Lie Algebras of Finite and Affine Type

Transcription

1 Lie Algebras of Finite and Affine Type R. W. CARTER Mathematics Institute University of Warwick CAMBRIDGE UNIVERSITY PRESS

2 Preface page xiii Basic concepts Elementary properties of Lie algebras Representations and modules Abelian, nilpotent and soluble Lie algebras 7 Representations of soluble and nilpotent Lie algebras Representations of soluble Lie algebras Representations of nilpotent Lie algebras 14 Cartan subalgebras Existence of Cartan subalgebras Derivations and automorphisms Ideas from algebraic geometry Conjugacy of Cartan subalgebras 33 The Cartan decomposition Some properties of root spaces The Killing form The Cartan decomposition of a semisimple Lie algebra The Lie algebra I n (C) 52 The root system and the Weyl group Positive systems and fundamental systems of roots The Weyl group Generators and relations for the Weyl group 65. vn

3 viii 6 The Cartan matrix and the Dynkin diagram The Cartan matrix The Dynkin diagram Classification of Dynkin diagrams Classification of Cartan matrices 80 7 The existence and uniqueness theorems Some properties of structure constants The uniqueness theorem Some generators and relations in a simple Lie algebra The Lie algebras L(A) and l(a) The existence theorem The simple Lie algebras Lie algebras of type A, Lie algebras of type D, Lie algebras of type B, Lie algebras of type C, Lie algebras of type G Lie algebras of type F Lie algebras of types E b, E 7, Properties of long and short roots Some universal constructions The universal enveloping algebra The Poincare-Birkhoff-Witt basis theorem Free Lie algebras Lie algebras defined by generators and relations Graph automorphisms of simple Lie algebras Irreducible modules for semisimple Lie algebras Verma modules Finite dimensional irreducible modules The finite dimensionality criterion Further properties of the universal enveloping algebra Relations between the enveloping algebra and the symmetric algebra Invariant polynomial functions The structure of the ring of polynomial invariants The Killing isomorphisms 222

4 ix 11.5 The centre of the enveloping algebra The Casimir element Character and dimension formulae Characters of L-modules Characters of Verma modules Chambers and roots Composition factors of Verma modules Weyl's character formula Complete reducibility Fundamental modules for simple Lie algebras An alternative form of Weyl's dimension formula Fundamental modules for A, Exterior powers of modules Fundamental modules for B, and D, 21A 13.5 Clifford algebras and spin modules Fundamental modules for C, Contraction maps Fundamental modules for exceptional algebras Generalised Cartan matrices and Kac-Moody algebras Realisations of a square matrix The Lie algebra L{A) associated with a complex matrix The Kac-Moody algebra L(A) The classification of generalised Cartan matrices A trichotomy for indecomposable GCMs Symmetrisable generalised Cartan matrices The classification of affine generalised Cartan matrices The invariant form, Weyl group and root system The invariant bilinear form The Weyl group of a Kac-Moody algebra The roots of a Kac-Moody algebra 377

5 x 17 Kac-Moody algebras of affine type Properties of the affine Cartan matrix The roots of an affine Kac-Moody algebra The Weyl group of an affine Kac-Moody algebra Realisations of affine Kac-Moody algebras Loop algebras and central extensions Realisations of untwisted affine Kac-Moody algebras Some graph automorphisms of affine algebras Realisations of twisted affine algebras Some representations of symmetrisable Kac-Moody algebras The category 0 of L(A)-modules The generalised Casimir operator Kac'character formula Generators and relations for symmetrisable algebras Representations of affine Kac-Moody algebras Macdonald's identities Specialisations of Macdonald's identities Irreducible modules for affine algebras The fundamental modules for L(A t ) The basic representation Borcherds Lie algebras Definition and examples of Borcherds algebras Representations of Borcherds algebras The Monster Lie algebra 530 Appendix 540 Summary pages - explanation 540 Type A, 543 Type B, 545 Type C, 547 Type D, 549 Type Type E Type E s 555 Type F Type G 2 559