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

Abstract

In this paper, we will study matrix elements of unitary representations of the quantum group SUq (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 SUq (1,1). Throughout this paper, Z denotes the set of integers, ℕ the set of non-negative integers.