Lie Groups and Lie Algebras

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1. Front matter
   1. Introduction to chapters IV, V and VI
   2. To the reader
2. ILie Algebras
   1. Lie Algebras
   2. § 1. Definition of Lie algebras20
   3. § 2. Enveloping algebra of a Lie algebra22
   4. § 3. Representations24
   5. § 4. Nilpotent Lie algebras18
   6. § 5. Solvable Lie algebras27
   7. § 6. Semi-simple Lie algebras50
   8. § 7. Ado’s Theorem6
3. IIFree Lie algebras
   1. Free Lie algebras
   2. § 1. Enveloping bigebra of a Lie algebra24
   3. § 2. Free Lie algebras32
   4. § 3. Enveloping algebra of the free Lie algebra11
   5. § 4. Central filtrations12
   6. § 5. Magnus algebras14
   7. § 6. The Hausdorff series14
   8. § 7. Convergence of the Hausdorff series (real or complex case)4
   9. § 8. Convergence of the Hausdorff series (ultrametric case)9
   10. App. 0. Möbius function1
4. IIILie groups
   1. Lie groups
   2. § 1. Lie groups54
   3. § 2. Group of tangent vectors to a Lie group10
   4. § 3. Passage from a Lie group to its Lie algebra111
   5. § 4. Passage from Lie algebras to Lie groups40
   6. § 5. Formal calculations in Lie groups5
   7. § 6. Real and complex Lie groups71
   8. § 7. Lie groups over an ultrametric field26
   9. § 8. Lie groups over R or Qp9
   10. § 9. Commutators, centralizers and normalizers in a Lie group54
   11. § 10. The automorphism group of a Lie group13
   12. App. 0. Operations on linear representations
   13. Historical Note
5. IVCoxeter groups and Tits systems
   1. Coxeter groups and Tits systems
   2. § 1. Coxeter Groups30
   3. § 2. Tits Systems30
   4. App. 0. Graphs6
6. VGroups generated by reflections
   1. Groups generated by reflections
   2. § 1. Hyperplanes, chambers and facets18
   3. § 2. Reflections16
   4. § 3. Groups of displacements generated by reflections37
   5. § 4. The geometric representation of a Coxeter group25
   6. § 5. Invariants in the symmetric algebra24
   7. § 6. The Coxeter transformation14
   8. App. 0. Complements on linear representations5
7. VIRoot systems
   1. Root systems
   2. § 1. Root systems79
   3. § 2. Affine Weyl group19
   4. § 3. Exponential invariants13
   5. § 4. Classification of root systems19
   6. Historical Note
8. VIICartan subalgebras and regular elements
   1. Cartan subalgebras and regular elements
   2. § 1. Primary decomposition of linear representations25
   3. § 2. Cartan subalgebras and regular elements of a Lie algebra40
   4. § 3. Conjugacy theorems17
   5. § 4. Regular elements of a Lie group18
   6. § 5. Decomposable linear Lie algebras29
   7. App. 1. Polynomial maps and Zariski topology7
   8. App. 2. A connectedness property3
9. VIIISplit semi-simple Lie algebras
   1. Split semi-simple Lie algebras
   2. § 1. The Lie algebra sl(2, k) and its representations23
   3. § 2. Root system of a split semi-simple Lie algebra28
   4. § 3. Subalgebras of split semi-simple Lie algebras28
   5. § 4. Split semi-simple Lie algebra defined by a reduced root system23
   6. § 5. Automorphisms of a semi-simple Lie algebra25
   7. § 6. Modules over a split semi-simple Lie algebra17
   8. § 7. Finite dimensional modules over a split semi-simple Lie algebra46
   9. § 8. Symmetric invariants24
   10. § 9. The formula of Hermann Weyl12
   11. § 10. Maximal subalgebras of semi-simple Lie algebras8
   12. § 11. Classes of nilpotent elements and sl2 -triplets29
   13. § 12. Chevalley orders34
   14. § 13. Classical splittable simple Lie algebras2
10. IXCompact real Lie groups
    1. Compact real Lie groups
    2. § 1. Compact Lie algebras13
    3. § 2. Maximal tori of compact Lie groups32
    4. § 3. Compact forms of complex semi-simple Lie algebras17
    5. § 4. Root system associated to a compact group60
    6. § 5. Conjugacy classes26
    7. § 6. Integration on compact Lie groups25
    8. § 7. Irreducible representations of connected compact Lie groups28
    9. § 8. Fourier transform9
    10. § 9. Compact Lie groups operating on manifolds19
    11. App. 1. Structure of compact groups9
    12. App. 2. Representations of real, complex or quaternionic type6
11. Indexes
    1. Index of notation
    2. Index of notation
    3. Index of notation
    4. Index of terminology
    5. Index of terminology
    6. Index of terminology