Notes on Lie Algebras
(Cartan sub Lie algebra, roots, Weyl group, Dynkin diagram, . . . ) and the classification, as found by Killing and Cartan (the list of all semisimple Lie algebras consists of (1) the special- linear ones, i. e. all matrices (of any fixed dimension) with trace 0, (2) the orthogonal ones, i. e. all s...
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| Format: | eBook |
| Language: | English |
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New York, NY
Springer New York
1990, 1990
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| Edition: | 2nd ed. 1990 |
| Series: | Universitext
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| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 1 Generalities
- 1.1 Basic definitions, examples
- 1.2 Structure constants
- 1.3 Relations with Lie groups
- 1.4 Elementary algebraic concepts
- 1.5 Representations; the Killing form
- 1.6 Solvable and nilpotent
- 1.7 Engel’s theorem
- 1.8 Lie’s theorem
- 1.9 Cartan’s first criterion
- 1.10 Cartan’s second criterion
- 1.11 Representations of A1
- 1.12 Complete reduction for A1
- 2 Structure Theory
- 2.1 Cartan subalgebra
- 2.2 Roots
- 2.3 Roots for semisimple g
- 2.4 Strings
- 2.5 Cartan integers
- 2.6 Root systems, Weyl group
- 2.7 Root systems of rank two
- 2.8 Weyl-Chevalley normal form, first stage
- 2.9 Weyl-Chevalley normal form
- 2.10 Compact form
- 2.11 Properties of root systems
- 2.12 Fundamental systems
- 2.13 Classification of fundamental systems
- 2.14 The simple Lie algebras
- 2.15 Automorphisms
- 3 Representations
- 3.1 The Cartan-Stiefel diagram
- 3.2 Weights and weight vectors
- 3.3 Uniqueness and existence
- 3.4 Complete reduction
- 3.5 Cartan semigroup; representation ring
- 3.6 The simple Lie algebras
- 3.7 The Weyl character formula
- 3.8 Some consequences of the character formula
- 3.9 Examples
- 3.10 The character ring
- 3.11 Orthogonal and symplectic representations
- References
- Symbol Index