ON THE REPRESENTATION-THEORY OF QUANTUM HEISENBERG-GROUP AND ALGEBRA

We show that the quantum Heisenberg group H-q(1) and its *-Hopf algebra structure can be obtained by means of contraction from quantum SUq(2) group. Its dual Hopf algebra is the quantum Heisenberg algebra U-q(h(1)). We derive left and right regular rep resentations for U-q(h(1)) as acting on its dual H-q(1). Imposing conditions on the right representation, the left representation is reduced to an irreducible holomorphic representation with an associated quantum coherent state. Realized in the Bargmann-Hilbert space of analytic functions the unitarity of regular representation is also shown. By duality, left and right regular representations for quantum Heisenberg group with the quantum Heisenberg algebra as representation module are also constructed. As before reduction of group left representations leads to finite dimensional irreducible ones for which the intertwinning operator is also investigated.