The Full Classification of Orthogonal Easy Quantum Groups

Raum, Sven; Weber, Moritz

doi:10.1007/s00220-015-2537-z

Cited by 72 publications

(126 citation statements)

References 55 publications

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“…Namely if G is an easy quantum group, the invariant subspace of the k−tensorproduct representation is spanned by the vectors T p , as defined in Definition1.5, with p belonging to a subcategory of P(k). See [BS09], [RW13] for more informations on the subject, and [KS09], [FW] and [Bra12a] for some applications. In this case, the scalar product matrix has a simpler form.…”

Section: Weingarten Calculusmentioning

confidence: 99%

Free wreath product quantum groups: The monoidal category, approximation properties and free probability

Lemeux

Tarrago

2016

Journal of Functional Analysis

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Abstract. In this paper, we find the fusion rules for the free wreath product quantum groups G * S + N for all compact matrix quantum groups of Kac type G and N ≥ 4. This is based on a combinatorial description of the intertwiner spaces between certain generating representations of G * S + N . The combinatorial properties of the intertwiner spaces in G * S + N then allows us to obtain several probabilistic applications. We then prove the monoidal equivalence between G * S + N and a compact quantum group whose dual is a discrete quantum subgroup of the free product G * SUq(2), for some 0 < q ≤ 1. We obtain as a corollary certain stability results for the operator algebras associated with the free wreath products of quantum groups such as Haagerup property, weak amenability and exactness.

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Section: Weingarten Calculusmentioning

confidence: 99%

Free wreath product quantum groups: The monoidal category, approximation properties and free probability

Lemeux

Tarrago

2016

Journal of Functional Analysis

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“…Regarding easiness in general, we refer to [6,14,16]. In the context of the present paper, let us go back to the Schur-Weyl considerations in Section 4:…”

Section: The Easy Casementioning

confidence: 99%

Weingarten integration over noncommutative homogeneous spaces

Banica¹

2017

Annales Mathématiques Blaise Pascal

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195-224 (2017) Weingarten integration over noncommutative homogeneous spaces Teodor Banica AbstractWe discuss an extension of the Weingarten formula, to the case of noncommutative homogeneous spaces, under suitable "easiness" assumptions. The spaces that we consider are noncommutative algebraic manifolds, generalizing the spaces of type X = G/H ⊂ C N , with H ⊂ G ⊂ UN being subgroups of the unitary group, subject to certain uniformity conditions. We discuss various axiomatization issues, then we establish the Weingarten formula, and we derive some probabilistic consequences. Intégration de Weingarten sur les espaces homogènes non commutatifsRésumé On présente une extension de la formule d'intégration de Weingarten, pour les espaces homogènes non commutatifs, vérifiant des hypothèses « d'aisance » adéquates. Les espaces qu'on considère sont des variétés algebriques non commutatives, généralisant les espaces du type X = G/H ⊂ C N , avec H ⊂ G ⊂ UN étant des sous-groupes du groupe unitaire, vérifiant certaines conditions d'uniformité. On traite d'abord les questions d'axiomatisation, ensuite on établit la formule de Weingarten, et on finit avec quelques conséquences probabilistes.

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“…As a general framework, we use the theory of easy quantum groups [9], [29], in its modified "quizzy" version, from [1], [2]. The idea is that any intermediate easy quantum group H N ⊂ G ⊂ O + N can be q-deformed at q = −1, into a certain intermediate quantum group H N ⊂Ḡ ⊂ O + N .…”

Section: Introductionmentioning

confidence: 99%

“…To be more precise, here O × N are the various versions of O N , obtained via liberation and twisting, and H × N are various versions of the hyperoctahedral group H N , which are known to be equal to their own twists. There are in fact many such quantum groups H × N , as explained in [29], and the dotted arrows in the middle stand for that. Now back to our questions, the above diagram, fully covering the liberations and twists of H N , O N , is precisely what we need.…”

Section: Introductionmentioning

confidence: 99%

Higher orbitals of quizzy quantum group actions

Banica

2019

Advances in Applied Mathematics

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The hyperoctahedral group H N is known to have two natural liberations: the "good" one H + N , which is the quantum symmetry group of N segments, and the "bad" oneŌ N , which is the quantum symmetry group of the N -hypercube. We study here this phenomenon, in the general "quizzy" framework, which covers the various liberations and twists of H N , O N . Our results include: (1) an interpretation of the embeddinḡ O N ⊂ S + 2 N , as corresponding to the antisymmetric representation of O N , (2) a study of the liberations of H N , notably with the result < H + N ,Ō N >= O + N , and (3) a comparison of the k-orbitals for the inclusions H N ⊂ H + N and H N ⊂Ō N , for k ∈ N small.2010 Mathematics Subject Classification. 46L65 (46L54).

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The Full Classification of Orthogonal Easy Quantum Groups

Cited by 72 publications

References 55 publications

Free wreath product quantum groups: The monoidal category, approximation properties and free probability

Free wreath product quantum groups: The monoidal category, approximation properties and free probability

Weingarten integration over noncommutative homogeneous spaces

Higher orbitals of quizzy quantum group actions

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