Classification results for easy quantum groups

Banica, Teodor; Curran, Stephen; Speicher, Roland

doi:10.2140/pjm.2010.247.1

Cited by 49 publications

(135 citation statements)

References 36 publications

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“…We thank Teo Banica, Stephen Curran and Roland Speicher for sending us an unpublished draft [BCS12a] of their work on the definition and classification of unitary easy quantum groups. Some parts of this article may be found in their draft, too.…”

Section: Acknowledgementsmentioning

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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Section: Acknowledgementsmentioning

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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“…Their representation theory has been studied by T. Banica [3]. A wealth of new examples has been described over time, see [5,6,7,61] and references therein.…”

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confidence: 99%

Connected components of compact matrix quantum groups and finiteness conditions

Cirio

D’Andrea

Pinzari

et al. 2014

Journal of Functional Analysis

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ABSTRACT. We introduce the notion of identity component of a compact quantum group and that of total disconnectedness. As a drawback of the generalized Burnside problem, we note that totally disconnected compact matrix quantum groups may fail to be profinite. We consider the problem of approximating the identity component as well as the maximal normal (in the sense of Wang) connected subgroup by introducing canonical, but possibly transfinite, sequences of subgroups. These sequences have a trivial behaviour in the classical case. We give examples, arising as free products, where the identity component is not normal and the associated sequence has length 1.We give necessary and sufficient conditions for normality of the identity component and finiteness or profiniteness of the quantum component group. Among them, we introduce an ascending chain condition on the representation ring, called Lie property, which characterizes Lie groups in the commutative case and reduces to group Noetherianity of the dual in the cocommutative case. It is weaker than ring Noetherianity but ensures existence of a generating representation. The Lie property and ring Noetherianity are inherited by quotient quantum groups. We show that A u (F ) is not of Lie type. We discuss an example arising from the compact real form of U q (sl 2 ) for q < 0.

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“…. , x (s) 11 ), which completes the proof. 2 : 1 i, j n, 1 r s ⊗ C(G) to be the homomorphism determined by α t (r)…”

Section: Uniform R-cyclicitymentioning

confidence: 58%

Quantum invariant families of matrices in free probability

Curran

Speicher

2011

Journal of Functional Analysis

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We consider (self-adjoint) families of infinite matrices of noncommutative random variables such that the joint distribution of their entries is invariant under conjugation by a free quantum group. For the free orthogonal and hyperoctahedral groups, we obtain complete characterizations of the invariant families in terms of an operator-valued R-cyclicity condition. This is a surprising contrast with the Aldous-Hoover characterization of jointly exchangeable arrays.

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Classification results for easy quantum groups

Cited by 49 publications

References 36 publications

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

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

Connected components of compact matrix quantum groups and finiteness conditions

Quantum invariant families of matrices in free probability

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