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,∆) and its dual, the “quantum Heisenberg group” (Â, ∆̂). 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”. Introduction. The quantum double construction, which was originally introduced by Drinfeld in the mid-80’s for (finite-dimensional) Hopf algebras [7], is among the most celebrated methods of constructing non-commutative and non-cocommutative Hopf algebras. Even in the case of an ordinary group, equivalently for the algebra of (continuous) functions C(G), the quantum double construction leads to an interesting crossed product algebra C(G)⋊α G, where α is the conjugation [17], [14]. We wish to carry out a similar construction in the framework of (C∗algebraic) locally compact quantum groups. This is not totally a new endeavor: As early as in [19], Podles and Woronowicz has constructed their example of a quantum Lorentz group, by considering the quantum double of the compact quantum group SUμ(2); Baaj and Skandalis [1] have a version in the context of the multiplicative unitary operators; And more recently, Yamanouchi [22] has made this more systematic while Baaj and Vaes [2] considers a more generalized framework of double crossed products. On the other hand, as is the case for a lot of going-ons in the study of locally compact quantum groups (especially for the non-compact ones), there have been only a handful of work done on actual examples. In this paper, we will obtain the quantum double object of the “quantum Heisenberg group algebra” (A,∆) and the “quantum Heisenberg group” (Â, ∆̂), which are the specific non-compact quantum groups we constructed earlier (See [9], [12], [11].). The quantum double will be also a valid locally compact quantum group.

Cite this paper

@inproceedings{Kahng2004QuantumDC, title={Quantum Double Construction in the C∗-algebra Setting of Certain Heisenberg-type Quantum Groups}, author={B. Kahng}, year={2004} }