On compact bicrossed products

Abstract

We make a comprehensive and self-contained study of compact bicrossed products arising from matched pairs of discrete groups and compact groups. We exhibit an automatic regularity property of such a matched pair and describe the representation theory and the fusion rules of the associated bicrossed product G. We investigate the relative co-property (T ) and the relative co-Haagerup property of the pair comprising of the compact group and the bicrossed product, discuss property (T ) and Haagerup property of the discrete dual G, and review co-amenability of G as well. We distinguish two such non-trivial compact bicrossed products with relative coproperty (T ) and also provide an infinite family of pairwise non isomorphic non-trivial discrete quantum groups with property (T ), the existence of even one of the latter was unknown. Finally, we examine all the properties mentioned above for the crossed product quantum group given by an action by quantum automorphisms of a discrete group on a compact quantum group, and also establish the permanence of rapid decay and weak amenability and provide several explicit examples.

Property (T) discrete quantum groups and subfactors with triangle presentations

“…Although property (T) for discrete quantum groups was defined in [Fim08] much in the same way as for discrete groups, until now, there were no genuinely quantum examples. The known examples were either twists of property (T) groups in [Fim08] or equal up to finite index to a property (T) group in [FMP15]. The main goal of this paper is to construct such genuinely quantum examples of property (T) discrete quantum groups.…”

The Baum–Connes property for a quantum (semi-)direct product

“…For every class rγs P Γ{G, we define the following clopen subsets of G (see [10] for more details) A r,s :" tg P G : β g prq " su, for every r, s P rγs. Consider as well its characteristic function, say 1 Ar,s ": 1 r,s , for all r, s P rγs.…”

“…CpGq (see [10] for more details). By definition of the crossed product by a discrete group we have a unital faithful˚-homomorphism π : CpGq ÝÑ CpFq and a group homomorphism u : Γ ÝÑ UpCpFqq defined by u γ :" λ γ b id CpGq , for all γ P Γ such that CpFq " Γα 4.1 Remark.…”

“…Let X P KKpC r pFq, C m pFqq be its inverse, so that we have rτ F s b CpGq " C˚xπpf qu γ : f P CpGq, γ P Γy. So, with the help of the α-invariant character above, we identify CmpΓq with the subalgebra of C m pFq generated by tu γ : γ P Γu by universal property (see Remark 3.6 in [10] for more details). Hence, we consider the canonical injection ι m : CmpΓq ãÑ C m pFq, which is such that ε m Γ˝ι m " id CmpΓq .…”

Topological automorphism groups of compact quantum groups

“…of Hopf algebras in the sense of [1]. The construction is also a generalization of that of a bicrossed product as in [10], where the compact group is Z{2, and the discrete group is the quantum discrete dual to the free unitary group Uǹ (i.e. the discrete quantum group with group algebra A).…”