Lie Algebras of Finite and Affine Type

This book provides a thorough but relaxed mathematical treatment of Lie algebras.

Roger Carter (Author)

9780521851381, Cambridge University Press

Hardback, published 27 October 2005

652 pages, 10 b/w illus.
23.6 x 15.9 x 3.5 cm, 1.04 kg

"This monograph provides a crystal clear exposition of the theory of finite-dimensional simple Lie algebras and a nice introduction to (infinite-dimensional) Kac - Moody algebras of affine type. Also, an excellent course on Lie algebras could be constructed starting from the book."
Daniel Beltita, Institute of Mathematics, Romanian Academy, SIAM Review

Lie algebras have many varied applications, both in mathematics and mathematical physics. This book provides a thorough but relaxed mathematical treatment of the subject, including both the Cartan-Killing-Weyl theory of finite dimensional simple algebras and the more modern theory of Kac-Moody algebras. Proofs are given in detail and the only prerequisite is a sound knowledge of linear algebra. The first half of the book deals with classification of the finite dimensional simple Lie algebras and of their finite dimensional irreducible representations. The second half introduces the theory of Kac-Moody algebras, concentrating particularly on those of affine type. A brief account of Borcherds algebras is also included. An Appendix gives a summary of the basic properties of each Lie algebra of finite and affine type.

1. Basic concepts
2. Representations of soluble and nilpotent Lie algebras
3. Cartan subalgebras
4. The Cartan decomposition
5. The root systems and the Weyl group
6. The Cartan matrix and the Dynkin diagram
7. The existence and uniqueness theorems
8. The simple Lie algebras
9. Some universal constructions
10. Irreducible modules for semisimple Lie algebras
11. Further properties of the universal enveloping algebra
12. Character and dimension formulae
13. Fundamental modules for simple Lie algebras
14. Generalized Cartan matrices and Kac-Moody algebras
15. The classification of generalised Cartan matrices
16 The invariant form, root system and Weyl group
17. Kac-Moody algebras of affine type
18. Realisations of affine Kac-Moody algebras
19. Some representations of symmetrisable Kac-Moody algebras
20. Representations of affine Kac-Moody algebras
21. Borcherds Lie algebras
Appendix.

Subject Areas: Physics [PH], Algebra [PBF]