The free wreath product of a compact quantum group by a quantum automorphism group

Abstract

Let G be a compact quantum group and G aut (B, ψ) be the quantum automorphism group of a finite dimensional C*-algebra (B, ψ). In this paper, we study the free wreath product G ≀ * G aut (B, ψ). First of all, we describe its space of intertwiners and find its fusion semiring. Then, we prove some stability properties of the free wreath product operation. In particular, we find under which conditions two free wreath products are monoidally equivalent or have isomorphic fusion semirings. We also establish some analytic and algebraic properties of G ≀ * G aut (B, ψ). As a last result, we prove that the free wreath product of two quantum automorphism groups can be seen as the quotient of a suitable quantum automorphism group.

Free wreath product quantum groups and standard invariants of subfactors

“…We also prove an analogue of Theorem B for the partial generalization of Bichon's free wreath product due to Fima and Pittau [FP16], see Theorem 7.1. In particular, our approach allows us to unify both previous definitions and generalize them considerably (Definition 7.5 and Remark 7.6).…”

“…When F = S + n , the representation theory of G * S + n has been studied in [LT16]. Moreover, Bichon's construction was partially generalized in [FP16] to the situation in which the right input F is replaced by the universal compact quantum group G aut (A, tr) acting on a finite-dimensional C * -algebra A in a Markov trace-preserving way. In fact, the definition in [FP16] is slightly more general, see [FP16] for details.…”

“…We recall some facts on G * G aut (A, tr) for later use. For any finite-dimensional unitary representation u ∈ B(H u ) ⊗ Pol(G), one can define a unitary representation a(u) ∈ B(H ⊗ H u ) ⊗ Pol(G * G aut (A, tr)) in the following way (see [FP16,Section 8]). For any x ∈ Irr(G) appearing as a subobject of u, choose a family of isometries S x,k ∈ Mor(u x , u), 1 ≤ k ≤ dim Mor(u x , u) such that S x,k S * x,k are pairwise orthogonal projections with x⊂u k S x,k S * x,k = id Hu .…”

“…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.…”

On the monoidal invariance of the cohomological dimension of Hopf algebras

“…Free wreath products. In this subsection we assume that the base field is k = C, since the monoidal equivalences on which we rely [18,25] were obtained in this framework. Before going to the general setting of Theorem 7.4, we feel it is probably worth to present a particular example.…”