Lie Algebras, Part 2, Volume 7

Preface. 18. Extensions of Lie algebras. 19. Explicit construction of affine Kac-Moody algebras. 20. Representations - enveloping algebra techniques. 21. The Weyl group and integrable representations. 22. More on representations. 23. Characters and multiplicities. 24. Quarks, leptons and gauge fields. 25. Lie algebras of infinite matrices. 26. Representations of loop algebras. 27. KP-hierarchies. 28. Conformal symmetry.

This is the long awaited follow-up to Lie Algebras, Part I which covered a major part of the theory of Kac-Moody algebras, stressing primarily their mathematical structure. Part II deals mainly with the representations and applications of Lie Algebras and contains many cross references to Part I.

The theoretical part largely deals with the representation theory of Lie algebras with a triangular decomposition, of which Kac-Moody algebras and the Virasoro algebra are prime examples. After setting up the general framework of highest weight representations, the book continues to treat topics as the Casimir operator and the Weyl-Kac character formula, which are specific for Kac-Moody algebras.

The applications have a wide range. First, the book contains an exposition on the role of finite-dimensional semisimple Lie algebras and their representations in the standard and grand unified models of elementary particle physics. A second application is in the realm of soliton equations and their infinite-dimensional symmetry groups and algebras. The book concludes with a chapter on conformal field theory and the importance of the Virasoro and Kac-Moody algebras therein.