Lie Algebras of Finite and Affine Type (Cambridge Studies in Advanced Mathematics, Series Number 96)

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. Cover......Page 1 Half-title......Page 2 Title......Page 5 Copyright......Page 6 Dedication......Page 7 Contents......Page 9 Preface......Page 15 1.1 Elementary properties of Lie algebras......Page 21 1.2 Representations and modules......Page 25 1.3 Abelian, nilpotent and soluble Lie algebras......Page 27 2.1 Representations of soluble Lie algebras......Page 31 2.2 Representations of nilpotent Lie algebras......Page 34 3.1 Existence of Cartan subalgebras......Page 43 3.2 Derivations and automorphisms......Page 45 3.3 Ideas from algebraic geometry......Page 47 3.4 Conjugacy of Cartan subalgebras......Page 53 4.1 Some properties of root spaces......Page 56 4.2 The Killing form......Page 59 4.3 The Cartan decomposition of a semisimple Lie algebra......Page 65 4.4 The Lie algebra.........Page 72 5.1 Positive systems and fundamental systems of roots......Page 76 5.2 The Weyl group......Page 79 5.3 Generators and relations for the Weyl group......Page 85 6.1 The Cartan matrix......Page 89 6.2 The Dynkin diagram......Page 92 6.3 Classification of Dynkin diagrams......Page 94 6.4 Classification of Cartan matrices......Page 100 7.1 Some properties of structure constants......Page 108 7.2 The uniqueness theorem......Page 113 7.3 Some generators and relations in a simple Lie algebra......Page 116 7.4 The Lie algebras L(A) and L(A)......Page 118 7.5 The existence theorem......Page 125 8 The simple Lie algebras......Page 141 8.1 Lie algebras of type A1......Page 142 8.2 Lie algebras of type D1......Page 144 8.3 Lie algebras of type B1......Page 148 8.4 Lie algebras of type C1......Page 152 8.5 Lie algebras of type G2......Page 155 8.6 Lie algebras of type F4......Page 158 8.7 Lie algebras of types E6, E7, E8......Page 160 8.8 Properties of long and short roots......Page 165 9.1 The universal enveloping algebra......Page 172 9.2 The Poincaré–Birkhoff–Witt basis theorem......Page 175 9.3 Free Lie algebras......Page 180 9.4 Lie algebras defined by generators and relations......Page 183 9.5 Graph automorphisms of simple Lie algebras......Page 185 10.1 Verma modules......Page 196 10.2 Finite dimensional irreducible modules......Page 206 10.3 The finite dimensionality criterion......Page 210 11.1 Relations between the enveloping algebra and the symmetric algebra......Page 221 11.2 Invariant polynomial functions......Page 227 11.3 The structure of the ring of polynomial invariants......Page 236 11.4 The Killing isomorphisms......Page 242 11.5 The centre of the enveloping algebra......Page 246 11.6 The Casimir element......Page 258 12.1 Characters of L-modules......Page 261 12.2 Characters of Verma modules......Page 264 12.3 Chambers and roots......Page 266 12.4 Composition factors of Verma modules......Page 275 12.5 Weyl’s character formula......Page 278 12.6 Complete reducibility......Page 282 13.1 An alternative form of Weyl’s dimension formula......Page 287 13.2 Fundamental modules for A1......Page 288 13.3 Exterior powers of modules......Page 290 13.4 Fundamental modules for B1 and D1......Page 294 13.5 Clifford algebras and spin modules......Page 301 13.6 Fundamental modules for C1......Page 312 13.7 Contraction maps......Page 315 13.8 Fundamental modules for exceptional algebras......Page 323 14.1 Realisations of a square matrix......Page 339 14.2 The Lie algebra L(A) associated with a complex matrix......Page 342 14.3 The Kac–Moody algebra L(A)......Page 351 15.1 A trichotomy for indecomposable GCMs......Page 356 15.2 Symmetrisable generalised Cartan matrices......Page 364 15.3 The classification of affine generalised Cartan matrices......Page 371 16.1 The invariant bilinear form......Page 380 16.2 The Weyl group of a Kac–Moody algebra......Page 391 16.3 The roots of a Kac–Moody algebra......Page 397 17.1 Properties of the affine Cartan matrix......Page 406 Summary......Page 412 17.2 The roots of an affine Kac–Moody algebra......Page 414 17.3 The Weyl group of an affine Kac–Moody algebra......Page 424 18.1 Loop algebras and central extensions......Page 436 18.2 Realisations of untwisted affine Kac–Moody algebras......Page 441 18.3 Some graph automorphisms of affine algebras......Page 446 18.4 Realisations of twisted affine algebras......Page 449 Comments on notation......Page 471 19.1 The category of L(A)-modules......Page 472 19.2 The generalised Casimir operator......Page 479 19.3 Kac’ character formula......Page 486 19.4 Generators and relations for symmetrisable algebras......Page 494 20.1 Macdonald’s identities......Page 504 20.2 Specialisations of Macdonald’s identities......Page 511 20.3 Irreducible modules for affine algebras......Page 514 20.4 The fundamental modules for L(Ã 1)......Page 524 20.5 The basic representation......Page 528 21.1 Definition and examples of Borcherds algebras......Page 539 21.2 Representations of Borcherds algebras......Page 544 21.3 The Monster Lie algebra......Page 550 Summary pages – explanation......Page 560 Notation......Page 630 Bibliography of books on Lie algebras......Page 639 Bibliography of articles on Kac–Moody algebras......Page 641 Index......Page 649 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 Appendix provides a summary of the basic properties of each Lie algebra of finite and affine type. Lie algebras have many varied applications, both in mathematics and mathematical physics. This book provides a thorough but relaxed mathematical treatment of the subject. Proofs are given in detail and the only prerequisite is a sound knowledge of linear algebra. A detailed Appendix is included. A Lie algebra is a vector space L over a field k on which a multiplication L x L L (x, y) [xy] is defined satisfying the following axioms: (i) (x, y) [xy] is linear in x and in y; (ii) [xx] = 0 for all x L; (iii) [[xy]z] + [[yz]x] + [[zx]y] = 0 for all x, y, z L.