*-Representations of a Quantum Heisenberg Group Algebra

In our earlier work, we constructed a specific non-compact quantum group whose quantum group structures have been constructed on a certain twisted group C*-algebra. In a sense, it may be considered as a ``quantum Heisenberg group C*-algebra''. In this paper, we will find, up to equivalence, all of its irreducible *-representations. We will point out the Kirillov type correspondence between the irreducible representations and the so-called ``dressing orbits''. By taking advantage of its comultiplication, we will then introduce and study the notion of ``inner tensor product representations''. We will show that the representation theory satisfies a ``quasitriangular'' type property, which does not appear in ordinary group representation theory.