Universal T-matrix, representations of OSpq(1∕2) and little Q-Jacobi polynomials

Abstract

We obtain a closed form expression of the universal T-matrix encapsulating the duality between the quantum superalgebra Uq[osp(1∕2)] and the corresponding supergroup OSpq(1∕2). The classical q→1 limit of this universal T-matrix yields the group element of the undeformed OSpq(1∕2) supergroup. The finite dimensional representations of the quantum supergroup OSpq(1∕2) are readily constructed employing the above-mentioned universal T-matrix and the known finite dimensional representations of the dually related deformed Uq[osp(1∕2)] superalgebra. Proceeding further, we derive the product law, the recurrence relations, and the orthogonality of the representations of the quantum supergroup OSpq(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. Two mutually complementary singular maps of the universal T-matrix on the universal R-matrix are also presented.