Braided quantum SU(2) groups

Abstract

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.

Podleś spheres for the braided quantum SU(2)

“…the corresponding C˚-algebras are isomorphic. This was proved already in [23,Appendix A.2] for real q, then stated in [3, Section 3.1] and proved in [5,Section 2]. Similarly Podleś spheres are all isomorphic to a certain extension K ' K by CpTq, where K is the algebra of compact operators on a separable Hilbert space ( [19], see also [13,Proposition 4]) or to the minimal unitization of K (the standard quantum sphere).…”

“…Section 3 recalls the details of the braided version of SUp2q and in Section 4 we study the quotient sphere S 2 q " SU q p2q{T. Section 5 is devoted to the definition and study of the three-dimensional irreducible representation of SU q p2q found as an irreducible subrepresentation of the tensor square of the fundamental representation (cf. [5,Section 5]) and in Section 6 we find the commutation relations implied by existence of an action of the braided quantum group SU q p2q satisfying certain conditions suggested by the quotient construction from Section 4 (cf. Section 5.5).…”

“…See also [6], where braided quantum groups arise naturally in the von Neumann algebraic description of semidirect products of locally compact quantum groups. The connection with semidirect products was already hinted at in [26] (for real q) and developed fully in [5,Section 6].…”

“…for the braided quantum group SU q p2q. Later we will use the notion of a tensor product of representations of a braided quantum group from [5] to define an irreducible three-dimensional representation of SU q p2q and find all compact quantum spaces with an action of SU q p2q (in a braided sense, see Sections 4, 6 and 7) which resembles the action on the sphere obtained via the quotient construction. It turns out that these quantum spaces are exactly the quantum spaces found by Podleś for the real deformation parameter |q|.…”

“…This is, of course, of little importance, but this convention works better in the context in which all considered C˚-algebras come equipped with a left action of T which is necessary for the braided monoidal structure leading to the braided SU q p2q (cf. [5] and Section 2 and 3).…”