Ju l 2 00 6 Universal T - matrix , Representations of OSp q ( 1 / 2 ) and Little Q - Jacobi Polynomials

We obtain a closed form expression of the universal T-matrix encapsulating the duality of the quantum superalgebra U q [osp(1/2)] and the corresponding su-pergroup OSp q (1/2). The finite dimensional representations of the quantum su-pergroup OSp q (1/2) are readily constructed employing the said universal T-matrix and the known finite dimensional representations of the dually related deformed U q [osp(1/2)] superalgebra. Proceeding further, we derive the product law, the recurrence relations and the orthogonality of the representations of the quantum su-pergroup OSp q (1/2). It is shown that the entries of these representation matrices are expressed in terms of the little Q-Jacobi polynomials with Q = −q. Contrasting the present example, this polynomial structure just does not appear for the classical OSp(1/2). Two mutually complementary singular maps of the universal T-matrix on the universal R-matrix are also presented.