$L^2$-homology for compact quantum groups

Kyed, David

doi:10.7146/math.scand.a-15072

Cited by 20 publications

(29 citation statements)

References 20 publications

(23 reference statements)

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“…where β (2) k (G) is the kth ℓ 2 -Betti number of the compact quantum group G introduced by Kyed. See [26] and [28]. Putting all of this together, we obtain the following result.…”

Section: Free Entropy Dimensionmentioning

confidence: 75%

The Connes embedding property for quantum group von Neumann algebras

Brannan¹,

Collins²,

Vergnioux³

2016

Trans. Amer. Math. Soc.

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ABSTRACT. For a compact quantum group G of Kac type, we study the existence of a Haar tracepreserving embedding of the von Neumann algebra L ∞ (G) into an ultrapower of the hyperfinite II 1 -factor (the Connes embedding property for L ∞ (G)). We establish a connection between the Connes embedding property for L ∞ (G) and the structure of certain quantum subgroups of G, and use this to prove that the II 1 -factorsassociated to the free orthogonal and free unitary quantum groups have the Connes embedding property for all N ≥ 4. As an application, we deduce that the free entropy dimension of the standard generators of L ∞ (O + N ) equals 1 for all N ≥ 4. We also mention an application of our work to the problem of classifying the quantum subgroups of O + N .

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“…where β (2) k (G) is the kth ℓ 2 -Betti number of the compact quantum group G introduced by Kyed. See [26] and [28]. Putting all of this together, we obtain the following result.…”

Section: Free Entropy Dimensionmentioning

confidence: 75%

The Connes embedding property for quantum group von Neumann algebras

Brannan¹,

Collins²,

Vergnioux³

2016

Trans. Amer. Math. Soc.

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“…k stands for the k-th L 2 -Betti number [33] of the dual discrete quantum group U + n . It was shown in [47] that β…”

Section: 3mentioning

confidence: 99%

Cohomological dimensions of universal cosovereign Hopf algebras

Bichon

2018

Publicacions Matemàtiques

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We compute the Hochschild and Gerstenhaber-Schack cohomological dimensions of the universal cosovereign Hopf algebras, when the matrix of parameters is a generic asymmetry. Our main tools are considerations on the cohomologies of free product of Hopf algebras, and on the invariance of the cohomological dimensions under graded twisting by a finite abelian group.2010 Mathematics Subject Classification. 16T05, 16E40, 16E10.

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“…L 2 -Betti numbers have been generalized further to a variety of different settings, see [Sa03,CS04,Ky06], and we refer to [Lü02] for an extensive monograph on the subject.…”

Section: Introductionmentioning

confidence: 99%

𝐿²-Betti numbers of locally compact groups and their cross section equivalence relations

Kyed

Petersen

Vaes

2015

Trans. Amer. Math. Soc.

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We prove that the L 2 -Betti numbers of a unimodular locally compact group G coincide, up to a natural scaling constant, with the L 2 -Betti numbers of the countable equivalence relation induced on a cross section of any essentially free ergodic probability measure preserving action of G. As a consequence, we obtain that the reduced and un-reduced L 2 -Betti numbers of G agree and that the L 2 -Betti numbers of a lattice Γ in G equal those of G up to scaling by the covolume of Γ in G. We also deduce several vanishing results, including the vanishing of the reduced L 2 -cohomology for amenable locally compact groups.

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$L^2$-homology for compact quantum groups

Cited by 20 publications

References 20 publications

The Connes embedding property for quantum group von Neumann algebras

The Connes embedding property for quantum group von Neumann algebras

Cohomological dimensions of universal cosovereign Hopf algebras

𝐿²-Betti numbers of locally compact groups and their cross section equivalence relations

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