Bibliothèque de L'IMIST/CNRST

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.

Pt. 2 authors: E.A. de Kerf, G.G.A. Bäuerle, A.P.E. ten Kroode.

Pt. 2 lacks distributor statement.

Includes bibliographical references and indexes.

Description based on print version record.

Cover -- Contents -- Chapter 18. Extensions of Lie algebras -- 18.1 Generalities -- 18.2 2-Cocycles on Lie algebras -- 18.3 Structure constants and central extensions -- 18.4 Central extensions of simple Lie algebras -- 18.5 Central extensions of loop algebras -- 18.6 The Witt algebra and the Virasoro algebra -- 18.7 Projective representations and central extensions -- Chapter 19. Explicit construction of affine Kac-Moody algebras -- 19.1 Main features of affine Kac-Moody algebras -- 19.2 Loop algebras reconsidered -- 19.3 Chevalley generators of L -- 19.4 Realization of A1(1) -- Chapter 20. Representations-nenveloping algebr a techniques -- 20.1 Poincaré-Birkhoff-Witt theo rem -- 20.2 Highest weight modules -- 20.3 Existence of highest weight modules and Verma modules -- 20.4 More on highest weight modules -- 20.5 Example -- The highest weight representations of sl(2, C) -- Chapter 21. The Weyl group and integrable representations -- 21.1 The Weyl group revisited -- 21.2 Weyl chambers and the Tits cone -- 21.3 Integrable representations -- 21.4 Integrable highest weight representations -- Chapter 22. More on representations -- 22.1 Fundamental highest weight modules -- 22.2 Bilinear forms on semisimple Lie algebras -- 22.3 Casimir operators -- 22.4 Generalized Casimir operators -- 22.5 Lie algebras with a triangular decomposition -- 22.6 Lowest weight modules -- 22.7 Contravariant bilinear form BA -- 22.8 Hermitian form HA on L(A) -- Chapter 23. Characters and multiplicities -- 23.1 Freudenthal's formula -- 23.2 Characters -- 23.3 Weyl-Kac character formula -- 23.4 Multiplicities of roots -- 23.5 Generalized Kostant formula -- 23.6 Weyl's dimension formula -- 23.7 The q-dimension -- Chapter 24. Quarks, leptons and gauge fields -- 24.1 Particle multiplets and symmetries -- 24.2 Standard model -- 24.3 Complex and real representations -- 24.4 Unified models -- 24.5 Anomalies -- Chapter 25. Lie algebras of infinite matrices -- 25.1 The algebras sl(8, C) and gl(8, C) -- 25.2 Completions -- 25.3 The fundamental representations of sl(n, C) -- 25.4 The semi-infinite wedge space -- 25.5 Fermions -- 25.6 The energy spectrum of A8C8 -- 25.7 Bosons -- 25.8 Boson-fermion correspondence I -- 25.9 Boson-fermion correspondence II -- Chapter 26. Representations of loop algebras -- 26.1 Embedding of loop algebras -- 26.2 Principal Heisenberg subalgebra -- 26.3 Automorphisms of finite order -- 26.4 The principal realization of the fundamental modules -- 26.5 Other Heisenberg subalgebras -- 26.6 Realization of type n, I -- 26.7 Multicomponent boson-fermion correspondence -- 26.8 Realization of type n, II -- 26.9 Other loop algebras -- Chapter 27. KP-hierarchies -- 27.1 Finite-dimensional Grassmannians -- 27.2 Infinite-dimensional Grassmannians -- 27.3 Completions and extensions -- 27.4 The KP-hierarchy -- 27.5 Multicompo.

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