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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.

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In this excellent introduction to the theory of Lie groups and Lie algebras, three of the leading figures in this area have written up their lectures from an LMS/SERC sponsored short course in 1993. Together these lectures provide an elementary account of the theory that is unsurpassed. In the first part Roger Carter concentrates on Lie algebras and root systems. In the second Graeme Segal discusses Lie groups. And in the final part, Ian Macdonald gives an introduction to special linear groups. Anybody requiring an introduction to the theory of Lie groups and their applications should look no further than this book.

W)(Ug)=f(Ugx). LEMMA 3.1. SF is isomorphic, as left KG-module, to KG® KU M. Proof. The proof is straightforward. Let x be a 1-dimensional representation of H with values in K, viz. a linear character of H. Define the subspace SF X of SF by J$ = {/ £ P:f{Uhg) = X (h)f(Ug), heH,geG}; is a i£6r-submodule of &'. Now the linear character % of H can be extended to a linear character of B with U in the kernel. Let M v be the 1-dimensional left i£jB-module affording the character %. LEMMA 3.2. J^ is isomorphic, as left KG-module, to KG® KB M X .Proof. The proof is straightforward. LEMMA 3.3. Suppose that the characteristic of K is prime to H and that K is a splitting field for H. Then ^ = ® x^rx , summed over all l-dimensional representations % of H. Proof. Consider the space of K-valued functions on H. This space has two natural bases, the set {x} of linear characters of H and the set of characteristic functions {«/» A }, where fl if h' = h,

One of the main aims of workers in the theory of groups has always been the determination of all finite simple groups. For simple groups may be regarded as the fundamental building blocks out of which finite groups are constructed. The cyclic groups of prime order are trivial examples of simple groups, and are the only simple groups which are Abelian. The first examples of non-Abelian simple groups were discovered by Galois, who showed that the alternating group A n is simple if n ^ 5. The group A 5 of order 60 is the smallest non-Abelian simple group.Further examples of finite simple groups are the so-called classical groups, i.e. the linear, symplectic, orthogonal and unitary groups over finite fields, which were first introduced by Jordan [8] and studied in detail by Dickson [4]. These groups are defined as follows. GF(q) denotes the Galois field with q elements, where q is any prime power.(i) The linear groups. Let GL n (q) be the group of all non-singular nxn matrices over GF(q), SL n (q) the subgroup of matrices of determinant 1, and PSL n (q) the factor group of SL n (q) by its centre. Then PSL n (q) is simple when n^2. (ii) The symplectic groups. Let V be a finite dimensional vector space over GF(q) and f(x, y) be a non-singular bilinear form on V with values in GF{q) satisfying f(x, x) = 0 for all xeV. T h e n / is skew-symmetiic, i.e. f(y, x) --f{x, y) for all x,yeV. The existence of such a bilinear form implies that the dimension of V is even. Let dim V = 2n. Then a basis 2n 2n e l5 e 2 , ..., e 2n can be chosen for F such that, if x = 2 & ^ y -S ^ e i} then •i=i i=i n /(«» V) = .2 (£< Vn+i -Vi £n+i)-Let Sp 2n (q) be the group of all non-singular linear transformations of V into itself satisfying f(x, y)=f(Tx, Ty), and let PSp 2n (q) be the factor group of Sp 2n {q) by its centre. Then PSp 2n (q) is simple and PSp 2 (q) is isomorphic to PSL 2 (q).

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