Index of terminology

The reference numbers indicate respectively the chapter, paragraph and number (or, occasionally, exercise)

Adjoint group of a real or complex Lie group: III.6.4. linear mapping of an element of a Lie algebra: I.1.2. representation of a Lie algebra: I.3.1. representation of a Lie group: III.3.12. Ado’s Theorem: I.7.3. Algebra (not necessarily associative): I.1.1. derived from an algebra by extending the scalars: I.1.1. enveloping, of a Lie algebra: I.2.1. Lie: I.1.2. opposite: I.1.1. product: I.1.1. quotient: I.1.1. restricted enveloping, of a Lie $p$-algebra: I.2, Exercise 6. symmetric, of a module: I.2.5. Almost simple Lie group: III.9.8. Alternants of degree $n$: II.2.6. Antiautomorphism, principal, of the enveloping algebra of a Lie algebra: I.2.4. Associated (bilinear form) with a $g$-module (a representation): I.3.6. (law of infinitesimal operation) with a law of operation: III.3.7. Associative algebra, free: II.3. Automorphism, special, of a Lie algebra: I.6.8.

Basic commutators: II.5.4. family of a Lie algebra: II.2.3. Bieberbach’s Theorem: III.4, Exercise 13. Bigebra: II.1.2. Biinvariant section: III.3.13. Binomial polynomial: II.5, Exercise 4. Bracket: I.1.2.

Canonical differential form, left: III.3.13, III.3.18. Cartan, criterion: I.5.4. Casimir element: I.3.7. Central extension: I.1.7. filtration of a group: II.4.4. Centralizer: I.1.6, III.9.3. Centre of a Lie algebra: I.1.6. Characteristic ideal: I.1.4. Class, nilpotency: II.2.7. of simple representations: I.3.1. Coboundaries, cochains, cocycles with values in a $\mathfrak{g}$-module: I.3, Exercise 12. Cogebra: II.1.1. Commutative Lie algebra: I.1.3. Compatible group and manifold structures: III.1.1. Completely invariant bilinear form: I.3.6. reducible representation: I.3.1. Complex Lie group: III.1.1, III.8.1. Complexification of a real Lie group: III.6.10. Component, isotypical, of a $\mathfrak{g}$-module: I.3.1. simple, of a semi-simple Lie algebra: I.6.2. Conjugate Lie group of a complex Lie group: III.1.1. Constant term of an element of the enveloping algebra of a Lie algebra: I.2.1. term of an element of the Magnus algebra: II.5.2. Constants of structure of an algebra with respect to a basis: I.1.1. Containing a simple representation $n$ times (representation): I.3.1. Contragredient of an analytic representation: III.3.11. Convolution of a point distribution and a function: III.3.4. product: III.3.1, III.3.18. Counit of a cogebra: II.1.1. $C^r$-connected subset of a Lie group: III.6.2. Criterion, Cartan: I.5.4.

Derivation of an algebra: I.1.1. inner, of a Lie algebra: I.1.2. left invariant, relative to a formal group: I.1, Exercise 24. Derivative, partial, in the algebra of a free group: II.5, Exercise 2. Derived (algebra) from an algebra by extending the scalars: I.1.1. (representation) from a representation by extending the scalars: I.3.8. ideal: I.1.5. series: I.1.5. Differential, left, of a mapping into a Lie group: III.3.17, III.3.18. Dimension of a representation: I.3.1. Dual representation: I.3.3.

Element, Casimir: I.3.7. nilpotent: I.6.3. of degree $n$ in the graded algebra associated with the enveloping algebra of a Lie algebra: I.2.6. of filtration $\leq n$ in the enveloping algebra of a Lie algebra: I.2.6. semi-simple: I.6.3. Elements, permutable, in a Lie algebra: I.1.3. Elimination theorem: II.2.9. Engel’s Theorem: I.4.2. Enveloping algebra of a Lie algebra: I.2.1. bigebra: II.1.4. Equivalent extensions of a Lie algebra: I.1.7. Exhaustive filtration: II.4.1. Exponential: II.6.1, III.4.3, III.6.4. Extension, central: I.1.7. inessential: I.1.7. of a $g$-module, of a representation of $g$: I.7.2. of a Lie algebra by a Lie algebra: I.1.7. trivial: I.1.7. Extensions, equivalent: I.1.7.

Faithful representations: I.3.1. Field of point distributions: III.3.5, III.3.18. Filtered bigebra: II.1.3. Filtration, real, on a group: II.4.1. First species, canonical chart of the: III.4.3. species, system of canonical coordinates of the: III.4.3. Foliation, left, associated with a Lie subalgebra: III.4.1. Form, bilinear, associated with a $g$-module (a representation of $g$): I.3.6. bilinear, completely invariant: I.3.6. bilinear, invariant: I.3.6. Killing: I.3.6. Formulae, Jacobson: I.1, Exercise 19. Free Lie algebra: II.2.2. Lie $p$-algebra: II.3, Exercise 4. magma: II.2.1. Function, order, associated with a filtration: II.4.2.

G-bundle, vector: III.1.8. g-module, left (right): I.3.1. pure, of species N: I.3.1. trivial: I.3.1.

Graded group associated with a filtered group: II.4.3. Lie algebra, associated: II.4.4. Graduation, total, of L(I): II.2.6. Group commutative formal: I.1, Exercise 24. formal: I.1, Exercise 24. formal linear: I.1, Exercise 25. formal orthogonal: I.1, Exercise 26. formal symplectic: I.1, Exercise 26. germ, Lie: III.1.10. germ, Lie, defined by a Lie algebra: III.4.2.

Hall basis: II.2.11. formula: II.5, Exercise 9. set: II.2.10. Hausdorff function: II.7.2, II.8.3. group: II.6.2. inversion formula: II.6, Exercise 4. series: II.6.4. Holomorph of a Lie algebra: I.1, Exercise 16. Homomorphism, algebra: I.1.1. canonical, of the enveloping algebra of a Lie subalgebra of g into the enveloping algebra of g: I.2.3. canonical, of the symmetric algebra of g onto the graded algebra associated with the enveloping algebra of g: I.2.6. formal: I.1, Exercise 24.

Ideal, characteristic: I.1.4. derived: I.1.5. left (right, two-sided) of an algebra: I.1.1. of a Lie algebra: I.1.4. Identity, Jacobi: I.1.2. Induced Lie group structure: III.4.5. Inessential extension: I.1.7. Infinitesimal automorphism: III.10.1. Inner derivation: I.1.2. Integral filtration of a group: II.4.1. subgroup of a Lie group: III.6.2. Invariant bilinear form: I.3.6. left, point distributions, field of: III.3.6. of a g-module (a representation): I.3.5. section: III.3.13.

Inverse image Lie group structure: III.1.9. Irreducible representation: I.3.1. Isomorphic representations: I.3.1. Isomorphism, canonical, of the symmetric algebra of $\mathfrak{g}$ onto the underlying vector space of its enveloping algebra: I.2.7. Isotypical component of a $\mathfrak{g}$-module: I.3.1.

Jacobi identity: I.1.2. Jordan’s Theorem: III.4, Exercise 11.

Kernel of an extension: I.1.7. Killing form: I.3.6.

Largest nilpotency ideal of a $\mathfrak{g}$-module (a representation): I.4.3. Largest nilpotent ideal of a Lie algebra: I.4.4. Law chunk of operation: III.1.11. formal group: I.1, Exercise 24. of infinitesimal operation: III.3.7, III.3.18. Length of an element of a free magma: II.2.1. Levi subalgebra: I.6.8. Levi-Malcev Theorem: I.6.8. Lie algebra: I.1.2. algebra, commutative: I.1.3. algebra, characteristically nilpotent: I.4, Exercise 19. algebra, nilpotent: I.4.1. algebra of a formal group: I.1, Exercise 24. algebra of a Lie group: III.3.7. algebra of a Lie group germ: III.3.18. algebra of endomorphisms of a module: I.1.2. algebra, reductive: I.6.4. algebra, semi-simple: I.6.1. algebra, simple: I.6.2. algebra, solvable: I.5.1. group: III.1.1. Lie’s Theorem: I.5.3. Locally isomorphic group germ: III.1.10. Logarithm: II.6.1, III.7.6. Lower central series: I.1.5.

Magnus algebra: II.5.1. group: II.5.2. Mapping, canonical, of a Lie algebra into its enveloping algebra: I.2.1. Maurer-Cartan formulae: III.3.14, III.3.18.

Möbius function: II, Appendix. inversion formula: II, Appendix. Morphism, Lie group: III.1.2. Lie group germ: III.1.10. Multigraduation of L(I): II.2.6.

Nilpotent Lie algebra: I.4.1, II.2.7. radical of a Lie algebra: I.5.3. Normable algebra: II.7. Normalizer: I.1.4, III.9.4. Normed Lie algebra: II.7, II.8.2.

Opposite algebra: I.1.1. Order of an element in a filtered group: II.4.2.

p-adic Lie group: III.1.1, III.8.1. p-algebra, Lie: I.1, Exercise 20. p-unipotent Lie: I.4, Exercise 23. p-core of a commutative Lie p-algebra: I.1, Exercise 23. p-derivation: I.2, Exercise 7. P-envelope of a nilpotent group: II.4, Exercise 15. p-homomorphism: I.1, Exercise 20. p-ideal: I.1, Exercise 22. P-integer: II.4, Exercise 14. p-mapping: I.1, Exercise 20. p-polynomial: I.7, Exercise 5. P-saturation of a subgroup of a nilpotent group: II.4, Exercise 14. P-torsion group, P-torsion-free group: II.4, Exercise 14. Permutable elements: I.1.3, III.9.3. Poincaré-Birkhoff-Witt Theorem: I.2.7. Polynomial, Lie: II.2.4. mapping: II.2.4. Presentation of a Lie algebra: II.2.3. Primitive element of a bigebra: II.1.2. Product algebra: I.1.1. of Lie groups: III.1.4. semi-direct, of Lie algebras: I.1.8. tensor, of representations: I.3.2, III, Appendix. Pure g-module of species N: I.3.1. representation: I.3.1.

Quasi-subgroup, Lie: III.1.3.

Quotient algebra: I.1.1. of a Lie group: III.1.6. representation: I.3.1.

Radical, nilpotent, of a Lie algebra: I.5.3. of a Lie algebra: I.5.2. of a Lie group: III.9.7. Real Lie group: III.1.1, III.8.1, III.8.2. Reductive Lie algebra: I.6.4. (Lie subalgebra) in a Lie algebra: I.6.6. Relators: II.2.3. Replica of an endomorphism: I.5, Exercise 14. Representation, adjoint: I.3.1. analytic linear, of a Lie group: III.1.2. completely reducible: I.3.1. containing a simple representation $n$ times: I.3.1. dual: I.3.3. faithful: I.3.1. irreducible: I.3.1. obtained by extending the ring of scalars: I.3.8. of a Lie algebra: I.3.1. pure, of species $\sigma$: I.3.1. quotient: I.3.1. semi-simple: I.3.1. simple: I.3.1. Representations, isomorphic: I.3.1. similar: I.3.1. Restriction of scalars, Lie group derived from a Lie group by: III.1.1. Roots of a solvable Lie algebra: III.9, Exercise 17.

Second species, canonical chart of the: III.4.3. species, system of canonical coordinates of the: III.4.3. Section of a vector bundle: III.1.8. Semi-direct product of Lie algebras: I.1.8. product of Lie groups: III.1.4. Semi-simple Lie algebra: I.6.1. Lie group: III.9.8. representation: I.3.1. Separated filtration: II.4.1. Series, composition, joining two subalgebras: I.1, Exercise 14. derived: I.1.5. Lie formal power: II.6.3. lower central, of a Lie algebra: I.1.5, II.2.7. upper central, of a Lie algebra: I.1.6.

Similar representations: I.3.1. Simple component of a semi-simple Lie algebra: I.6.2. Lie algebra: I.6.2. representation: I.3.1. Solvable Lie algebra: I.5.1. Space, cohomology, with values in a $g$-module: I.3, Exercise 12. Lie homogeneous: III.1.6. Special automorphism: I.6.8. Species of an isotypical component, of a pure $g$-module: I.3.1. Standard group: III.7.3. Subalgebra: I.1.1. Lie: I.6.8. reductive in a Lie algebra: I.6.6. subinvariant: I.1, Exercise 14. Subgroup germ, Lie: III.1.10. Lie: III.1.3. Subrepresentation: I.3.1. Sum, direct, of representations: I.3.1. Symmetric algebra of a module: I.2.5.

$t$-th power mapping: III.4.3. Tangent law of composition: III.2.1. Lie subgroup: III.4.5. Theorem, Ado’s: I.7.3. Engel’s: I.4.2. Levi-Malcev: I.6.8. Lie’s: I.5.3. Poincaré-Birkhoff-Witt: I.2.7. Weyl’s: I.6.2. Zassenhaus’s: I.7.2. Transporter: I.1, Exercise 1. Triangular group, upper strict: II.4.6. Trivial extension: I.1.7. $g$-module: I.3.1. vector G-bundle: III.1.8. Trivialization, right (resp. left), of $T(G)$: III.2.1, III.2.2. Two-sided ideal, I.1.1. Type (N), real Lie group of: III.9, Exercise 29.

$u$-primitive elements of a cogebrad: II.1.1. Unipotent endomorphism: III.9.5. Universal covering of a connected Lie group: III.1.9.

Upper central series of a group: II.4, Exercise 18. central series of a Lie algebra: I.1.6.

Weyl’s Theorem: I.6.2.

Zassenhaus’s Theorem: I.7.2.