§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)
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