Quantum double construction in the C*-algebra setting of certain Heisenberg-type quantum groups

Abstract

In this paper, we carry out the ``quantum double construction'' of the specific quantum groups we constructed earlier, namely, the ``quantum Heisenberg group algebra'' (A,Δ) and its dual, the ``quantum Heisenberg group'' (\hat{A},\hatΔ). Our approach will be by constructing a suitable multiplicative unitary operator. In this way, we are able to retain the C*-algebra framework, and thus able to carry out our construction within the category of locally compact quantum groups. This construction is a kind of a generalized crossed product. To establish that the quantum double we obtain is indeed a locally compact quantum group, we will also discuss the dual of the quantum double and the Haar weights on both of these double objects. Towards the end, we also include a construction of a (quasitriangular) ``quantum universal R-matrix''.