Casimir invariants and characteristic identities for generators of the general linear, special linear and orthosymplectic graded Lie algebras

Abstract

We present the commutation and anticommutation relations, satisfied by the generators of the graded general linear, special linear and orthosymplectic Lie algebras, in canonical two-index matrix form. Tensor operators are constructed in the enveloping algebra, including powers of the matrix of generators. Traces of the latter are shown to yield a sequence of Casimir invariants. The transformation properties of vector operators under these algebras are also exhibited. The eigenvalues of the quadratic Casimir invariants are given for the irreducible representations of ggl(m ‖ n), gsl(m ‖ n), and osp(m ‖ n) in terms of the highest-weight vector. In such representations, characteristic polynomial identities of order (m+n), satisfied by the matrix of generators, are obtained in factorized form. These are used in each case to determine the number of independent Casimir invariants of the trace form.