Wreath products of finite groups by quantum groups

Abstract

We introduce a notion of partition wreath product of a finite group by a partition quantum group, a construction motivated on the one hand by classical wreath products and on the other hand by the free wreath product of J. Bichon. We identify the resulting quantum group in several cases, establish some of its properties and show that when the finite group in question is abelian, the partition wreath product is itself a partition quantum group. This allows us to compute its representation theory, using earlier results of the first named author.2010 Mathematics Subject Classification. 20E22, 05E10, 05A18, 20G42.

New products and ℤ 2 -extensions of compact matrix quantum groups

“…Then we describe concrete applications. Note that colored partitions were already used to describe unitary quantum groups [21] or wreath product of quantum groups [8]. However, we will use it in a bit more primitive way to describe quantum groups with fundamental representation in the form of a direct sum.…”

“…So far the only results in this direction, which are known to us, are the above mentioned tensor and free products defined by Wang [25,26]. Note that there are many results concerning generalizations of the group semidirect product [1,12,13,17,24] and wreath product [4,8]. Let us also mention here the glued product construction [5,21].…”

New products and $\mathbb{Z}_2$-extensions of compact matrix quantum groups

“…Then we describe concrete applications. Note that colored partitions were already used to describe unitary quantum groups [TW17] or wreath product of quantum groups [FS18]. However, we will use it in a bit more primitive way to describe quantum groups with fundamental representation in the form of a direct sum.…”

“…So far the only results in this direction, which are known to us, are the above mentioned tensor and free products defined by Wang [Wan95a,Wan95b]. Note that there are many results concerning generalizations of the group semidirect product [Maj90, Maj91, VV03, BV05, MRW17] and wreath product [Bic04,FS18]. Let us also mention here the glued product construction [TW17,CW16].…”

Free wreath product quantum groups and standard invariants of subfactors

“…It will be the main focus of this paper. While Bichon's definition was given directly on the C * -algebraic level, it has become apparent in the recent works [LT16] and [FP16] (see also [FS15]) that the construction is at its core a categorical one.…”