Basic hypergeometric functions and covariant spaces for even-dimensional representations of Uq[osp(1/2)

Abstract

Representations of the quantum superalgebra Uq[osp(1/2)] and their relations to the basic hypergeometric functions are investigated. We first establish Clebsch–Gordan decomposition for the superalgebra Uq[osp(1/2)] in which the representations having no classical counterparts are incorporated. Formulae for these Clebsch–Gordan coefficients are derived, and is observed that they may be expressed in terms of the Q-Hahn polynomials. We next investigate representations of the quantum supergroup OSpq(1/2) which are not well defined in the classical limit. Employing the universal -matrix, the representation matrices are obtained explicitly, and found to be related to the little Q-Jacobi polynomials. Characteristically, the relation Q = −q is satisfied in all cases. Using the Clebsch–Gordan coefficients derived here, we construct new noncommutative spaces that are covariant under the coaction of the even-dimensional representations of the quantum supergroup OSpq(1/2).