Matrix Elements of Unitary Representation of the Quantum Group SU_q (1,1) and the Basic Hypergeometric Functions

Abstract

In this paper, we will study matrix elements of unitary representations of the quantum group SU_q (1,1). We begin with classification of real forms of the universal quantum enveloping algebra U _q (sl(2)) (§1), and next consider the structure of a topological quantum group A associated with this algebra (§2). In §3, we “exponentiate” a family of infinite dimensional representations of U _q(sl(2)), and determine the matrix elements in A in terms of the basic hypergeometric functions in the q-analogue analysis. The differential representation of U _q (sl(2)) in A provides us a fundamental tool for this procedure. In § 4, introducing the unitary structure of the real form U _q (su(1, 1)), we classify series of unitary representations of the quantum group SU_q (1,1). Throughout this paper, Z denotes the set of integers, ℕ the set of non-negative integers.