Equivariant Poincaré duality for quantum group actions

Nest, Ryszard; Voigt, Christian

doi:10.1016/j.jfa.2009.10.015

Cited by 47 publications

(137 citation statements)

References 42 publications

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“…For more information we refer to [2], [16], [17], [26], [36]. Our notation and conventions will mainly follow [21], [33]. Let φ be a normal, semifinite and faithful weight on a von Neumann algebra M .…”

Section: Preliminaries On Quantum Groupsmentioning

confidence: 99%

“…There is a canonical map G f A → G r A, and this map is an isomorphism if G is amenable. We refer to [21] for more details. Finally, let us recall the definition of the Drinfeld double of a locally compact quantum group G, see [3].…”

Section: Preliminaries On Quantum Groupsmentioning

confidence: 99%

“…The braided tensor product is naturally a G-C * -algebra, and it is a natural replacement of the minimal tensor product of G-C * -algebras in the group case, see [21] for more information. We will be interested in the case that G is a discrete quantum group and work with D(G)-C * -algebras obtained from algebraic actions and coactions.…”

Section: Preliminaries On Quantum Groupsmentioning

confidence: 99%

“…We shall essentially follow Kasparov and Skandalis [14], in addition we take into account the natural Yetter-Drinfeld structures in the quantum setting. In the sequel we assume that the reader is familiar with equivariant KK-theory for quantum groups, our notation will follow [21], [33]. Let G 0 and G 1 be discrete quantum groups and let G = G 0 * G 1 be their free product.…”

Section: The Dirac and Dual Dirac Elementsmentioning

confidence: 99%

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The $$K$$ -theory of free quantum groups

Vergnioux¹,

Voigt²

2013

Math. Ann.

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Abstract. In this paper we study the K-theory of free quantum groups in the sense of Wang and Van Daele, more precisely, of free products of free unitary and free orthogonal quantum groups. We show that these quantum groups are K-amenable and establish an analogue of the Pimsner-Voiculescu exact sequence. As a consequence, we obtain in particular an explicit computation of the K-theory of free quantum groups. Our approach relies on a generalization of methods from the Baum-Connes conjecture to the framework of discrete quantum groups. This is based on the categorical reformulation of the Baum-Connes conjecture developed by Meyer and Nest. As a main result we show that free quantum groups have a γ-element and that γ = 1. As an important ingredient in the proof we adapt the Dirac-dual Dirac method for groups acting on trees to the quantum case. We use this to extend some permanence properties of the Baum-Connes conjecture to our setting.

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Section: Preliminaries On Quantum Groupsmentioning

confidence: 99%

Section: Preliminaries On Quantum Groupsmentioning

confidence: 99%

Section: Preliminaries On Quantum Groupsmentioning

confidence: 99%

Section: The Dirac and Dual Dirac Elementsmentioning

confidence: 99%

See 2 more Smart Citations

The $$K$$ -theory of free quantum groups

Vergnioux¹,

Voigt²

2013

Math. Ann.

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“…[1,3,5,6,8,12,17,18,19,21,25] and the references therein. But still some basic questions remain untouched, e.g.…”

Section: Introductionmentioning

confidence: 99%

A residue formula for the fundamental Hochschild class on the Podleś sphere

Krähmer¹,

Wagner²

2013

J K-Theor

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Towards a Classification of Compact Quantum Groups of Lie Type

Neshveyev

Yamashita

2016

Operator Algebras and Applications

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Abstract. This is a survey of recent results on classification of compact quantum groups of Lie type, by which we mean quantum groups with the same fusion rules and dimensions of representations as for a compact connected Lie group G. The classification is based on a categorical duality for quantum group actions recently developed by De Commer and the authors in the spirit of Woronowicz's Tannaka-Krein duality theorem. The duality establishes a correspondence between the actions of a compact quantum group H on unital C * -algebras and the module categories over its representation category Rep H. This is further refined to a correspondence between the braided-commutative Yetter-Drinfeld H-algebras and the tensor functors from Rep H. Combined with the more analytical theory of Poisson boundaries, this leads to a classification of dimension-preserving fiber functors on the representation category of any coamenable compact quantum group in terms of its maximal Kac quantum subgroup, which is the maximal torus for the q-deformation of G if q = 1. Together with earlier results on autoequivalences of the categories Rep Gq, this allows us to classify up to isomorphism a large class of quantum groups of G-type for compact connected simple Lie groups G. In the case of G = SU(n) this class exhausts all non-Kac quantum groups.

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Equivariant Poincaré duality for quantum group actions

Cited by 47 publications

References 42 publications

The $$K$$ -theory of free quantum groups

The $$K$$ -theory of free quantum groups

A residue formula for the fundamental Hochschild class on the Podleś sphere

Towards a Classification of Compact Quantum Groups of Lie Type

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