A note on reduced and von Neumann algebraic free wreath products

Wahl, Jonas

doi:10.48550/arxiv.1411.4861

Cited by 3 publications

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“…As a final comment, we wish to point out that our result, which expresses certain quantum automorphism groups of finite graphs as free wreath products, will be useful to study the representation theory and operator algebraic properties of these quantum groups, thanks to general results on quantum groups obtained as free wreath product recently proved in [10,11,15].…”

Section: Introductionmentioning

confidence: 86%

Quantum automorphism group of the lexicographic product of finite regular graphs

Chassaniol¹

2016

Journal of Algebra

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Section: Introductionmentioning

confidence: 86%

Quantum automorphism group of the lexicographic product of finite regular graphs

Chassaniol¹

2016

Journal of Algebra

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“…Strictly following Bichon's article, the free wreath product comes without a specification of the fundamental unitary in the case of compact matrix quantum groups. The one that is commonly used nowadays is the one above (see for instance [Wah14], [Lem14] or [BV09]). In this sense, the free wreath product is already in a "glued version".…”

Section: Free Unitary Easy Quantum Groupsmentioning

confidence: 99%

Unitary Easy Quantum Groups: The Free Case and the Group Case

Tarrago

Weber

2016

Int Math Res Notices

163

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Abstract. Easy quantum groups have been studied intensively since the time they were introduced by Banica and Speicher in 2009. They arise as a subclass of (C * -algebraic) compact matrix quantum groups in the sense of Woronowicz. Due to some Tannaka-Krein type result, they are completely determined by the combinatorics of categories of (set theoretical) partitions. So far, only orthogonal easy quantum groups have been considered in order to understand quantum subgroups of the free orthogonal quantum group O + n . We now give a definition of unitary easy quantum groups using colored partitions to tackle the problem of finding quantum subgroups of U + n . In the free case (i.e. restricting to noncrossing partitions), the corresponding categories of partitions have recently been classified by the authors by purely combinatorial means. There are ten series showing up each indexed by one or two discrete parameters, plus two additional quantum groups. We now present the quantum group picture of it and investigate them in detail. We show how they can be constructed from other known examples using generalizations of Banica's free complexification. For doing so, we introduce new kinds of products between quantum groups.We also study the notion of easy groups. IntroductionIn order to provide a new notion of symmetries adapted to the situation in operator algebras, Woronowicz introduced compact matrix quantum groups in 1987 in [Wor87]. The idea is roughly to take a compact Lie group G ⊆ M n (C) and to pass to the algebra C(G) of continuous functions over it. It turns out that this algebra fulfills all axioms of a C * -algebra with the special feature that the multiplication is commutative. If we now also dualize main properties of the group law µ : G×G → G to properties of a map ∆ : C(G) → C(G×G) ∼ = C(G)⊗C(G) (the comultiplication), and consider noncommutative C * -algebras equiped with such a map ∆, we are right in the heart of the definition of a compact matrix quantum group (see Section 3.1 for details). It generalizes the notion of a compact matrix group. The reader not familiar with quantum groups might also first jump to Section 5 and read it as a motivation.Date: December 2, 2015. 2010 Mathematics Subject Classification. 20G42 (Primary); 05A18, 46L54 (Secondary).

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“…In the other case, when the decomposition has only one factor, we can generalise the proof presented by Lemeux in [Lem14, Theorem 3.5]. The assumption of Lemeux on the minimum number of irreducible representations of G can be removed by making use of a trick introduced by Wahl in [Wah14].…”

Section: The Free Wreath Productmentioning

confidence: 87%

“…In [Lem14], Lemeux proved the simplicity and uniqueness of the trace for the reduced C*-algebra in the discrete case. His argument, based on the so-called Powers method and on the simplicity and uniqueness of the trace of S + n (n ≥ 8) proved by Brannan in [Bra13], was extended by Wahl in [Wah14] to the general case of a matrix pseudogroup of Kac type.…”

Section: Introductionmentioning

confidence: 99%

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

Fima

Pittau

2016

Journal of Functional Analysis

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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.

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A note on reduced and von Neumann algebraic free wreath products

Cited by 3 publications

References 0 publications

Quantum automorphism group of the lexicographic product of finite regular graphs

Quantum automorphism group of the lexicographic product of finite regular graphs

Unitary Easy Quantum Groups: The Free Case and the Group Case

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

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