Property (T) discrete quantum groups and subfactors with triangle presentations

Vaes, Stefaan; Valvekens, Matthias

doi:10.1016/j.aim.2019.01.023

Cited by 8 publications

(4 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since Rep f (K) has vanishing L 2 -Betti numbers, our criterion yields that the L 2 -Betti numbers of C f (K < G) also vanish. Our results also imply that certain bicrossed product discrete quantum groups (in particular those considered in [VV18]) have vanishing L 2 -Betti numbers.…”

Section: Introductionsupporting

confidence: 62%

$L^2$-Betti numbers of $C^*$-tensor categories associated with totally disconnected groups

Valvekens¹

2020

Preprint

Self Cite

View full text Add to dashboard Cite

We prove that the L 2 -Betti numbers of a rigid C * -tensor category vanish in the presence of an almost-normal subcategory with vanishing L 2 -Betti numbers, generalising a result of [BFS12]. We apply this criterion to show that the categories constructed from totally disconnected groups in [AV16] have vanishing L 2 -Betti numbers. Given an almost-normal inclusion of discrete groups Λ < Γ, with Γ acting on a type II 1 factor P by outer automorphisms, we relate the cohomology theory of the quasi-regular inclusion P Λ ⊂ P Γ to that of the Schlichting completion G of Λ < Γ. If Λ < Γ is unimodular, this correspondence allows us to prove that the L 2 -Betti numbers of P Λ ⊂ P Γ are equal to those of G.

show abstract

Section: Introductionsupporting

confidence: 62%

$L^2$-Betti numbers of $C^*$-tensor categories associated with totally disconnected groups

Valvekens¹

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…We summarize the main results of this work in the following table. There are several other classes of infinite diagram algebras appearing in the work of Banica and Speicher [5], Weber [42] and others, for example, [12,28,33], for which the trace classification problem is still open and for which in many cases even the branching graphs are unknown. In the companion article [40], we classify the extremal traces on the next prominent class of infinite diagram algebras of [5], namely those containing the crossing partition.…”

Section: Infinite Diagram Algebra 𝐴 ( #+ 𝛿) (∞) and Infinite Fuss-ca...mentioning

confidence: 99%

Traces on diagram algebras I: Free partition quantum groups, random lattice paths and random walks on trees

Wahl¹

2022

Journal of London Math Soc

View full text Add to dashboard Cite

We classify extremal traces on the seven direct limit algebras of noncrossing partitions arising from the classification of free partition quantum groups of Banica–Speicher [5] and Weber [42]. For the infinite‐dimensional Temperley–Lieb algebra (corresponding to the quantum group ON+$O^+_N$) and the Motzkin algebra (BN+$B^+_N$), the classification of extremal traces implies a classification result for well‐known types of central random lattice paths. For the 2‐Fuss–Catalan algebra (HN+$H_N^+$), we solve the classification problem by computing the minimal or exit boundary (also known as the absolute) for central random walks on the Fibonacci tree, thereby solving a probabilistic problem of independent interest, and to our knowledge the first such result for a nonhomogeneous tree. In the course of this article, we also discuss the branching graphs for all seven examples of free partition quantum groups, compute those that were not already known, and provide new formulae for the dimensions of their irreducible representations.

show abstract

“…As generalisations of discrete groups, many avenues of research in the theory of rigid C * -tensor categories are motivated by known results about discrete groups. In [VV18], we develop a spectral criterion to detect property (T) in the setting of rigid C * -tensor categories, inspired by Żuk's criterion for discrete groups [ Żuk01]. Using this criterion, we construct a family of examples of discrete quantum groups with property (T) given by relations coming from triangle presentations [CMSZ91] These discrete quantum groups are in some sense very different from classical discrete groups with property (T).…”

Section: Strong Locality Beyond Linear Energy Boundsmentioning

confidence: 99%

Subfactors and Applications

et al. 2020

View full text Add to dashboard Cite

The theory of subfactors connects diverse topics in mathematics and mathematical physics such as tensor categories, vertex operator algebras, quantum groups, quantum topology, free probability, quantum field theory, conformal field theory, statistical mechanics, condensed matter physics and, of course, operator algebras. We invited an international group of researchers from these areas and many fruitful interactions took place during the workshop.

show abstract

Property (T) discrete quantum groups and subfactors with triangle presentations

Cited by 8 publications

References 33 publications

$L^2$-Betti numbers of $C^*$-tensor categories associated with totally disconnected groups

$L^2$-Betti numbers of $C^*$-tensor categories associated with totally disconnected groups

Traces on diagram algebras I: Free partition quantum groups, random lattice paths and random walks on trees

Subfactors and Applications

Contact Info

Product

Resources

About