT
Proposition 5 Let P be a closed subset of R. ↩ 1 00ZU
Proposition 6 Let R₁, . . . ,Rₚ be the irreducible components of R. For i= 1, . . . , p, put 𝔤ᵢ=𝔥_Rᵢ+𝔤^Rᵢ. Then 𝔤₁, . . . ,𝔤ₚ are the simple components of 𝔤. ↩ 1 00ZV
Corollary 1 𝔤 is simple if and only if R is irreducible (in other words, its Dynkin graph is connected). ↩ 3 01IZ
Corollary 2 A splittable simple Lie algebra is absolutely simple. 00ZW
Proposition 7 Let 𝔟=𝔥+𝔤ᴾ be a subalgebra of 𝔤 containing 𝔥. The following conditions are equivalent: ↩ 3 00ZX
Definition 1 A subalgebra of 𝔤 containing 𝔥 and satisfying the equivalent condition in Prop. 7 is called a Borel subalgebra of (𝔤,𝔥). ↩ 1 00ZY
Remark The map which associates R₊(C) to a chamber C of R is injective (Chap. VI, §1, no. 7, Cor. 1 of Prop. 20). Consequently, C → 𝔥+𝔤^R₊(C) is a bijection from the set of chambers of R… 00ZZ
Proposition 8 Let 𝔟 be a subalgebra of 𝔤,k' an extension of k. Then 𝔟⊗ₖk' is a Borel subalgebra of (𝔤⊗ₖk',𝔥⊗ₖk') if and only if 𝔟 is a Borel subalgebra of (𝔤,𝔥). ↩ 1 0100
Proposition 9 Let 𝔟 be the Borel subalgebra of (𝔤,𝔥) defined by a chamber C of R. Let 𝔫=𝔤^R₊(C)=∑_α∈ R,α >0𝔤^α. Let l= dim𝔥. ↩ 7 0101
Corollary Let 𝔟 be a Borel subalgebra of 𝔤. ↩ 2 0102
Proposition 10 Let 𝔟,𝔟' be Borel subalgebras of 𝔤. There exists a splitting Cartan subalgebra of 𝔤 contained in 𝔟∩ 𝔟'. ↩ 6 0103
Corollary The group Autₑ(𝔤) operates transitively on the set of pairs (𝔱,𝔟) where 𝔱 is a splitting Cartan subalgebra of 𝔤 and 𝔟 is a Borel subalgebra of (𝔤,𝔱). ↩ 4 0104
Proposition 11 Let 𝔭=𝔥+𝔤ᴾ be a subalgebra of 𝔤 containing 𝔥. The following conditions are equivalent: ↩ 1 0105
Definition 2 A subalgebra of 𝔤 containing 𝔥 and satisfying the equivalent conditions of Prop. 11 is called a parabolic subalgebra of (𝔤,𝔥). A parabolic subalgebra of 𝔤 is a parabolic subalgebra… ↩ 1 0106
Remark Let B be a basis of R, and 𝔟 the corresponding Borel subalgebra. If Σ ⊂ B, denote by Q_Σ the set of roots that are linear combinations of elements of Σ with coefficients ≤ 0; put 𝔭… 0107
Lemma 3 Let V be a finite dimensional real vector space, S a root system in V^*,P the set of parabolic subsets of S; let H be the set of Ker α for α ∈ S, and F the set of facets of V relat… ↩ 1 0108
Proposition 12 Let H be the set of hyperplanes of 𝔥_R consisting of the kernels of the roots in R. Let F be the set of facets of 𝔥_R relative to H. Let S be the set of parabolic subalgebras of (𝔤… 0109
Example The facets corresponding to the parabolic subalgebras of (𝔤,𝔥) containing a Borel algebra 𝔟 are the facets contained in the closure of the chamber associated to 𝔟 (cf. the Remark a… 010A
Proposition 13 Let 𝔭=𝔥+𝔤ᴾ be a parabolic subalgebra of (𝔤,𝔥), Q the set of α ∈ P such that -α ∉ P, and 𝔰=𝔥+𝔤^P∩(-P). Then 𝔭=𝔰⊕ 𝔤^Q,𝔰 is reductive in 𝔤, and 𝔤^Q is the largest nilpotent ideal of 𝔭… ↩ 2 010B
Proposition 14 Let k' be an extension of k and 𝔤'=𝔤⊗ₖk'. Let 𝔪 be a subalgebra of 𝔤 and 𝔪'=𝔪⊗ₖk'. If 𝔪' is a parabolic (resp. Borel ) subalgebra of 𝔤',𝔪 is a parabolic (resp. Borel ) subalgebra o… ↩ 2 010C
Definition 3 Let 𝔞 be a semi-simple (or more generally reductive) Lie algebra and overlinek an algebraic closure of k. A subalgebra 𝔪 of 𝔞 is said to be parabolic (resp. Borel ) if 𝔪⊗ₖoverlinek… 010D
Proposition 15 Let 𝔞 be a reductive Lie algebra, k' an extension of k, and 𝔪 a subalgebra of 𝔞. Then 𝔪 is a parabolic (resp. Borel ) subalgebra of 𝔞 if and only if 𝔪⊗ₖk' is a parabolic (resp. Bor… 010E

§ 4. Split semi-simple Lie algebra defined by a reduced root system

Proposition 1 Let (𝔤,𝔥) be a split semi-simple Lie algebra, R its root system, B a basis of R, and (n(α , β ))_α ,β∈ B the corresponding Cartan matrix. For all α ∈ B, let X_α∈ 𝔤^α, X_-α∈ 𝔤^-α. T… ↩ 3 010T
Definition 1 Let (𝔤,𝔥) be a split semi-simple Lie algebra, R its root system. A framing of (𝔤,𝔥) is a pair (B,(X_α)_α∈ B), where B is a basis of R, and where, for all α ∈ B,X_α is a non-zero el… 010U
Lemma 1 For all α , β ∈ B, we have ↩ 1 010V
Lemma 2 The endomorphisms X_α⁰,H_β⁰,X_-γ⁰, where α , β , γ ∈ B, are linearly independent. 010W
Lemma 3 The canonical images in 𝔞 of the elements x_α, h_β, x_-γ, where α , β , γ ∈ B, are linearly independent. 010X
Lemma 4 There exists a unique involutive automorphism θ of 𝔞 such that 010Y
Lemma 5 Let z∈ 𝔞. Then z∈ 𝔞^μ if and only if [h_α, z] =⟨ μ, α^∨⟩ z for all α ∈ B. ↩ 2 010Z
Remark It follows from Lemma 5 that every ideal of 𝔞 is homogeneous, since it is stable under the ad h_α. 0110
Proposition 2 (i) The Lie algebra 𝔞₊ is generated by the family (x_α)_α∈ B. ↩ 1 0111
Proposition 3 The Lie algebra 𝔞₊ (resp. 𝔞₋) is a free Lie algebra with basic family (x_α)_α∈ B (resp. (x_-α)_α∈ B) (cf. Chap. II, §2, no. 3). ↩ 1 0112
Lemma 6 Let α , β ∈ B with α not=β. Then ↩ 1 0113
Lemma 7 The ideal 𝔫 of 𝔞₊ generated by the x_α β(α , β ∈ B, α not=β ) is an ideal of 𝔞. The ideal of 𝔞₋ generated by the y_α β(α , β ∈ B, α not=β ) is an ideal of 𝔞 and is equal to θ (𝔫). 0114
Theorem 1 Let R be a reduced root system, B a basis of R. Let 𝔤 be the Lie algebra defined by the generating family ((X_α, H_α, X_-α))_α∈ B and the relations (16) to (22). Let 𝔥=∑_α∈ BkH_α… ↩ 3 0115
Lemma 8 Let α ∈ B∪ (-B). Then ad X_α is locally nilpotent.¹ ↩ 1 0116
Lemma 9 Let μ, ν ∈ Q and w∈ W(R) be such that wμ=ν. There exists an automorphism of 𝔤 that takes 𝔤^μ to 𝔤^ν. ↩ 1 0117
Lemma 10 Let μ∈ Q, and assume that μ is not a multiple of a root. There exists w∈ W(R) such that certain of the coordinates of wμ with respect to the basis B are >0 and certain of them are… ↩ 1 0118
Lemma 11 Let μ∈ Q. If μ ∉ R∪ 0, then 𝔤^μ= 0. If μ∈ R, then dim𝔤^μ= 1. 0119
Proposition 4 Let (𝔤,𝔥,B,(X_α)_α∈ B) be a framed semi-simple Lie algebra. Let (n(α , β ))_α ,β∈ B and (X_α)_α∈ B∪(-B) be the corresponding Cartan matrix and generating family. ↩ 3 011A
Corollary Every framed semi-simple Lie algebra is obtained from a framed semi-simple Q-Lie algebra by extension of scalars from Q to k. ↩ 1 011B
Theorem 2 Let (𝔤,𝔥,B,(X_α)_α∈ B) and (𝔤',𝔥',B',(X'_α)_α∈ B') be framed semi-simple Lie algebras, let R and R' be the root systems of (𝔤,𝔥) and (𝔤',𝔥'), let (n(α , β ))_α ,β∈ B (resp. (n'(α… ↩ 4 011C
Scholium To any splittable semi-simple Lie algebra 𝔤 is associated a Dynkin graph, which determines 𝔤 up to isomorphism (Th. 2 (iii)). This graph is non-empty and connected if and only if 𝔤… 011D
Proposition 5 Let (𝔤,𝔥,B,(X_α)_α∈ B) be a framed semi-simple Lie algebra, and (X_α)_α∈ B∪(-B) the corresponding generating family. There exists a unique automorphism θ of 𝔤 such that θ (X_α) =X_… ↩ 1 011E
Corollary Let (𝔤,𝔥) be a split semi-simple Lie algebra. Then (𝔤,𝔥) possesses a Chevalley system (§2, no. 4, Def. 3). ↩ 3 011F

§ 5. Automorphisms of a semi-simple Lie algebra

Proposition 1 Let (𝔤,𝔥,B,(X_α)_α∈ B) be a framed semi-simple Lie algebra, and R the root system of (𝔤,𝔥). Let G be the set of s∈ Aut(𝔤,𝔥) that leave B stable, and such that s(X_α) =X_ε(s)α for a… ↩ 4 011N
Proposition 2 The sequence of homomorphisms ↩ 7 011O
Corollary 1 Let (B,(X_α)_α∈ B) be a framing of (𝔤,𝔥). Let G be the set of s∈ Aut(𝔤,𝔥) that leave the framing invariant. Then Aut(𝔤,𝔥) is the semi-direct product of G and ε⁻¹(W(R)). ↩ 2 011P
Corollary 2 The group ε⁻¹(W(R)) operates simply-transitively on the set of framings of (𝔤,𝔥). ↩ 1 011Q
Corollary 3 Let B be a basis of R. The group Ker ε =f(T_Q) operates simply-transitively on the set of framings of (𝔤,𝔥) of the form (B,(X_α)_α∈ B). 011R
Lemma 1 Let α ∈ R and t∈ k^*. Let φ be the homomorphism λ → t^λ(H_α) from Q(R) to k^*. Then f(φ ) =θ_α(t)θ_α(-1). ↩ 1 011S
Proposition 3 The image of the composite homomorphism ↩ 3 011T
Lemma 2 Let 𝔥 be a splitting Cartan subalgebra of 𝔤, and s∈ Aut₀(𝔤,𝔥). Assume that the restriction of s to ∑_α∈ R𝔤^α does not have 1 as an eigenvalue. ↩ 1 011U
Lemma 3 (i) Let m= (P(R) : Q(R)). If φ is the mth power of an element of T_Q, then φ ∈ q(T_P). ↩ 1 011V
Proposition 4 We have f(T_Q)⊂Aut₀(𝔤,𝔥) and ε⁻¹(W(R)) = Aut₀(𝔤,𝔥). ↩ 6 011W
Corollary — Let B be a basis of R. The group Aut(mathfrak g,mathfrak h) is isomorphic to the semi-direct product of the groups Aut(R,B) and Aut₀(mathfrak g,mathfrak h). 01J0
Remark Let ε',ε'' be the restrictions of ε to Aut₀(mathfrak g,mathfrak h), Autₑ(mathfrak g,mathfrak h). Let f' be the homomorphism from T_P to Autₑ(mathfrak g,mathfrak h) induced by f via… 011X
Proposition 5 — Assume that mathfrak g is splittable. The group Aut₀(mathfrak g) operates simply-transitively on the set of framings of mathfrak g. ↩ 5 01J1
Corollary 1 — Let (mathfrak g,mathfrak h,B,(X_α)_α∈ B) be a framing of mathfrak g, and G the group (isomorphic to Aut(R,B)) of automorphisms of mathfrak g that leave this framing invariant. Th… ↩ 5 01J2
Corollary 2 Aut(𝔤) = Aut₀(𝔤) when 𝔤 is a splittable simple Lie algebra of type A₁,Bₙ(n≥ 2), Cₙ(n≥ 2), E₇,E₈,F₄,G₂. The quotient Aut(𝔤)/Aut₀(𝔤) is of order 2 when 𝔤 is of type Aₙ(n≥ 2), Dₙ(n≥ 5… ↩ 2 01J3
Remark 1 Let e₁= (𝔤,𝔥₁,B₁,(X_α¹)_α∈ B₁),e₂= (𝔤,𝔥₂,B₂,(X_α²)_α∈ B₂), e'₂= (𝔤,𝔥₂,B₂,(Y_α²)_α∈ B₂) be framings of 𝔤, and s (resp. s') an element of Aut₀(𝔤) that transforms e₁ to e₂ (resp. e'₂)… ↩ 1 011Z
Remark 2 Let X be the set of pairs (𝔥,B) where 𝔥 is a splitting Cartan subalgebra of 𝔤 and B a basis of the root system of (𝔤,𝔥). If x= (𝔥,B) and x'= (𝔥',B') are two elements of X, there ex… ↩ 6 0120
Proposition 6 Let 𝔥 be a splitting Cartan subalgebra of 𝔤. We have, with the notations in no. 1, Aut₀(𝔤) = Autₑ(𝔤).Ker ε = Autₑ(𝔤).f(T_Q). ↩ 2 0121
Remark 3 Prop. 6 shows that the canonical homomorphism ↩ 1 0122
Proposition 7 Let 𝔤 be a splittable semi-simple Lie algebra, 𝔟 a Borel subalgebra of 𝔤, and 𝔭₁ and 𝔭₂ distinct parabolic subalgebras of 𝔤 containing 𝔟. Then 𝔭₁ and 𝔭₂ are not conjugate under Aut… ↩ 1 0123
Remark 4 Let 𝔤 be a splittable semi-simple Lie algebra, P the set of parabolic subalgebras of 𝔤, a set on which Aut₀(𝔤) operates. Retain the notations of Remark 2. Let Σ be a subset of B(𝔤)… 0124
Proposition 8 Let V be the set of endomorphisms of the vector space 𝔤. Then Aut(𝔤) is closed in V for the Zariski topology (Chap. VII, App. I). 0125
Proposition 9 Let 𝔥 be a splitting Cartan subalgebra of 𝔤. ↩ 1 0126
Proposition 10 Assume that 𝔤 is splittable. 0127
Proposition 11 Assume that k is R,C or a non-discrete complete ultrametric field. Let 𝔥 be a splitting Cartan subalgebra of 𝔤. 0128

§ 6. Modules over a split semi-simple Lie algebra

Lemma 1 Let V be a 𝔤-module and v∈ V. The following conditions are equivalent: ↩ 1 012M
Definition 1 Let V be a 𝔤-module and v∈ V. Then v is said to be a primitive element of V if vnot= 0 and v satisfies the conditions of Lemma 1. 012N
Proposition 1 Let V be a 𝔤-module, v a primitive element of V and ω the weight of v. Assume that V is generated by v as a 𝔤-module. ↩ 13 012O
Definition 2 The homomorphism χ of Prop. 1 (iv) is called the central character of the 𝔤-module V. ↩ 1 012P
Proposition 2 Let V be a 𝔤-module generated by a primitive element e of weight ω, and X a semi-simple 𝔤-module. Let Φ be the set of homomorphisms from the 𝔤-module V to the 𝔤-module X. Then φ →… ↩ 1 012Q
Lemma 2 Let V be a simple 𝔤-module, ω a weight of V. The following conditions are equivalent: ↩ 3 012R
Proposition 3 Let V be a simple 𝔤-module with a highest weight ω. ↩ 3 012S
Corollary If V is finite dimensional, the canonical homomorphism U(𝔤)→ End(V) is surjective. 012T
Proposition 4 Let V be a simple 𝔤-module with a highest weight ω, X a semi-simple 𝔤-module, and X' the isotypical component of type V in X. Then X' is the submodule of X generated by X^ω_π. Its… ↩ 2 012U
Proposition 5 (i) The element e of Z(λ ) is primitive of weight λ and generates the 𝔤-module Z(λ ). ↩ 1 012V
Theorem 1 Let λ ∈ 𝔥^*. The 𝔤-module E(λ ) is simple and has highest weight λ. Every simple 𝔤-module of highest weight λ is isomorphic to E(λ ). ↩ 1 012W
Proposition 6 Let V be a 𝔤-module, λ an element of 𝔥^* and v a primitive element of V of weight λ. ↩ 3 012X
Corollary 1 Let λ ∈ 𝔥^* and α ∈ B be such that λ (H_α) + 1∈ N. Then Z(-α +s_αλ ) is isomorphic to a 𝔤-submodule of Z(λ ). ↩ 2 012Y
Corollary 2 Let ρ =1/2∑_α∈ R₊α, and λ , μ∈ 𝔥^*. Assume that λ +ρ is a dominant weight in R, and that there exists w∈ W with μ+ρ =w(λ +ρ ). Then Z(μ) is isomorphic to a submodule of Z(λ ). ↩ 1 012Z
Lemma 3 Put L = (𝔫₋U)∩ U⁰. 0130
Proposition 7 Let λ ∈ 𝔥^*, E a 𝔤-module generated by a primitive element of weight λ ,χ the central character of E, and φ the Harish-Chandra homomorphism from U⁰ to V. Then, χ (z) = (φ (z))(λ )… ↩ 3 0131
Corollary Let ⟨ · ,· ⟩ be a non-degenerate invariant symmetric bilinear form on 𝔤, C the Casimir element associated to ⟨ · ,· ⟩. Denote also by ⟨ · ,· ⟩ the inverse form on 𝔥^* of the restri… ↩ 3 0132

§ 7. Finite dimensional modules over a split semi-simple Lie algebra

Proposition 1 Let V be a finite dimensional 𝔤-module. ↩ 2 0138
Corollary Let ρ be a finite dimensional representation of 𝔤 and Φ the bilinear form associated to ρ. ↩ 1 0139
Lemma 1 Let V be a 𝔤-module and ρ the corresponding representation of 𝔤. ↩ 3 013A
Proposition 2 Let V be a finite dimensional 𝔤-module and s∈ Aut₀(𝔤). ↩ 8 013B
Remark 1 Let S satisfy the condition in Prop. 2 (i), and let 𝔥'=s(𝔥); denote by s^* the isomorphism λ → λ ∘ s⁻¹ from 𝔥^* to 𝔥^'*. It is clear that 013C
Corollary 1 The isomorphism s^* takes the weights of V with respect to 𝔥 to those of V with respect to 𝔥'; corresponding weights have the same multiplicity. ↩ 1 013D
Corollary 2 Let w∈ W. For all λ ∈ 𝔥^*, the vector subspaces V^λ and V^wλ have the same dimension. The set of weights of V is stable under W. ↩ 3 013E
Remark 2 By Cor. 1 of Prop. 2 and §5, no. 3, Remark 2, it makes sense to speak of the weights of V with respect to the canonical Cartan subalgebra of 𝔤, and of their multipicities. 01L8
Remark 3 Lemma 1 (i) and Prop. 2 remain valid, with the same proof, even if 𝔤 is not assumed to be splittable. 013F
Theorem 1 A simple 𝔤-module is finite dimensional if and only if it has a highest weight belonging to P₊₊. ↩ 1 013G
Corollary 1 If λ ∈ 𝔥^* and λ ∉ P₊₊, the 𝔤-module E(λ ) (§6, no. 3) is infinite dimensional. 013H
Corollary 2 The 𝔤-modules E(λ ) for λ ∈ P₊₊ constitute a set of representatives of the classes of finite dimensional simple 𝔤-modules. ↩ 2 013I
Remark 1 Let λ ∈ P₊₊,ρ_λ the representation of 𝔤 on E(λ ),s∈ Aut(𝔤), and σ the canonical image of s in Aut(R,B) (§5, no. 3, Cor. 1 of Prop. 5). Then ρ_λ∘ s is equivalent to ρ_σ λ; indeed, i… ↩ 1 013J
Proposition 3 Let V be a finite dimensional 𝔤-module and X its set of weights. Let λ ∈ X,α ∈ R, I the set of t∈ Z such that λ +tα ∈ X,p (resp. -q) the largest (resp. smallest) element of I. Let… ↩ 4 013K
Corollary 1 If λ ∈ X and λ (H_α)≥ 1, then λ -α ∈ X. If λ +α ∈ X and λ (H_α) = 0, then λ ∈ X and λ -α ∈ X. 013L
Corollary 2 Let μ∈ P₊₊ and ν ∈ Q₊. If μ+ν ∈ X, then μ∈ X. ↩ 2 01J4
Corollary 3 Let v∈ V be primitive of weight ω. Let Σ be the set of α ∈ B such that ω (H_α) = 0. The stabilizer in 𝔤 of the line kv is the parabolic subalgebra 𝔭_Σ associated to Σ (§3, no. 4, R… 013M
Proposition 4 Let V be a finite dimensional 𝔤-module and d an integer ≥ 1. The set of weights of V of multiplicity ≥ d is R-saturated. ↩ 1 013N
Proposition 5 Let V be a finite dimensional simple 𝔤-module, ω its highest weight, X its set of weights. Choose a W-invariant non-degenerate positive symmetric bilinear form (· |· ) on 𝔥^*_R, an… ↩ 7 013O
Corollary Let X be a finite R-saturated subset of P. There exists a finite dimensional 𝔤-module whose set of weights is X. 013P
Remark 2 Recall (Chap. VI, §1, no. 6, Cor. 3 of Prop. 17) that there exists a unique element w₀ of W that transforms B into -B; we have w₀²= 1. and -w₀ respects the order relation on P. Wit… ↩ 1 013Q
Proposition 6 Let λ ∈ P, and X the smallest R-saturated subset of P containing λ. Choose a norm | · | as in Prop. 5. The following conditions are equivalent: ↩ 4 013R
Proposition 7 Let V be a finite dimensional simple 𝔤-module, X the set of weights of V, and λ the highest element of X (cf. Prop. 5 (i)). Conditions (i), (ii) and (iii) of Prop. 6 are equivalent… ↩ 1 013S
Proposition 8 Assume that 𝔤 is simple. Denote by α₁, . . . , αₗ the elements of B. Let varpi₁, . . . , varpiₗ be the corresponding fundamental weights. Let H = n₁H_α₁+⋯ +nₗH_αₗ be the highest ro… ↩ 1 013T
Definition 1 Assume that 𝔤 is simple. A minuscule weight of (𝔤,𝔥) is an element of P₊₊ -- 0 which satisfies the equivalent conditions (i), (ii), (iii), (iv), (v) and (vi) of Prop. 6, 7 and 8. 013U
Remark Assume that 𝔤 is simple. Let Σ^'∨ be the Coxeter graph of the affine Weyl group Wₐ(R^∨). Recall that the vertices of Σ^'∨ are the vertices of the Coxeter graph Σ^∨ of W(R^∨), toget… 013V
Proposition 9 Let E,F be finite dimensional simple 𝔤-modules, with highest weights λ , μ, respectively. ↩ 3 013W
Remark Retain the notations of Prop. 9. Let C be the isotypical component of E⊗ F of highest weight λ +μ. Then C depends only on E and F and not on the choice of 𝔥 and the basis B. In oth… 013X
Corollary 1 Let (varpi_α)_α∈ B be the family of fundamental weights relative to B. Let λ =∑_α∈ Bm_αvarpi_α∈ P₊₊. For all α ∈ B, let E_α be a simple 𝔤-module of highest weight varpi_α. In the 𝔤… ↩ 1 01J5
Corollary 2 Assume that k is R or C or a non-discrete complete ultrametric field. Let G be a Lie group with Lie algebra 𝔤. Assume that, for any fundamental representation ρ of 𝔤, there exists… 013Y
Proposition 10 Let λ , μ∈ P₊₊. Let E,F,G be simple 𝔤-modules with highest weights λ , μ, λ +μ. Let X (resp. X',X'') be the set of weights of E (resp. F,G). Then X''=X+X'. ↩ 3 013Z
Proposition 11 Let E be a finite dimensional 𝔤-module, and consider the 𝔤-module E^*= Homₖ(E, k). ↩ 3 0140
Remark 1 Let E,E^* be as in Prop. 11, and σ ∈ Aut(𝔤,𝔥) be such that ε (σ ) =-w₀ in the notations of §5, no. 1 (§5, no. 2, Prop. 2). Let ρ , ρ ' be the representations of 𝔤 associated to E,E… 0141
Remark 2 Assume that w₀=-1. Then, for any finite dimensional 𝔤-module E, E is isomorphic to E^*. Recall that, if 𝔤 is simple, w₀=-1 in the following cases: 𝔤 of type A₁,Bₗ(l≥ 2), Cₗ(l≥ 2)… 0142
Lemma 2 Let h⁰=∑_α∈ R₊H_α. Then h⁰=∑_α∈ Ba_αH_α, where the a_α are integers ≥ 1. Let (b_α)_α∈ B,(c_α)_α∈ B be families of scalars such that b_αc_α=a_α for all α ∈ B. Put x=∑_α∈ Bb_αX_α, y=… ↩ 2 0143
Proposition 12 Let E be a finite dimensional simple 𝔤-module, ω its highest weight, and B the vector space of 𝔤-invariant bilinear forms on E. Let m be the integer ∑_α∈ R₊ω (H_α), so that m/2 is… ↩ 4 0144
Definition 2 A finite dimensional irreducible representation ρ of mathfrak g is said to be orthogonal (resp. symplectic) if there exists on E a non-degenerate symmetric (resp. alternating) bili… 01J6
Lemma 3 Let G be an abelian group written additively, and φ :F_𝔞→ G a map; by abuse of notation, we denote by φ (F) the image under φ of the class of any finite dimensional 𝔞-module F. Ass… ↩ 2 0145
Example Take G =Z and φ (F) = dim F. The corresponding homomorphism from R(𝔞) to Z is denoted by dim. Let c be the class of a trivial 𝔞-module of dimension 1, and let ψ be the homomorphism… 0146
Lemma 4 There exists on the additive group R(𝔞) a unique multiplication distributive over addition such that [E][F] = [E⊗ F] for all finite dimensional 𝔞-modules E, F. In this way R(𝔞) is… 0147
Lemma 5 There exists a unique involutive automorphism X → X^* of the ring R(𝔞) such that [E]^*= [E^*] for every finite dimensional 𝔞-module E. 0148
Example Assume that Δ=N. Let T be an indeterminate. There exists a unique isomorphism from the algebra Z[N] to the algebra Z[T] that takes eⁿ to Tⁿ for all n∈N. For any finite dimensional… 01JX
Remark Every element of P defines a simple 𝔥-module of dimension 1, hence a homomorphism from the group Z[P] to the group R(𝔥), which is an injective homomorphism of rings. It is immediat… 0149
Lemma 6 If λ ∈ P₊₊, then ch[λ ]∈ Z[P]ᵂ. The unique maximal term of ch[λ ] (Chap. VI, §3, no. 2, Def. 1) is e^λ. ↩ 1 014A
Theorem 2 (i) Let (varpi_α)_α∈ B be the family of fundamental weights relative to B. Let (T_α)_α∈ B be a family of indeterminates. The map f → f(([varpi_α])_α∈ B) from Z[(T_α)_α∈ B] to R(𝔤)… ↩ 3 014B
Corollary Let E,E' be finite dimensional 𝔤-modules. Then E is isomorphic to E' if and only if ch E = ch E'. 014C

§ 8. Symmetric invariants

Lemma 1 Let Π be a subgroup of V^* that generates the vector space V^*, and m an integer ≥ 0. Then prₘ(exp Π ) generates the vector space Sᵐ(V^*). ↩ 1 0154
Proposition 1 Let Sᵐ(𝔥^*)ᵂ be the set of elements of Sᵐ(𝔥^*) invariant under W. Then prₘ(ψ (k[P]ᵂ)) =Sᵐ(𝔥^*)ᵂ. ↩ 3 0155
Proposition 2 Let E be a finite dimensional 𝔤-module. Let U(𝔥) =S(𝔥) be the enveloping algebra of 𝔥. If u∈ U(𝔥), then ↩ 1 0156
Corollary 1 — Let U(mathfrak g) be the enveloping algebra of mathfrak g. Let the homomorphism ζ:U(mathfrak g)^*longrightarrowU(mathfrak h)^*=S(mathfrak h)^* be the transpose of the canonical i… 01J7
Corollary 2 — Let m be an integer geqslant 0. Every element of Sᵐ(mathfrak h^*)ᵂ is a linear combination of polynomial functions on mathfrak h of the form x↦Tr(ρ(x)ᵐ), where ρ is a finite dime… ↩ 2 01J8
Lemma 2 Let ρ be a finite dimensional linear representation of 𝔞, and m an integer ≥ 0. The function x → Tr(ρ (x)ᵐ) on 𝔞 is an invariant polynomial function. ↩ 1 0157
Lemma 3 Let E be a finite dimensional 𝔤-module, and x∈ E. Then x is an invariant element of the 𝔤-module E if and only if (exp a_E).x=x for every nilpotent element a of 𝔤. ↩ 1 0158
Proposition 3 Let f be a polynomial function on 𝔤. The following conditions are equivalent: ↩ 1 0159
Theorem 1 Let I(𝔤^*) be the algebra of invariant polynomial functions on 𝔤. Let i:S(𝔤^*)→ S(𝔥^*) be the restriction homomorphism. ↩ 10 015A
Corollary 1 Assume that 𝔤 is simple. Let m₁, . . . , mₗ be the exponents of the Weyl group of 𝔤. There exist elements P₁, . . . ,Pₗ of I(𝔤^*), homogeneous of degrees ↩ 1 015B
Corollary 2 Let B be a basis of R, R₊ (resp. R₋) the set of positive (resp. negative) roots of (𝔤,𝔥) relative to B,𝔫₊=∑_α∈ R₊𝔤^α,𝔫₋=∑_α∈ R₋𝔤^α, S(𝔥) the symmetric algebra of 𝔥, and J the ideal… ↩ 2 015C
Proposition 4 Let 𝔞 be a semi-simple Lie algebra, l its rank. Let I (resp. I') be the set of elements of S(𝔞^*) (resp. S(𝔞)) invariant under the representation induced by the adjoint representat… ↩ 2 015D
Lemma 4 ² Let A =⨁_n≥ 0Aⁿ be a graded k-algebra, k' an extension of k, and A'= A⊗ₖk'. Assume that A' is a graded polynomial algebra over k'. Then A is a graded polynomial algebra over k. ↩ 1 01JY
Lemma 5 A is a graded polynomial algebra if and only if P is finite dimensional and g is bijective. 015E
Proposition 5 We retain the notations of Prop. 4, and denote by 𝔭 the ideal of S(𝔞^*) generated by the homogeneous elements of I of degree ≥ 1. Let x∈ 𝔞. Then x is nilpotent if and only if f(x)… ↩ 1 015F
Remark 1 \Let P₁, . . . ,Pₗ be algebraically independent homogeneous elements of I that generate the algebra I. Then (P₁, . . . ,Pₗ) is an S(𝔞^*)-regular sequence (Chap. V, §5, no. 5). Inde… ↩ 1 01L5
Remark 2 The algebra S(𝔞^*) is a graded free module over I. Indeed, this follows from Prop. 4, Remark 1, and Chap. V, §5, no. 5, Lemma 5.\ ↩ 2 01L6
Lemma 6 Let V be a finite dimensional vector space, G a finite group of automorphisms of V, and v and v' elements of V such that v'∉ Gv. There exists a G-invariant polynomial function f on… ↩ 2 015G
Proposition 6 Let 𝔞 be a semi-simple Lie algebra and s∈ Aut(𝔞). The following conditions are equivalent: ↩ 4 015H
Corollary The group Aut₀(𝔞) is open and closed in Aut(𝔞) in the Zariski topology. ↩ 1 015I
Theorem 2 Let U be the enveloping algebra of 𝔤, Z its centre, V⊂ U the enveloping algebra of 𝔥(identified with S(𝔥)), U⁰ the commutant of V in U, φ the Harish-Chandra homomorphism (§6, no. 4… ↩ 3 015J
Corollary 1 For all λ ∈ 𝔥^*, let χ_λ be the homomorphism z → (φ (z))(λ ) from Z to k. ↩ 3 015K
Corollary 2 Let E,E' be finite dimensional simple 𝔤-modules, and χ , χ ' their central characters. If χ =χ ', E and E' are isomorphic. 015L
Proposition 7 For any class γ of finite dimensional simple 𝔤-modules, let U_γ be the isotypical component of type γ of the 𝔤-module U (for the adjoint representation of 𝔤 on U). Let γ₀ be the cl… 015M

§ 9. The formula of Hermann Weyl

Lemma 1 In the ring Z⟨ P⟩, we have K.∏_α∈ R₊(1-e^-α) = Ke^-ρd= 1. ↩ 2 015T
Lemma 2 Let λ ∈ 𝔥^*. The module Z(λ ) (§6, no. 3) admits a character that is an element of Z⟨ P⟩, and we have d. ch Z(λ ) =e^λ+ρ. 015U
Lemma 3 Let M be a 𝔤-module which admits a character ch(M) whose support is contained in a finite union of the sets μ-P₊. Let U be the enveloping algebra of 𝔤, Z the centre of U,λ₀∈ 𝔥^*, a… ↩ 1 015V
Theorem 1 (Character Formula of H. Weyl) Let M be a finite dimensional simple 𝔤-module, and λ its highest weight. Then ↩ 5 015W
Example Take 𝔤=𝔰𝔩(2, k),𝔥=kH. Let α be the root of (𝔤,𝔥) such that α (H) = 2. The 𝔤-module V(m) has highest weight (m/2)α. Hence 015X
Theorem 2 Let E be a finite dimensional simple 𝔤-module, λ its highest weight and (· | · ) a W-invariant non-degenerate positive symmetric bilinear form on 𝔥^*. Then: ↩ 5 015Y
Example 1 In the Example of no. 1, we find that 015Z
Example 2 Take 𝔤 to be the splittable simple Lie algebra of type G₂ and adopt the notations of Chap. VI, Plate IX. Give 𝔥^*_R the W-invariant positive symmetric form (· | · ) such that (α₁|α… 0160
Proposition 1 Let ω ∈ P₊₊. For all λ ∈ P, the multiplicity of λ in E(ω ) is ↩ 2 0161
Corollary If λ is a weight of E(ω ) distinct from ω, 0162
Proposition 2 Let λ , μ∈ P₊₊. In R(𝔤), we have ↩ 3 0163
Example We return to the Example of no. 1. Let λ = (n/2)α , μ= (p/2)α, ν = (q/2)α with n≥ p. We have 0164

§ 10. Maximal subalgebras of semi-simple Lie algebras

Theorem 1 Let V be a finite dimensional vector space, 𝔤 a reductive Lie subalgebra in 𝔤𝔩(V),𝔮 a Lie subalgebra of 𝔤 and Φ the bilinear form (x, y) → Tr(xy) on 𝔤× 𝔤. Assume that the orthogona… ↩ 3 016L
Lemma 1 Let 𝔤 be a semi-simple Lie algebra, V a finite dimensional vector space, ρ a linear representation of 𝔤 on V, D a vector subspace of V,𝔥 a Cartan subalgebra of 𝔤,𝔰 (resp. 𝔰') the s… ↩ 1 016M
Definition 1 Let 𝔮 be a Lie subalgebra of the semi-simple Lie algebra 𝔤. Then 𝔮 is said to be decomposable in 𝔤 if, for all x∈ 𝔮, the semi-simple and nilpotent components of x in 𝔤 belong to 𝔮… 016N
Theorem 2 Let 𝔤 be a semi-simple Lie algebra, 𝔫 a subalgebra of 𝔤 consisting of nilpotent elements, 𝔮 the normalizer of 𝔫 in 𝔤. Assume that 𝔫 is the set of nilpotent elements of the radical… ↩ 1 016O
Corollary 1 Let 𝔮 be a maximal element of the set of subalgebras of 𝔤 distinct from 𝔤. Then 𝔮 is either parabolic or reductive in 𝔤. 016P
Corollary 2 Let 𝔫 be a subalgebra of 𝔤 consisting of nilpotent elements. There exists a parabolic subalgebra 𝔮 of 𝔤 with the following properties: ↩ 5 016Q
Theorem 3 Assume that k is algebraically closed. Let 𝔤 be a semi-simple Lie algebra. Let 𝔞 be a solvable subalgebra of 𝔤. There exists a Borel subalgebra of 𝔤 containing 𝔞. 016R
Corollary If k is algebraically closed, every maximal solvable subalgebra of 𝔤 is a Borel subalgebra. 016S

§ 11. Classes of nilpotent elements and sl2 -triplets

Definition 1 An 𝔰𝔩₂-triplet in 𝔤 is a sequence (x, h, y) of elements of 𝔤, distinct from (0,0,0), such that 0170
Lemma 1 Let x, h, y, y'∈ 𝔤. If (x, h, y) and (x, h, y') are 𝔰𝔩₂-triplets in 𝔤, then y=y'. ↩ 3 0171
Lemma 2 Let 𝔫 be a subalgebra of 𝔤 such that, for all n∈ 𝔫, ad_𝔤(n) is nilpotent. Let h∈ 𝔤 be such that [h,𝔫] =𝔫. Then e^ad_𝔤𝔫.h=h+𝔫. 0172
Lemma 3 Let x∈ 𝔤,𝔭= Ker(ad x),𝔮= Im(ad x). Then [𝔭,𝔮]⊂ 𝔮, and 𝔭∩ 𝔮 is a subalgebra of 𝔤. 0173
Lemma 4 Let (x, h, y) and (x, h', y') be 𝔰𝔩₂-triplets in 𝔤. There exists z∈ 𝔤 such that ad_𝔤z is nilpotent and such that ↩ 1 0174
Proposition 1 Let G be a group of automorphisms of 𝔤 containing Autₑ(𝔤). Let (x, h, y) and (x', h', y') be 𝔰𝔩₂-triplets in 𝔤. Let ↩ 3 0175
Remark The three conditions of Prop. 1 are equivalent if we assume only that k=k² (cf. Exerc. 1). 0176
Lemma 5 Let V be a finite dimensional vector space, A and B endomorphisms of V. Assume that A is nilpotent and that [A,[A,B]] = 0. Then AB is nilpotent. ↩ 2 0177
Lemma 6 Let h, x∈ 𝔤 be such that [h, x] = 2x and h∈ (ad x)(𝔤). Then there exists y∈ 𝔤 such that (x, h, y) is either (0,0,0) or an 𝔰𝔩₂-triplet. ↩ 2 0178
Proposition 2 (Jacobson-Morozov) Assume that 𝔤 is semi-simple. Let x be a non-zero nilpotent element of 𝔤. There exist h, y∈ 𝔤 such that (x, h, y) is an 𝔰𝔩₂-triplet. ↩ 3 0179
Corollary Assume that 𝔤 is semi-simple. Let G be a group of automorphisms of 𝔤 containing Autₑ(𝔤). The map which associates to any 𝔰𝔩₂-triplet (x, h, y) in 𝔤 the nilpotent element x defines… ↩ 1 017A
Lemma 7 Let K be a commutative field with at least 4 elements. Let G be the group of matrices (α 0α β₋₁) where α ∈ K^*,β ∈ K. Let G' be the group ↩ 1 017B
Proposition 3 Assume that 𝔤 is semi-simple. The group Autₑ(𝔤) is equal to its derived group. If 𝔤 is splittable, Autₑ(𝔤) is the derived group of Aut₀(𝔤). ↩ 5 017C
Definition 2 An element h of 𝔤 is said to be simple if there exist x, y∈ 𝔤 such that (x, h, y) is an 𝔰𝔩₂-triplet in 𝔤. 017D
Proposition 4 Let h be a non-zero element of 𝔤. Then h is simple if and only if there exists x∈ 𝔤 such that [h, x] = 2x and h∈ (ad x)(𝔤). 017E
Proposition 5 Assume that 𝔤 is splittable semi-simple. Let 𝔥 be a splitting Cartan subalgebra of 𝔤, R the set of roots of (𝔤,𝔥), and B a basis of R. Let h be a simple element of 𝔤 belonging to 𝔥… ↩ 3 017F
Corollary If k is algebraically closed and if 𝔤 is semi-simple of rank l, the number of conjugacy classes of simple elements of 𝔤, relative to Autₑ(𝔤), is at most 3ˡ. ↩ 2 017G
Lemma 8 Assume that k is algebraically closed and that 𝔤 is semi-simple. Let h be a semi-simple element of 𝔤 such that the eigenvalues of ad h are rational. Let 𝔤⁰= Ker(ad h),𝔤²= Ker(ad h-… ↩ 1 017H
Proposition 6 Assume that k is algebraically closed and that 𝔤 is semi-simple. Let G be a group of automorphisms of 𝔤 containing Autₑ(𝔤). Let (x, h, y) and (x', h', y') be 𝔰𝔩₂-triplets in 𝔤. The… ↩ 1 017I
Corollary 1 The map which associates to any 𝔰𝔩₂-triplet its simple element defines by passage to the quotients a bijection from the set of G-conjugacy classes of 𝔰𝔩₂-triplets to the set of G-c… ↩ 2 017J
Corollary 2 If rk(𝔤) =l, the number of conjugacy classes, relative to Autₑ(𝔤), of non-zero nilpotent elements of 𝔤 is at most 3ˡ. 017K
Corollary 3 If rk(𝔤) =l, the number of conjugacy classes, relative to Autₑ(𝔤), of subalgebras of 𝔤 isomorphic to 𝔰𝔩(2, k) is at most 3ˡ. 017L
Definition 3 Assume that 𝔤 is semi-simple. 017M
Proposition 7 Assume that 𝔤 is semi-simple. Let (x, h, y) be an 𝔰𝔩₂-triplet in 𝔤. The following conditions are equivalent: 017N
Proposition 8 Assume that 𝔤 is semi-simple not= 0. Let 𝔥 be a splitting Cartan subalgebra of 𝔤, R the root system of (𝔤,𝔥), B a basis of R,h⁰ the element of 𝔥 such that α (h⁰) = 2 for all α ∈ B. ↩ 3 017O
Corollary In a splittable semi-simple Lie algebra, there exist principal nilpotent elements. 017P
Proposition 9 Assume that k is algebraically closed and that 𝔤 is semi-simple. All the principal simple (resp. nilpotent) elements of 𝔤 are conjugate under Autₑ(𝔤). ↩ 1 017Q
Lemma 9 With the notations of Prop. 8, put 𝔤ᵖ= Ker(ad h⁰-p) for p∈ Z. Let 𝔤²_* be the set of elements of 𝔤²=∑_α∈ B𝔤^α that have a non-zero component ↩ 1 017R
Proposition 10 Assume that 𝔤 is semi-simple. Let 𝔥 be a splitting Cartan subalgebra of 𝔤, R the root system of (𝔤,𝔥), B a basis of R, R₊ the set of positive roots relative to B, and 𝔫₊=∑_α∈ R₊𝔤^α… ↩ 3 017S

§ 12. Chevalley orders

Proposition 1 Let A be a bigebra, with coproduct c, and x a primitive element (Chap. II, §1, no. 2) of A. Then ↩ 2 017V
Theorem 1 Let U be a biorder in the bigebra U(𝔤). Let G=U∩ 𝔤, which is an order in the Lie algebra 𝔤. Let (xᵢ)_i∈ I be a basis of G. Give I a total order, and assume that we are given, for a… ↩ 3 017W
Lemma 1 Let n∈ Nᴵ. If |n|< r, then c⁺ᵣ([n]) = 0. If |n|=r, then 017X
Proposition 2 (i) (^V_Z) is a biorder in the bigebra S(V), and (^V_Z)∩ V =V. ↩ 2 017Y
Corollary Let X be an indeterminate. The polynomials (^Xₙ), where n∈ N, form a basis of the Z-module consisting of the polynomials P∈ k[X] such that P(Z)⊂ Z. ↩ 2 017Z
Lemma 2 If x, y∈ A and n∈ N, ↩ 3 0180
Lemma 3 Let x, h∈ A and λ ∈ k be such that (ad h)x=λ x. For all n∈ N, and all P∈ k[X], we have ↩ 2 0181
Lemma 4 Let x, y, h∈ A be such that ↩ 2 0182
Definition 1 A lattice H in 𝔥 is said to be permissible (relative to 𝔤) if, for all α ∈ R, we have H_α∈ H and α (H)⊂ Z. ↩ 3 0183
Remark 1 Let B be a basis of R. A lattice H in 𝔥 is permissible if and only if H_α∈ H and α (H)⊂ Z for all α ∈ B. 0184
Remark 2 Let 𝔠 be the centre of 𝔤. Then, a lattice H in 𝔥 is permissible if and only if Q(R^∨)⊂ H⊂ P(R^∨)⊕ 𝔠. The lattice H∩ D𝔤 is then permissible in the Cartan subalgebra 𝔥∩ D𝔤 of D𝔤. The… 0185
Remark 3 If 𝔤 is semi-simple, the permissible lattices in 𝔥 are the subgroups H of 𝔥 such that Q(R^∨)⊂ H⊂ P(R^∨). 0186
Lemma 5 (i) U₊ is a lattice in the vector space U(𝔫₊). ↩ 1 0187
Proposition 3 Let H be a permissible lattice in 𝔥. Let U₊,U₋,U₀ be the Z-subalgebras of U(𝔤) generated respectively by the elements x_α⁽ⁿ⁾ (α ∈ B,n∈ N),y_α⁽ⁿ⁾(α ∈ B, n∈ N),(ʰₙ)(h∈ H, n∈ N). Let… ↩ 3 0188
Remark 4 By Prop. 5 of §4, no. 4, there exists a unique automorphism θ of 𝔤 such that θ (x_α) =y_α and θ (y_α) =x_α for all α ∈ B, and θ (h) =-h for all h∈ 𝔥; we have θ²= 1. By construction… 0189
Corollary 1 Put G=U∩ 𝔤. Then G is an order in the Lie algebra 𝔤, stable under θ. We have G=H+∑_α∈ R(G∩ 𝔤^α). For all α ∈ B and all n∈ N, the maps (ad x_α)ⁿ/n!, (ad y_α)ⁿ/n! leave U and G stabl… ↩ 2 018A
Corollary 2 Let w∈ W. There exists an elementary automorphism φ of 𝔤 that commutes with θ, leaves G and U stable, and extends w. ↩ 2 018B
Corollary 3 Let α ∈ R. If x∈ G∩ 𝔤^α and n∈ N, we have x⁽ⁿ⁾∈ U, and (ad x)ⁿ/n! leaves G and U stable. ↩ 1 018C
Corollary 4 There exists a Chevalley system (X_α)_α∈ R in (𝔤,𝔥) (§2, no. 4, Def. 3) such that X_α=x_α and X_-α=y_α for α ∈ B. For every Chevalley system (X'_α)_α∈ R having these properties, an… ↩ 1 018D
Remark 5 Let (X_α)_α∈ R be the Chevalley system constructed above. If α , β , α +β ∈ R and if we put [X_α, X_β] = N_α ,βX_α+β, we have [X_α, X_β]∈ G∩ 𝔤^α+β, and we recover the fact that N_α… 018E
Remark 6 We have obtained in passing a new proof of the existence of Chevalley systems (cf. §4, no. 4, Cor. of Prop. 5), independent of Lemma 4, §2. 018F
Theorem 2 The following conditions are equivalent: ↩ 2 018G
Definition 2 When conditions (i) to (iv) of Th. 2 are satisfied, G is said to be a Chevalley order in (𝔤,𝔥). 018H
Remark Chevalley orders in (𝔤,𝔥) always exist. Indeed, the Chevalley orders are the sets of the form H⊕ ∑_α∈ RZX_α, where (X_α)_α∈ R is a Chevalley system in (𝔤,𝔥) and H is a lattice in 𝔥… 018I
Theorem 3 We retain the notations at the beginning of no. 7, and assume that G is a Chevalley order in (𝔤,𝔥). ↩ 2 018J
Lemma 6 Let M be a free Z-module of finite type, u an endomorphism of M, and v the endomorphism u⊗ 1 of M⊗_ZQ. Assume that (ᵛₙ)(M)⊂ M for all n∈ N. Then u is diagonalizable. ↩ 1 018K
Remark 1 If we only assume that v is diagonalizable with integer eigenvalues, u is not necessarily diagonalizable (for example, take M =Z² and u(x, y) = (y, x) for all (x, y)∈ M). 018L
Definition 3 Let E be a 𝔤-module. A lattice E in E is said to be admissible (relative to G) if the following conditions are satisfied: ↩ 2 018M
Remark 2 Let ρ be the adjoint representation of 𝔤 on U(𝔤). Let α , x, n, h be as in (ii) above. We have ρ (x⁽ⁿ⁾).U⊂ U by Lemma 2. On the other hand, if p∈ N, 018N
Remark 3 Let E be a finite dimensional 𝔤-module, E an admissible lattice in E, 𝔠 the centre of 𝔤. By Lemma 6, every element of 𝔠 defines a diagonalizable endomorphism of E. Hence E is semi-… ↩ 1 018O
Remark 4 If 𝔤 is semi-simple and H= Q(R^∨), conditions (i) and (ii) of Def. 3 are equivalent, by Th. 3 (ii), to 018P
Remark 5 Let B be a basis of R; in conditions (i) and (ii) above, “α ∈ R” can be replaced by “α ∈ B∪ (-B)” (loc. cit). 018Q
Theorem 4 Let E be a finite dimensional 𝔤-module. The following conditions are equivalent: 018R
Corollary If 𝔤 is semi-simple and H= Q(R^∨), every finite dimensional 𝔤-module has an admissible lattice. 018S

§ 13. Classical splittable simple Lie algebras

Lemma 1 Let V be a finite dimensional vector space, Q a non-degenerate quadratic form on V,Ψ the symmetric bilinear form associated to Q, C(Q) the Clifford algebra of V relative to Q,f₀ th… ↩ 3 018T
Lemma 2 For 1≤ r≤ℓ, let Fᵣ be the subspace of bigwedgeʳ V generated by the isotropic r-vectors. Then 01K8

IX. Compact real Lie groups

§ 1. Compact Lie algebras

Proposition 1 Let 𝔤 be a (real) Lie algebra. The following conditions are equivalent: ↩ 6 019D
Definition 1 A compact Lie algebra⁴ is a Lie algebra that has properties (i) to (v) of Proposition 1. ↩ 1 019E
Proposition 2 a) The product of a finite number of Lie algebras is a compact Lie algebra if and only if each factor is compact. 019F
Proposition 3 Let G be a Lie group of which the group of connected components is finite. The following conditions are equivalent: ↩ 4 019G
Theorem 1 (H. Weyl) Let G be a connected Lie group whose Lie algebra is compact semi-simple. Then G is compact and its centre is finite. ↩ 2 019H
Proposition 4 Let G be a connected Lie group whose Lie algebra is compact. There exist a torus T, a simply-connected compact semi-simple Lie ↩ 4 019I
Corollary 1 Let G be a connected compact Lie group. Then C(G)₀ is a torus, D(G) is a connected compact semi-simple Lie group and the morphism (x, y) → xy from C(G)₀× D(G) to G is a finite cove… ↩ 9 019J
Corollary 2 The centre and the fundamental group of a connected compact semi-simple Lie group are finite. Its universal covering is compact. ↩ 3 019K
Corollary 3 The fundamental group of a connected compact Lie group G is a Z-module of finite type, of rank equal to the dimension of C(G). 019L
Corollary 4 Let G be a connected compact Lie group. The following conditions are equivalent: 019M
Corollary 5 Let G be a connected compact Lie group. Then Int(G) is the identity component of the Lie group Aut(G) (Chap. III, §10, no. 2). ↩ 1 019N
Proposition 5 Let G be a Lie group whose Lie algebra is compact. ↩ 1 019O
Corollary If N satisfies the conditions of b) (iv), and if K₁ and K₂ are two maximal compact subgroups of G, there exists n∈ N such that nK₁n⁻¹= K₂. 019P

§ 2. Maximal tori of compact Lie groups

Lemma 1 Let G be a Lie group, K a compact subgroup of G, and F an invariant bilinear form on L(G). Let x, y∈ L(G). There exists an element k of K such that F(u,[(Ad k)(x), y]) = 0 for all… ↩ 1 019Z
Theorem 1 Let 𝔤 be a compact Lie algebra. The Cartan subalgebras of 𝔤 (Chap. VII, §2, no. 1, Def. 1) are its maximal commutative subalgebras; in particular, 𝔤 is the union of its Cartan suba… ↩ 5 01A0
Corollary Let 𝔱 and 𝔱' be Cartan subalgebras of 𝔤,𝔞 a subset of 𝔱, and u an automorphism of 𝔤 that takes 𝔞 into 𝔱'. There exists an element v of Int(𝔤) such that u∘ v takes 𝔱 to 𝔱', and coin… ↩ 1 01A1
Theorem 2 Let G be a connected compact Lie group. ↩ 12 01A2
Corollary 1 a) The exponential map of G is surjective. 01A3
Remark 1 There exists a compact subset K of L(G) such that exp_G(K) = G. Indeed, if T is a maximal torus of G, there exists a compact subset C⊂ L(T) such that exp_T(C) = T; it suffices to t… 01A4
Corollary 2 The intersection of the maximal tori of G is the centre of G. ↩ 4 01A5
Corollary 3 Let g∈ G, and let C be its centralizer. Then g belongs to C₀; the group C₀ is the union of the maximal tori of G containing g. ↩ 4 01A6
Corollary 4 Let g∈ G. If g is regular (Chap. VII, §4, no. 2, Def. 2), it belongs to a unique maximal torus, which is the identity component of its centralizer. Otherwise, it belongs to infinit… ↩ 2 01A7
Corollary 5 a) Let S be a torus of G. The centralizer of S is connected; it is the union of the maximal tori of G containing S. ↩ 5 01A8
Remark 2 It follows from Cor. 5 that a maximal torus of G is a maximal commutative subgroup. The converse is not true: for example, in the group SO(3,R), the maximal tori are of dimension 1… ↩ 1 01A9
Corollary 6 The maximal tori of G are their own centralizers, and are the fixers of their Lie algebras. ↩ 2 01AA
Corollary 7 Let T and T' be two maximal tori of G, A a subset of T and s an automorphism of G that takes A into T'. There exists g∈ G such that s∘ (Int g) takes T to T' and coincides with s on… ↩ 1 01AB
Corollary 8 Let H be a compact Lie group, T a maximal torus of H. Then H = N_H(T).H₀, and the injection of N_H(T) into H induces an isomorphism from N_H(T)/N_H₀(T) to H/H₀. ↩ 1 01AC
Remark 3 Let G be a connected Lie group whose Lie algebra is compact. The Cartan subgroups of G are the integral subgroups whose Lie algebras are the Cartan subalgebras of L(G) (the Cartan… ↩ 1 01AD
Proposition 1 Let G and G' be two connected compact Lie groups. ↩ 4 01AE
Remark 1 Proposition 1 generalizes immediately to connected groups with compact Lie algebras. In particular, if G is a connected Lie group whose Lie algebra is compact, the Cartan subgroups… 01AF
Remark 2 Let G be a connected compact Lie group, widetildeD(G) the universal covering of the group D(G) and f: widetildeD(G)→ G the composite of the canonical morphisms from widetildeD(G) t… ↩ 1 01AG
Proposition 2 Let f : G → G' be a surjective morphism of connected compact Lie groups whose kernel is central. The maps H → f(H) and H' → f⁻¹(H') are inverse bijections between the set of connec… ↩ 1 01AH
Proposition 3 Let G be a connected compact Lie group, and H a connected closed subgroup of maximal rank. ↩ 2 01AI
Proposition 4 Let G be a compact Lie group, H a connected closed subgroup of G of maximal rank and N the normalizer of H in G. Then H is of finite index in N and is the identity component of N. ↩ 3 01AJ
Remark 1 Every integral subgroup H of G such that rk H = rk G is closed: indeed, the preceding proof shows that H is the identity component of its normalizer, which is a closed subgroup of… ↩ 1 01AK
Remark 2 With the notations of Prop. 4, every closed subgroup H' of G containing H and such that (H': H) is finite normalizes H, and hence is contained in N; similarly, the normalizer of H'… ↩ 1 01AL
Proposition 5 a) Every conjugacy class of G meets T. 01J9
Corollary 1 The canonical injection of T into G defines by passage to the quotient a homeomorphism from T/W_G(T) to the space G/Int(G) of conjugacy classes of G. ↩ 4 01JA
Corollary 2 Let E be a subset of G stable under inner automorphisms. Then E is open (resp. closed, resp. dense) in G if and only if E∩ T is open (resp. closed, resp. dense) in T. ↩ 1 01JB
Proposition 6 a) Every orbit of G on 𝔤 (for the adjoint representation) meets 𝔱. ↩ 1 01JC
Corollary The canonical injection of 𝔱 into 𝔤 defines by passage to the quotient a homomorphism from 𝔱/W_G(T) to 𝔤/Ad(G). ↩ 1 01JD
Proposition 7 Let H be a closed subgroup of G containing T. ↩ 2 01JE
Lemma 2 Let H be a Lie group, f_T: T→ H and widetildef: widetildeD(G)→ H morphisms of Lie groups such that f_T(p(t)) = widetildef(t) for all t∈ widetildeT. There exists a unique morphism o… ↩ 1 01AO
Proposition 8 Let G be a connected compact Lie group, T a maximal torus of G, H a Lie group and φ : L(G)→ L(H) a homomorphism of Lie algebras. There exists a morphism of Lie groups f: G→ H such… ↩ 2 01AP
Proposition 9 Let G be a connected compact Lie group, T a maximal torus of G, H a Lie group and f: G→ H a morphism. Then f is injective if and only if its restriction to T is injective. ↩ 1 01AQ

§ 3. Compact forms of complex semi-simple Lie algebras

Definition 1 Let 𝔞 be a complex Lie algebra. A real form of 𝔞 is a real subalgebra 𝔤 of 𝔞 that is an R-structure on the C-vector space 𝔞(Algebra, Chap. II, §8, no. 1, Def. 1). 01B1
Proposition 1 a) Let 𝔤 be a real form of 𝔞 and σ the conjugation of 𝔞 relative to 𝔤. Then: ↩ 3 01JF
Proposition 2 a) The real vector subspace 𝔞₀=𝔥₀+∑_α∈ RRX_α of 𝔞 is a ↩ 1 01JG
Remark With the preceding notations, we have the following formulas: 01B2
Proposition 3 Let 𝔤 be a compact real form of 𝔞 such that 𝔤∩ 𝔥=i𝔥₀. There exists a homomorphism γ : Q(R)→ R^*₊ such that 𝔤=f(γ )(𝔞ᵤ). ↩ 2 01B3
Theorem 1 Let 𝔞 be a complex semi-simple Lie algebra. ↩ 2 01B4
Remark We shall see much later a general classification of real forms of a complex semi-simple Lie algebra. 01B5
Corollary 1 Let 𝔤 and 𝔤' be two compact real Lie algebras. Then 𝔤 and 𝔤' are isomorphic if and only if the complex Lie algebras 𝔤_(C) and 𝔤'_(C) are isomorphic. ↩ 1 01B6
Corollary 2 Let 𝔞 be a complex Lie algebra. The following conditions are equivalent: 01B7
Corollary 3 Let 𝔞₁ and 𝔞₂ be two complex semi-simple Lie algebras. The compact real forms of 𝔞₁× 𝔞₂ are the products 𝔤₁× 𝔤₂, where, for i= 1,2,𝔤ᵢ is a compact real form of 𝔞ᵢ. 01B8
Proposition 4 a) The compact real forms of the complex Lie algebra 𝔰𝔩(V) are the algebras 𝔰𝔲(Φ ), where Φ belongs to the set of separating positive hermitian forms on the complex vector space V. 01B9
Corollary Every compact simple real Lie algebra of type Aₙ(n≥ 1) is isomorphic to 𝔰𝔲(n+ 1,C). ↩ 1 01BA
Remark 1 We have 𝔤𝔩(V) =𝔰𝔩(V)× C.1_V,𝔲(Φ ) =𝔰𝔲(Φ )× R.i1_V; the compact real forms of 𝔤𝔩(V) are the 𝔰𝔲(Φ )× R.α 1_V,α ∈ C^*. 01BB
Remark 2 If the complex Lie algebra 𝔞=𝔰𝔩(n,C) is equipped with the splitting and Chevalley system introduced in Chap. VIII, §13, no. 1 (IX), then, with the notations in no. 2, 01BC
Proposition 5 Every compact simple real Lie algebra of type Bₙ,n≥ 1 (resp. of type Dₙ,n≥ 3) is isomorphic to 𝔬(2n+ 1,R) (resp. 𝔬(2n,R)). ↩ 1 01BD
Proposition 6 Every compact semi-simple real Lie algebra of rank 1 is isomorphic to 𝔰𝔲(2,C) and to 𝔬(3,R). Every connected semi-simple compact Lie group of rank 1 is isomorphic to SU(2,C) if it… ↩ 1 01BE
Remark We have seen above that SU(2,C) is simply-connected and that π₁(SO(3,R)) is of order 2. We shall see much later that these results generalize to SU(n,C) (n≥ 1) and SO(n,R) (n≥ 3)… 01BF

§ 4. Root system associated to a compact group

Proposition 1 — For any compact Lie group H, the Z-module X(H) is of finite type. It is free if H is connected. 01JH
Proposition 2 — Let H be a commutative compact Lie group, and (aᵢ)_i∈ I a family of elements of X(H); the aᵢ generate X(H) if and only if the intersection of the Keraᵢ reduces to the identity el… ↩ 1 01JI
Proposition 3 The bilinear form (a, X) → ⟨ a, X⟩ on X(S)× Γ (S) is invertible. ↩ 1 01BP
Proposition 4 Let S and S' be tori. Denote by M(S,S') the group of morphisms of Lie groups from S to S'. The maps f → X(f) and f → Γ (f) are isomorphisms of groups from M(S,S') to Hom_Z(X(S'),X(… ↩ 2 01BQ
Remark 1 Let f : S → S' be a morphism of tori. The snake diagram (Algebra, Chap. X, §1, no. 2) associated to (1) gives an exact sequence ↩ 1 01BR
Remark 2 By Prop. 4, the map f → Γ (f)(2π i) from M(U,S) to Γ (S) is an isomorphism; if a∈ X(S) = M(S,U) and f∈ M(U,S), then the composite a∘ f∈ M(U,U) is the endomorphism u → uʳ, where r=⟨… 01BS
Remark 3 To the exact sequence 0→ Γ (S)→ L(S)longrightarrow^exp_SS→ 0 is associated an isomorphism from Γ (S) to the fundamental group of S, called canonical in the sequel. For any morphism… ↩ 1 01BT
Remark 4 The homomorphisms of Z-modules δ : X(S)→ Hom_C(L(S)_(C),C) and ι :Γ (S)→ L(S)_(C)(ι is induced by the canonical injection of Γ (S) into L(S)) extend to isomorphisms of C-vector spa… 01BU
Lemma 1 If G is commutative, widetildeV is the direct sum of the widetildeV_λ(G) for λ ∈ X(G). ↩ 1 01BV
Definition 1 The weights of the representation ρ of G, relative to a maximal torus T of G, are the elements λ of X(T) such that widetildeV_λ(T)not= 0. 01BW
Proposition 5 The set P(ρ ,T) is stable under the operation of the Weyl group W. Let n∈ N_G(T), and let w be its class in W; for λ ∈ X(T), we have ρ (n)( widetildeV_λ(T)) = widetildeV_wλ(T) and… ↩ 2 01BX
Proposition 6 The homomorphism ρ : G→ GL(V) is injective if and only if P(ρ ,T) generates the Z-module X(T). ↩ 2 01BY
Proposition 7 a) For all λ ∈ X(T), we have widetildeV_λ(T) = widetildeV_δ(λ)(𝔱_C). ↩ 2 01BZ
Proposition 8 Let Q(R) be the subgroup of X(T) generated by the roots of G. ↩ 5 01C0
Proposition 9 Every automorphism of the Lie group G that induces the identity on T is of the form Int t, with t∈ T. ↩ 3 01C1
Corollary Let u be an automorphism of G and H the closed subgroup of G consisting of the fixed points of u. Then, the automorphism u is inner if and only if H₀ is of maximal rank. 01C2
Lemma 2 Let S be a closed subgroup of T and Z(S) its normalizer in G. ↩ 2 01C3
Theorem 1 Let α ∈ R(G,T). The centralizer Z_α of the kernel of α is a connected closed subgroup of G; its centre is Ker α; its derived group D(Z_α) = S_α is a connected closed semi-simple su… ↩ 2 01C4
Corollary — There exists a morphism of Lie groups ν:SU(2, C)→ G with the following properties: ↩ 1 01C7
Remark For all x∈ R, we have 01C5
Proposition 10 The map u → L(u)_(C) from Aut(T) to GL(𝔱_C) induces an isomorphism from W to the Weyl group of the split reductive Lie algebra (𝔤_C,𝔱_C). For all α ∈ R, W_Z_α(T) is of order 2, and… ↩ 1 01C6
Corollary Assume that G is semi-simple. Then every element of G is the commutator of two elements of G. ↩ 2 01JJ
Proposition 11 The homomorphism f(G,T) :Γ (T)→ π₁(G) is surjective. Its kernel is the subgroup N(G,T) of Γ (T) generated by the family of nodal vectors (K_α)_α∈ R(G,T). ↩ 3 01C8
Corollary 1 G is simply-connected if and only if the family (K_α)_α∈ R(G,T) generates Γ (T). 01C9
Corollary 2 Let H be a connected closed subgroup of G containing T; there is an exact sequence 01CA
Remark It can be shown (cf. Exercise 2 of §5) that π₂(G/H) is zero. The exactness of the preceding sequence then gives an isomorphism from π₂(G/H) to N(G,T)/N(H,T). 01CB
Corollary 3 The homomorphism π₁(D(G))→ π₁(G) corresponding to the inclusion of D(G) into G induces an isomorphism from π₁(D(G)) to the torsion subgroup of π₁(G). ↩ 1 01CC
Proposition 12 Let H be the set of connected closed subgroups of G containing T, ordered by inclusion. The map H → R(H,T) is an increasing bijection from H to the set of symmetric closed subsets… ↩ 1 01CD
Corollary 1 There are only finitely-many closed subgroups of G containing T. ↩ 1 01CE
Corollary 2 Let H be a connected closed subgroup of G containing T, and let Wᴴ_G(T) be the stabilizer in W_G(T) of the subset R(H,T) of R. The group N_G(H)/H is isomorphic to the quotient grou… 01CF
Remark 1 The group Wᴴ_G(T) is also the stabilizer in W_G(T) of the subgroup C(H) of T: this follows from Prop. 8 of no. 4. 01CG
Proposition 13 Let H be a connected closed subgroup of G of maximal rank, and C its centre. Then C contains the centre of G, and H is the identity component of the centralizer of C. ↩ 2 01CH
Remark 2 Say that a subgroup C of G is radical if there exists a maximal torus S of G and a subset P of R(G,S) such that C =⋂_α∈ P Kerα. It follows from Prop. 13 and Lemma 2 of no. 5 that t… 01CI
Corollary The set of g∈ G such that T∩ gTg⁻¹not= C(G) is the union of a finite number of closed analytic submanifolds of G distinct from G. 01CJ
Proposition 14 Let X be a subset of T, and let R_X be the set of roots α ∈ R(G,T) such that α (X) =1. The group Z_G(X)/Z_G(X)₀ is isomorphic to the quotient of the subgroup of W_G(T) fixing X by… ↩ 1 01CK
Definition 2 A root diagram (or simply a diagram, if there is no risk of confusion) is a triple D = (M,M₀,R) where: 01CL
Example 1 For every free Z-module of finite type M, the triple (M,M,∅ ) is a root diagram. 01CM
Example 2 If D = (M,M₀,R) is a root diagram, let M^*₀ be the orthogonal complement of V(R) in M^*, and let R^∨ be the set of inverse roots of D. Then D^∨= (M^*,M^*₀,R^∨) is a root diagram, c… 01CN
Example 3 Let (𝔤,𝔥) be a split reductive Q-Lie algebra, and M⊂ 𝔥 a permissible lattice (Chap. VIII, §12, no. 6, Def. 1). Let M₀ be the subgroup of M orthogonal to the roots of (𝔤,𝔥) and R^∨… 01CO
Example 4 Let V be a vector space over Q and R a root system in V; denote by P(R) the group of weights of R and by Q(R) the group of radical weights of R (Chap. VI, §1, no. 9). Then (Q(R),0… 01CP
Theorem 2 a) (X(T),X(T/(T∩ D(G)),R(G,T))) is a reduced root diagram; its Weyl group consists of the X(w) for w∈ W; the group X(C(G)) is isomorphic to the quotient of X(T) by the subgroup gen… ↩ 1 01CQ
Example 1 If G is semi-simple of rank 1, then D^*(G,T) and D_*(G,T) are necessarily isomorphic to one of the two diagrams Δ₂= (Z,0,2,-2), Δ₁= (Z,0,1,-1). If G is isomorphic to SU(2,C), D_*(G… 01CR
Example 2 If G and G' are two connected compact Lie groups, with maximal tori T and T', respectively, and if D^*(G,T) = (M,M₀,R) and D^*(G',T') = (M',M'₀,R'), then D^*(G× G',T× T') can be id… ↩ 1 01CS
Example 3 Let N be a closed subgroup of T, central in G, and let (M,M₀,R) be the contravariant diagram of G relative to T. Then the contravariant diagram of G/N relative to T/N can be identi… ↩ 1 01CT
Example 4 Similarly, let N be a finite abelian group, and φ :π₁(G)→ N a surjective homomorphism. Let G' be the covering of G associated to this homomorphism; this is a connected compact Lie… 01CU
Remark 1 Let 𝔠 be the centre of 𝔤_C; then 𝔠= L(C(G))_(C). We have the following relations between the diagrams of G relative to T and the direct and inverse root systems of the split reduct… 01CV
Remark 2 Assume that the group G is semi-simple; denote by R (resp. R^∨) the root system R(G,T) (resp. R^∨(G,T)), so that we have inclusions 01CW
Proposition 15 Let φ be an isomorphism from D^*(G',T') to D^*(G,T) (resp. from D_*(G,T) to D_*(G',T')). There exists an isomorphism f: G→ G' such that f(T) = T' and φ = D^*(f) (resp. φ = D_*(f))… ↩ 2 01CX
Corollary Two connected compact Lie groups G and G' are isomorphic if and only if the diagrams D^*(G) and D^*(G') (resp. D_*(G) and D_*(G')) are equal. 01CY
Proposition 16 For every reduced root diagram D, there exists a connected compact Lie group G such that D^*(G) (resp. D_*(G)) is isomorphic to D. 01CZ
Scholium The classification of connected compact Lie groups up to isomorphism is thus reduced to that of reduced root diagrams. The connected compact semi-simple Lie groups correspond to th… 01D0
Remark 3 Let T' be another maximal torus of G, B (resp. B') a basis of the root system R(G,T) (resp. R(G',T')) (Chap. VI, §1, no. 5, Def. 2). There exist elements g∈ G such that Int g maps… 01D1
Proposition 17 The homomorphism Aut(G,T)→ Aut(D) is surjective, with kernel Int_G(T). 01D2
Proposition 18 The group Aut(G)/Int(G) is isomorphic to Aut(D)/W(D). 01K9
Corollary If G is simply-connected, or if C(G) reduces to the identity element, the group Aut(G)/Int(G) is isomorphic to the group of automorphisms of the Dynkin graph of R(G,T). 01D3
Definition 3 A framing of (G,T) is a pair (B,(U_α)_α∈ B), where B is a basis of R(G,T) (Chap. VI, §1, no. 5, Def. 2) and where, for all α ∈ B,U_α is an element of V(α ) such that K(U_α) =-1. ↩ 1 01D4
Lemma 3 Let B₀ be a basis of R(G,T). The group Int_G(T) operates simply-transitively on the set of framings of (G,T) of the form (B₀,(U_α)_α∈ B₀). ↩ 2 01D5
Proposition 19 The group Int(G) operates simply-transitively on the set of framings of G. ↩ 1 01D6
Corollary Let e be a framing of (G,T) and let E be the group of automorphisms of G that leave e stable. Then Aut(G) is the semi-direct product of E by Int(G), and Aut(G,T) is the semi-direct… 01D7
Remark Let G and G' be two connected compact Lie groups, and let e= (T,B,(U_α)) and e'= (T',B',(U'_α)) be framings of G and G', respectively. Let X be the set of isomorphisms from G to G'… 01D8

§ 5. Conjugacy classes

Proposition 1 Put n= dim G. There exists a compact real-analytic manifold V of dimension n-3 and an analytic map φ : V→ G whose image is G -- Gᵣ. ↩ 2 01DV
Remark Call an element g of G very regular if Z(g) is a maximal torus of G. If g∈ T,g is very regular if and only if w(g)not=g for every non-identity element w of W_G(T) (§4, no. 7, Prop… 01DW
Proposition 2 a) The group Wₐ (resp. W'ₐ) is the semi-direct product of W by N(G,T) (resp. Γ (T)); the subgroup Wₐ of W'ₐ is normal. 01DX
Corollary 1 Let A be an alcove of 𝔱,overlineA its closure, and H_A the stabilizer of A in W'ₐ. ↩ 5 01DY
Remark 1 The group H_A can be identified naturally with Γ (T)/N(G,T), hence also with π₁(G). Thus, it reduces to the identity element when G is simply-connected. 01DZ
Remark 2 Let x∈ A; then exp x∈ Tᵣ, so Z(exp x)₀= T. Further, exp x is very regular (no. 1, Remark) if and only if w(x)not=x for all w∈ W'ₐ distinct from the identity. By Cor. 1, this also m… ↩ 2 01E0
Remark 3 The special points of Wₐ (Chap. VI, §2, no. 2) are the elements x of 𝔱 such that δ (α )(x)∈ 2π iZ for all α ∈ R(G,T) (loc. cit., Prop. 3), that is such that exp x∈ C(G) (§4, no. 4… 01E1
Corollary 2 Let C be a chamber of 𝔱 and overlineC its closure. The canonical injections overlineC→ 𝔱→ 𝔤 induce by passage to the quotient homeomorphisms 01E2
Remark 4 Denote by 𝔤_reg the set of regular elements of 𝔤 (Chap. VII, §2, no. 2, Def. 2) and put 𝔱_reg=𝔱∩ 𝔤_reg. For x∈ 𝔱, we have 01E3
Corollary 3 Assume that G is simply-connected; let g be a regular element of G. There exist a maximal torus S of G, and an alcove A of L(S), both uniquely determined, such that g∈ exp(A) and 0… ↩ 1 01E4
Lemma 1 Let u be an automorphism of G, and H the set of its fixed points. ↩ 1 01E5
Lemma 2 Let x be an element of T and S a subtorus of T. If the identity component of Z(x)∩ Z(S) reduces to T, there exists an element s of S such that xs is regular. ↩ 1 01E6
Lemma 3 Assume that G is simply-connected. Let C be a chamber of 𝔱, and u an automorphism of G such that T and C are stable under u. Then the set of points of T fixed by u is connected. ↩ 1 01E7
Proposition 3 Let u be an automorphism of G, and let x be a point of G fixed by u. ↩ 2 01E8
Corollary Let 𝔰 be a compact Lie algebra, and let φ be an automorphism of 𝔰. There exists a regular element of 𝔰 fixed by φ. 01E9
Theorem 1 Let u be an automorphism of a connected compact Lie group G. ↩ 2 01EA
Corollary 1 Assume that the Z-module π₁(G) is torsion-free. 01EB
Corollary 2 Let Γ be a compact subgroup of Aut(G) with the following property: ↩ 1 01EC
Corollary 3 Let H be a Lie group and Γ a compact subgroup of H. Assume that H₀ is compact and that Γ satisfies condition () of Cor. 2. Then there exists a maximal torus T of H₀ such that Γ ⊂ N… ↩ 1 01ED
Corollary 4 Every nilpotent subgroup of a compact Lie group is contained in the normalizer of a maximal torus. 01EE
Example Take G =U(n,C). We shall see later that the subgroup of diagonal matrices in G is a maximal torus of G and that its normalizer is the set of monomial matrices (Algebra, Chap. II, §… 01EF
Lemma 4 Let g∈ G,t∈ T, and let overlineg be the image of g in G/T. Identify the tangent space of G/T (resp. T, resp. G) at overlineg (resp. t, resp. gtg⁻¹) with 𝔤/𝔱 (resp. 𝔱, resp. 𝔤) by m… ↩ 2 01EG
Proposition 4 a) Let g∈ G, t∈ T, x∈ 𝔱, and let overlineg be the image of g in G/T. The following conditions are equivalent: 01EH
Remark 1 By Prop. 3 of §2, no. 4, the manifold (G/T)× A is simply-connected. It follows that φ_A is a universal covering of Gᵣ; since π₁(Gᵣ) is canonically isomorphic to π₁(G) (no. 1, Prop… 01EI
Remark 2 The restriction of φ_A to W× A⊂ (G/T)× A makes W× A a principal covering of Tᵣ with group H_A. We thus recover Cor. 1 of Prop. 2 of no. 2. 01EJ
Remark 3 Denote by 𝔤ᵣ the inverse image of Gᵣ under the exponential map and by ε :𝔤ᵣ→ Gᵣ the map induced by exp_G. The map (g, x) → (Ad g)(x) from G× 𝔱ᵣ to 𝔤ᵣ defines by passage to the quot… 01EK

§ 6. Integration on compact Lie groups

Lemma 1 There exists an A-bilinear map from Altˢ(M'')× Altʳ(M') to Altˢ⁺ʳ(M), denoted by (u, v) → u∩ v, and characterized by either of the following two properties: ↩ 2 01EX
Proposition 1 Let ω_G, ω_G/T and ω_T be invariant differential forms on G,G/T and T, respectively, of respective degrees n, n-r and r. If ω_G= ω_G/T∩ ω_T, then ↩ 1 01EY
Lemma 2 If ω_G=ω_G/T∩ ω_T, then ↩ 1 01EZ
Lemma 3 The inverse image on (G/T)× Tᵣ of the measure dg on Gᵣ under the local homeomorphism fᵣ(Integration, Chap. V, §6, no. 6) is the measure μ⊗ δ_Gdt, where μ is the unique G-invariant… ↩ 1 01F0
Theorem 1 (H. Weyl) The measure dg on G is the image under the map (g, t) → gtg⁻¹ from G× T to G of the measure dg⊗ 1/w(G)δ_Gdt, where ↩ 1 01F1
Corollary 1 (i) Let φ be an integrable function on G with values in a Banach space or in overlineR. For almost all t∈ T, the function g → φ (gtg⁻¹) on G is integrable for dg. The function t →… ↩ 1 01F2
Corollary 2 Let φ be a central function on G (that is, such that φ (gh) = φ (hg) for all g and h in G) with values in a Banach space or in overlineR. ↩ 4 01F3
Corollary 3 Let H be a connected closed subgroup of G containing T, 𝔥 its Lie algebra, and dh the Haar measure on H of total mass 1. Let φ be an integrable central function on G, with values i… ↩ 1 01F4
Remark 1 If we take φ = 1 in Cor. 3, we obtain 01F5
Remark 2 Let ν be the measure on the quotient T/W defined by 01F6
Remark 3 Assume that G is simply-connected. Let A be an alcove of 𝔱, and dx the Haar measure on 𝔱 such that ∫_Adx= 1. Then the measure ν can also be obtained by transporting the measure 1/w… 01F7
Example Take G to be the group SU(2,C) and T to be the subgroup of diagonal matrices (§3, no. 6); identify 𝔱 with R by the choice of basis iH of 𝔱(loc. cit.). Put A = 0, π; this is an alco… 01F8
Proposition 2 Let H be a (real) Lie group of dimension m,𝔥 its Lie algebra. Let ω_H be a right-invariant differential form of degree m on H, and let ω_𝔥 be the translation-invariant differential… 01F9
Remark Consider the function λ_𝔤 associated to a compact Lie group G; in view of §2, no. 1, Th. 1, to calculate λ_𝔤 it suffices to know its values on 𝔱. But, for x∈ 𝔱, the endomorphism ad… 01FA
Proposition 3 Let ω_𝔤,ω_𝔱,ω_G/T be invariant differential forms on 𝔤,𝔱, G/T, respectively, of respective degrees n, r, n-r. If ω_𝔤=ω_G/T∩ ω_𝔱, we have 01FB
Proposition 4 The measure dz on 𝔤 is the image under the proper map (g, x) → (Ad g)(x) from G× 𝔱 to 𝔤 of the measure dg⊗ 1/w(G)π_𝔤dx. ↩ 1 01FC
Lemma 4 The law of operation H× Sʳ(X; E)→ Sʳ(X; E) is continuous. ↩ 2 01FD
Proposition 5 Assume that the group H is compact and denote by dh the Haar measure on H of total mass 1. Let s be a section of E of class Cʳ. Denote by s^sharp the vector integral ∫_Hʰs dh. Then… 01FE
Corollary 1 Let F be a Banach space, ρ : H→ GL(F) an analytic linear representation, f∈ Cʳ(X; F). For x∈ X, put ↩ 1 01FF
Corollary 2 Let F be a Banach space, f∈ Cʳ(X; F); put ↩ 1 01FG
Corollary 3 Let F be a Banach space, p an integer ≥ 0,ᵏΩᵖ(X; F) the space of differential forms of degree p on X, with values in F, and of class Cᵏ(2≤ k+ 1≤ r). For ω ∈ᵏΩᵖ(X; F), put ω^sharp=∫… ↩ 1 01FH
Theorem 2 The canonical injection ι :Ω (X)ᴳ→ Ω (X) is a homotopism of complexes (Algèbre, Chap. X, p. 33, déf. 5); the map ω → ω^sharp=∫_Gτ_g^*ω dg is a homotopism, inverse to it up to homot… ↩ 2 01FI
Corollary 1 a) In the diagram (18), all the morphisms are homotopisms. 01FJ
Remark Consider the subcomplexes Z(Ω (G)) and B(Ω (G)) of Ω (G) (Algèbre, Chap. X, p. 25). It follows from the formula giving the differential of the product of two forms (Differentiable… 01FK
Corollary 2 Let H be a closed subgroup of G. 01FL

§ 7. Irreducible representations of connected compact Lie groups

Lemma 1 Let λ be an element of X(T). The following conditions are equivalent: ↩ 1 01FZ
Lemma 2 Let φ be a linear representation of the complex Lie algebra 𝔤_C on a finite dimensional complex vector space V. There exists a representation τ of G on V such that L(τ )_(C)=φ if a… ↩ 1 01G0
Theorem 1 a) Let τ : G→ GL(V) be an irreducible representation of G. Then the set of weights of τ (relative to T) has a largest element λ, which is dominant, and the space V_λ(T) is of dimen… ↩ 7 01G1
Corollary The group G has a (finite dimensional) faithful linear representation. 01G2
Remark 1 Let 𝔫₊ be the subalgebra of 𝔤_C that is the sum of the 𝔤^α for α >0. Let τ : G→ GL(V) be an irreducible representation, λ ∈ X₊₊ its highest weight and τ ':𝔤_C→ 𝔤𝔩(V) the representa… ↩ 1 01G3
Remark 2 Let Θ (G) be the algebra of continuous representative functions on G with values in C(Algebra, Chap. VIII). Let G operate on Θ (G) by left and right translations. For each λ ∈ X₊₊… 01G4
Remark 3 Let τ : G→ GL(V) be an irreducible representation. There exists an element ν of X(C(G)) such that τ (s)v=ν (s)v for all s∈ C(G), v∈ V: indeed, τ (C(G)) is contained in the commutan… ↩ 1 01G5
Remark 4 The definitions and statements of Chap. VIII, §7, nos. 2 to 5 can be generalized without difficulty to the present situation; we leave the details to the reader. 01G6
Proposition 1 Let τ : G→ GL(V) be an irreducible representation of G of highest weight λ ∈ X₊₊. Let m be the integer ∑_α∈ R₊⟨ λ , K_α⟩, and let w₀ 01G7
Proposition 2 The homomorphism Ch induces an isomorphism from R(G) to Z[X(T)]ᵂ. 01G8
Lemma 3 For each λ ∈ X₊₊, let C_λ be an element of Z[X(T)]ᵂ having unique maximal term e^λ. Then the family (C_λ)_λ∈ X₊₊ is a basis of Z[X(T)]ᵂ. 01G9
Corollary The restriction map r: ZΘ (G)→ Θ (T)ᵂ is bijective. ↩ 1 01GA
Proposition 3 The elements J(λ ρ )/surdoverlinew(G), for λ ∈ X₊₊, form an orthonormal basis of the subspace of anti-invariant elements of L²(T), and an algebraic basis of the subspace of anti-in… ↩ 2 01GB
Lemma 4 If φ and ψ are two continuous central functions on G, ↩ 2 01GC
Theorem 2 (H. Weyl) For all λ ∈ X₊₊, we have J(ρ ).χ_λ|T = J(λ ρ ). ↩ 3 01GD
Corollary 1 With the notations of no. 3, we have in Z[X(T)], 01GE
Corollary 2 For all λ ∈ X₊₊ and every regular element t of T, ↩ 2 01GF
Corollary 3 The map which associates to any continuous central function φ on G the function w(G)^-1/2J(ρ )(φ |T) induces an isomorphism from the space of central representative functions on G… ↩ 1 01GG
Corollary 4 Let φ be a continuous central function on G. Then, ↩ 2 01GH
Remark 1 For all w∈ W, put ρ_w=ʷρ /ρ; then 01GI
Remark 2 Similarly, the first equality of Cor. 4 remains valid without the assumption that ρ ∈ X(T). 01GJ
Remark 3 We can deduce Th. 2 from the analogous infinitesimal statement (Chap. VIII, §9, no. 1, Th. 1); the same is true for Th. 3 of the next number (which is the analogue of Th. 2 of loc… 01GK
Theorem 3 The dimension of the space of an irreducible representation of G of highest weight λ is given by ↩ 2 01GL
Lemma 5 We have 01GM
Corollary 1 Let | | be a norm on X(T)⊗ R. For all λ ∈ X₊₊, let d(λ ) be the dimension of the space of an irreducible representation of G of highest weight λ. ↩ 2 01GN
Corollary 2 Assume that G is semi-simple; let d be an integer. The set of classes of representations of G of dimension ≤ d is finite. 01GO
Proposition 4 Let Γ be the Casimir element of G. ↩ 2 01GP
Remark Let x∈ 𝔤. There exists a strictly positive real number A such that, for every irreducible representation τ : G→ GL(V) and every Hilbert structure on V invariant under G, 01GQ

§ 8. Fourier transform

Proposition 1 Every square integrable complex function f on G is the sum in the Hilbert space L²(G) of the family (fᵤ)_u∈widehatG where, for all h∈ G and all u∈ widehatG, ↩ 1 01GZ
Proposition 2 Let m → |m| be a norm on the R-vector space R⊗ X(T) and Γ a Casimir element of G. Let φ be a function on widehatG with positive real values. 01H0
Example The function u → d(u) is moderately increasing (§7, no. 5, Cor. 1 of Th. 3); for any norm | | on R⊗ X(T), the function u → |λ (u)| is moderately increasing. For any Casimir element… 01H1
Proposition 3 For all t∈ U(G), the functions u → |u(t)|_∞ and u → |u(t)|₂ on widehatG are moderately increasing. ↩ 1 01H2
Theorem 1 a) Let f be an infinitely-differentiable complex function on G. Then the family (fᵤ)_u∈widehatG, where fᵤ(g) =⟨ u(g)|u(f)⟩, is uniformly summable on G and, for all h∈ G, 01H3
Corollary The maps overlineF:f → (u(f))_u∈widehatG and F: A → ∑_u∈widehatG⟨ u(g)|Aᵤ⟩ induce inverse isomorphisms between the topological vector spaces C^∞(G;C) and S( widehatG). 01H4
Proposition 4 Let f∈ L²(G). Then f is central if and only if u(f) is a homothety for all u∈ widehatG. In that case 01H5
Proposition 5 Let f be a continuous central function on G. Then f is infinitely-differentiable if and only if the function u → |⟨ χᵤ|f⟩ | is rapidly decreasing on widehatG; in that case, ↩ 1 01H6
Corollary Let f be a continuous central function on G. Then f is infinitely-differentiable if and only if the restriction of f to T is infinitely-differentiable. ↩ 1 01H7

§ 9. Compact Lie groups operating on manifolds

Lemma 1 Let T and T' be two topological spaces, A and A' compact subsets of T and T', respectively, W a neighbourhood of A× A' in T× T'. There exists an open neighbourhood U of A in T and… ↩ 1 01HE
Proposition 1 Let φ : X→ Y be a morphism of class Cʳ, A a compact subset of X. The following conditions are equivalent: 01HF
Proposition 2 Assume that the manifold Y is paracompact; let A be a subset of X, and let φ :X→ Y be a morphism of class Cʳ that induces a homeomorphism from A to φ (A), and that is étale at ever… ↩ 1 01HG
Proposition 3 Let A be a compact subset of X. The set P of morphisms φ ∈ Cʳ(X;Y) that are embeddings in the neighbourhood of A is open in Cʳ(X;Y) in the topology of compact Cʳ-convergence (§ 6… ↩ 1 01HH
Lemma 2 Let x be a point of X,φ : X→ Y a morphism of class C¹ that is an immersion at x. There exists a neighbourhood Ω of φ in C¹(X;Y) and a neighbourhood B of x in X such that, for all ψ… 01HI
Proposition 4 Let A be a compact subset of X. There exist a finite dimensional vector space E and a morphism φ ∈ Cʳ(X;E) (rnot=ω ) that is an embedding in the neighbourhood of A. ↩ 3 01HJ
Lemma 3 Let G be a compact topological group operating continuously on a topological space X; let A be a subset of X, stable under G, and W a neighbourhood of A. Then, there exists an open… ↩ 2 01HK
Theorem 1 Let G be a compact Lie group, (g, x) → gx a law of left operation of class Cʳ of G on X, and A a compact subset of X. There exists an analytic linear representation ρ of G on a fin… ↩ 1 01HL
Corollary 1 Assume that X is compact. There exists an analytic linear representation ρ of G on a finite dimensional vector space E and an embedding φ : X→ E such that φ (gx) =ρ (g)φ (x) for g∈… ↩ 1 01HM
Corollary 2 Let H be a closed subgroup of G. There exists an analytic linear representation of G on a finite dimensional vector space E and a point v∈ E with fixer H. ↩ 1 01HN
Corollary 3 Assume that X is paracompact. There exists a real Hilbert space E, a continuous unitary representation ¹⁰ρ of G on E and an embedding φ : X→ E of class Cʳ such that φ (gx) =ρ (g)φ… 01HO
Lemma 4 Let H be a compact Lie group, ρ : H→ GL(V) a continuous (hence analytic) representation of H on a finite dimensional real vector space, and W a neighbourhood of the origin in V. Th… ↩ 2 01HP
Proposition 5 Let H be a compact Lie group, (h, x) → hx a law of left operation of class Cʳ of H on X, and x a point of X fixed under the operation of H. Then the group H operates linearly on th… ↩ 1 01HQ
Proposition 6 Assume that the manifold X is paracompact; let x be a point of X, Gₓ its fixer. There exists a finite dimensional analytic linear representation τ : Gₓ→ GL(W), and an open embeddin… ↩ 5 01HR
Lemma 5 Let Z be a separated manifold of class Cʳ, equipped with a law of left operation m: G× Z→ Z of class Cʳ, and μ: Z→ X a morphism (of class Cʳ) compatible with the operations of G. L… 01HS
Lemma 6 Let G be a Lie group. 01HT
Theorem 2 Let G be a Lie group operating properly on X, such that the law of operation (g, x) → gx is of class Cʳ. Assume that X is paracompact. 01HU
Example 1 Let G be a connected compact Lie group, operating on itself by inner automorphisms. The fixer of an element x of G is simply the centralizer Z(x) of x in G; it contains every maxim… 01HV
Example 2 Let E be a real euclidean affine space, ℌ a set of hyperplanes of E, W the group of displacements of E generated by the orthogonal reflections with respect to the hyperplanes of ℌ… 01HW

App. 1. Structure of compact groups

Proposition 1 Every compact topological group G is isomorphic to a closed subgroup of a product of compact Lie groups. 01II
Corollary 1 Let V be a neighbourhood of the identity element of G. Then V contains a closed normal subgroup H of G such that the quotient G/H is a Lie group. ↩ 2 01IJ
Corollary 2 The canonical map G→ lim__← -G/H_α is an isomorphism of topological groups. α ↩ 1 01IK
Corollary 3 G is a Lie group if and only if there exists a neighbourhood of the identity element e of G that contains no normal subgroup distinct from e. 01IL
Lemma 1 Let (G_α, f_α β) be a projective system of topological groups relative to a filtered index set I, and G its limit. Assume that the canonical maps f_α: G→ G_α are surjective. ↩ 2 01IM
Lemma 2 Let (Sₐ)_a∈ A,(T_b)_b∈ B be two finite families of almost simple, simply-connected Lie groups (Chap. III, § 9, no. 8, Def. 3),u:∏_a∈ ASₐ→ ∏_b∈ BT_b a surjective morphism. Then ther… ↩ 1 01IN
Lemma 3 Under the hypotheses of Lemma 1, assume that the G_α are simply-connected compact Lie groups. Then, the topological group G is isomorphic to the product of a family of almost simpl… ↩ 1 01IO
Proposition 2 Let G be a connected compact group. 01IP
Corollary Every solvable connected compact group is commutative. 01IQ

App. 2. Representations of real, complex or quaternionic type

Proposition 1 Let A be an R-algebra (associative and unital) and V a finite dimensional simple A-module over R. Then, we must be in one of the following three situations: ↩ 1 01IS
Proposition 2 Let A be an R-algebra (associative and unital), and W a finite dimensional simple A_(C)-module over C. Then, we must be in one of the following three situations: ↩ 1 01IT
Proposition 3 Assume that ρ is irreducible. ↩ 1 01IU
Lemma 1 Let Wᴳ be the subspace of W consisting of the elements invariant under G. The endomorphism ∫_Gρ (g)dg of W is a projection with image Wᴳ, compatible with the operations of G. In pa… ↩ 1 01IV
Lemma 2 Let u be an endomorphism of a finite dimensional vector space E over a field K. Then ↩ 1 01IW
Proposition 4 Assume that ρ is irreducible. Then, ρ is of real (resp. complex, resp. quaternionic) type if and only if the integral ∫_G Trρ (g²)dg is equal to 1 (resp. 0, resp. -1). 01IX