We put two C * -algebras together in a noncommutative tensor product using quantum group coactions on them and a bicharacter relating the two quantum groups that act. We describe this twisted tensor product in two equivalent ways, based on certain pairs of quantum group representations and based on covariant Hilbert space representations, respectively. We establish basic properties of the twisted tensor product and study some examples.2010 Mathematics Subject Classification. 81R50 (46L05 46L55).
We construct a family of q-deformations of SU(2) for complex parameters q = 0. For real q, the deformation coincides with Woronowicz' compact quantum SUq(2) group. For q / ∈ R, SUq (2) is only a braided compact quantum group with respect to a certain tensor product functor for C * -algebras with an action of the circle group.2010 Mathematics Subject Classification . 81R50 (46L55, 46L06). Key words and phrases. braided compact quantum group; SUq(2); Uq(2). S. Roy was supported by a Fields-Ontario Postdoctoral fellowship. St.L. Woronowicz was supported by the Alexander von Humboldt-Stiftung.
Abstract. C * -quantum groups with projection are the noncommutative analogues of semidirect products of groups. Radford's Theorem about Hopf algebras with projection suggests that any C * -quantum group with projection decomposes uniquely into an ordinary C * -quantum group and a "braided" C * -quantum group. We establish this on the level of manageable multiplicative unitaries.
Abstract. For a quasitriangular C * -quantum group, we enrich the twisted tensor product constructed in the first part of this series to a monoidal structure on the category of its continuous coactions on C * -algebras. We define braided C * -quantum groups, where the comultiplication takes values in a twisted tensor product. We show that compact braided C * -quantum groups yield compact quantum groups by a semidirect product construction.
We construct a family of [Formula: see text] deformations of E(2) group for nonzero complex parameters [Formula: see text] as locally compact braided quantum groups over the circle group [Formula: see text] viewed as a quasitriangular quantum group with respect to the unitary [Formula: see text]-matrix [Formula: see text] for all [Formula: see text]. For real [Formula: see text], the deformation coincides with Woronowicz’s [Formula: see text] groups. As an application, we study the braided analogue of the contraction procedure between [Formula: see text] and [Formula: see text] groups in the spirit of Woronowicz’s quantum analogue of the classic Inönü–Wigner group contraction. Consequently, we obtain the bosonization of braided [Formula: see text] groups by contracting [Formula: see text] groups.
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