The classification of tensor categories of two-colored noncrossing partitions

Tarrago, Pierre; Weber, Moritz

doi:10.48550/arxiv.1509.00988

Cited by 2 publications

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“…Let us first briefly introduce the combinatorial objects underlying the unitary easy quantum groups. See [TW15] for more details concerning this section. The work in [TW15] is an extension of the well-known combinatorics of orthogonal easy quantum groups which were introduced by Banica and Speicher in [BS09].…”

Section: Categories Of Two-colored Partitionsmentioning

confidence: 99%

“…The cases of categories of non-crossing partitions. The classification of categories of noncrossing partitions in [TW15] is based on a division of the categories into four cases. By ↑…”

Section: Categories Of Two-colored Partitionsmentioning

confidence: 99%

“…In the unitary case (two colors) however, little is known. In a recent article [TW15], the authors classified all categories of (two-colored) noncrossing partitions. This is called the free case, since the corresponding quantum groups all sit in between Wang's free symmetric quantum group S + n and U + n .…”

Section: Introductionmentioning

confidence: 99%

“…While there are only seven free orthogonal easy quantum groups, the world of free unitary easy quantum groups is way richer. In the present article, we introduce the unitary easy quantum groups associated to the categories of partitions found in [TW15]. There are ten series of quantum groups each indexed by one or two parameters from the natural numbers, and two additional quantum groups.…”

Section: Introductionmentioning

confidence: 99%

“…We define these constructions as a generalization of Banica's free complexification [Ban08] by replacing Z by Z/dZ. Note that from the classification in [TW15], we know that the categories of noncrossing partitions fall into the class of globally colorized ones and the locally colorized ones. The first case is much easier (for instance the u ij are normal) and the quantum groups can be written as tensor d-complexifications.…”

Section: Introductionmentioning

confidence: 99%

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

Tarrago

Weber

2016

Int Math Res Notices

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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: Categories Of Two-colored Partitionsmentioning

confidence: 99%

“…The cases of categories of non-crossing partitions. The classification of categories of noncrossing partitions in [TW15] is based on a division of the categories into four cases. By ↑…”

Section: Categories Of Two-colored Partitionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Tarrago

Weber

2016

Int Math Res Notices

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163

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Fixed Point Algebras for Easy Quantum Groups

Gabriel¹,

Weber²

2016

SIGMA

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Abstract. Compact matrix quantum groups act naturally on Cuntz algebras. The first author isolated certain conditions under which the fixed point algebras under this action are Kirchberg algebras. Hence they are completely determined by their K-groups. Building on prior work by the second author, we prove that free easy quantum groups satisfy these conditions and we compute the K-groups of their fixed point algebras in a general form. We then turn to examples such as the quantum permutation group S + n , the free orthogonal quantum group O + n and the quantum reflection groups H s+ n . Our fixed point-algebra construction provides concrete examples of free actions of free orthogonal easy quantum groups, which are related to Hopf-Galois extensions.

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The classification of tensor categories of two-colored noncrossing partitions

Cited by 2 publications

References 0 publications

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

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

Fixed Point Algebras for Easy Quantum Groups

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