§1. Preliminaries A finite dimensional simple Lie algebra over the field C of complex numbers is a member of one of four infinite families of Lie algebras, {Ar | r 1}, {Br \ r 3}, {Cr \ r 2}, {Dr | r 4}, or it is one of five exceptional Lie algebras 6?2, -Fi, Ee, ET, Eg. In this work we will be concerned exclusively with algebras in the four infinite families. We call these algebras classical Lie algebras to distinguish them from the exceptional ones and to emphasize their con- nections with certain classical groups. In this first section we review the basic facts about the structure of the classical Lie algebras, their associated Lie groups, and their finite dimensional representations. We adopt a constructive approach, illustrating the results as much as possible by concrete matrix realizations. For more detailed expositions we refer the interested reader to Humphreys [Hu] and Miller [Mi]. Let Xr denote one of the finite dimensional complex simple Lie algebras of rank r and type X = A, B, C, or D. It is important for us to view each of these Lie algebras as the Lie algebra of a certain classical complex Lie group. The Lie group is a subgroup of the general linear group GL(n,C) of invertible n x n complex matrices for some n, and the Lie algebra is a subalgebra of the Lie algebra gl(n, C) of all n x n complex matrices under the commutator product [x, y] = xy yx. In particular, Ar is the Lie algebra of the normal subgroup SL(r -j-1, C) of all matrices in GL(r + 1,C) having determinant + 1 . The algebra Ar can be realized as the space s/(r-fl, C) of all (r-f 1) x (r + 1) traceless complex matrices. As such it has a natural action by matrix multiplication on the vector space V of (r + 1) X 1 complex matrices. In a similar fashion, the algebra 2?r, C r , or Dr admits a faithful representation on the space V = C 2 r + 1 , C 2 r , or C 2 r , respectively. We denote the dimension of the defining representation V for the algebra Xr by N(Xr) so that V is the C-vector space of N(Xr) x 1 matrices where f r + 1 (A) (1.1) N(Xr) = { 2r + 1 (B) { 2r (C) or (D). (Our equation labelling convention is to use (A) to indicate that the stated equation holds when the algebra is of type X = A, etc. and to use (X) to mean that the equation is true for all four types of algebras.) For X B,C, and D we define a bilinear form &(.,.) on V by specifying an N(Xr) x N(Xr) matrix J and setting (1.2) b(v, w) = vt J w for v,w G V, where the superscript "t" denotes the transpose operation. The matrix J for X = B is given by (1.3) 4
Previous Page Next Page