Casimir invariants, characteristic identities, and Young diagrams for color algebras and superalgebras

Abstract

The generalized commutation relations satisfied by generators of the general linear, special linear, and orthosymplectic color (super) algebras are presented in matrix form. Tensor operators, including Casimir invariants, are constructed in the enveloping algebra. For the general, special linear and orthosymplectic cases, eigenvalues of the quadratic and higher Casimir invariants are given in terms of the highest-weight vector. Correspondingly, characteristic polynomial identities, satisfied by the matrix of generators, are obtained in factorized form. Classes of finite-dimensional representations are identified using Young diagram techniques, and dimension, branching, and product rules for these are given. Finally, the connection between color (super) algebras and generalized particle statistics is elucidated.