scholarly journals MONOIDS, EMBEDDING FUNCTORS AND QUANTUM GROUPS

2008 ◽  
Vol 19 (01) ◽  
pp. 93-123 ◽  
Author(s):  
MICHAEL MÜGER ◽  
LARS TUSET
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Regular Representation ◽  
Tensor Category ◽  
Left Regular Representation ◽  
Left Regular ◽  
Group A ◽  
Absorbing Property

We show that the left regular representation πl of a discrete quantum group (A, Δ) has the absorbing property and forms a monoid [Formula: see text] in the representation category Rep (A, Δ). Next we show that an absorbing monoid in an abstract tensor *-category [Formula: see text] gives rise to an embedding functor (or fiber functor) [Formula: see text], and we identify conditions on the monoid, satisfied by [Formula: see text], implying that E is *-preserving. As is well-known, from an embedding functor [Formula: see text] the generalized Tannaka theorem produces a discrete quantum group (A, Δ) such that [Formula: see text]. Thus, for a C*-tensor category [Formula: see text] with conjugates and irreducible unit the following are equivalent: (1) [Formula: see text] is equivalent to the representation category of a discrete quantum group (A, Δ), (2) [Formula: see text] admits an absorbing monoid, (3) there exists a *-preserving embedding functor [Formula: see text].

L p {L_{p}} -representations of discrete quantum groups

2017 ◽  
Vol 2017 (732) ◽  
pp. 165-210 ◽  
Author(s):  
Michael Brannan ◽  
Zhong-Jin Ruan
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Regular Representation ◽  
Multiplier Algebra ◽  
Haagerup Property ◽  
Convolution Algebra ◽  
Left Regular Representation ◽  
Left Regular ◽  
Locally Compact Quantum Group ◽  
Compact Quantum Groups

Abstract Given a locally compact quantum group {\mathbb{G}} , we define and study representations and {\mathrm{C}^{\ast}} -completions of the convolution algebra {L_{1}(\mathbb{G})} associated with various linear subspaces of the multiplier algebra {C_{b}(\mathbb{G})} . For discrete quantum groups {\mathbb{G}} , we investigate the left regular representation, amenability and the Haagerup property in this framework. When {\mathbb{G}} is unimodular and discrete, we study in detail the {\mathrm{C}^{\ast}} -completions of {L_{1}(\mathbb{G})} associated with the non-commutative {L_{p}} -spaces {L_{p}(\mathbb{G})} . As an application of this theory, we characterize (for each {p\in[1,\infty)} ) the positive definite functions on unimodular orthogonal and unitary free quantum groups {\mathbb{G}} that extend to states on the {L_{p}} - {\mathrm{C}^{\ast}} -algebra of {\mathbb{G}} . Using this result, we construct uncountably many new examples of exotic quantum group norms for compact quantum groups.

scholarly journals Local rigidity for actions of Kazhdan groups on noncommutative Lp-spaces

2015 ◽  
Vol 26 (08) ◽  
pp. 1550064
Author(s):  
Bachir Bekka
Keyword(s):  
Von Neumann Algebra ◽  
Regular Representation ◽  
Von Neumann ◽  
Lp Spaces ◽  
Local Rigidity ◽  
Left Regular Representation ◽  
Left Regular ◽  
Property T ◽  
Finite Factor ◽  
Finite Index Subgroup

Let Γ be a discrete group and 𝒩 a finite factor, and assume that both have Kazhdan's Property (T). For p ∈ [1, +∞), p ≠ 2, let π : Γ →O(Lp(𝒩)) be a homomorphism to the group O(Lp(𝒩)) of linear bijective isometries of the Lp-space of 𝒩. There are two actions πl and πr of a finite index subgroup Γ+ of Γ by automorphisms of 𝒩 associated to π and given by πl(g)x = (π(g) 1)*π(g)(x) and πr(g)x = π(g)(x)(π(g) 1)* for g ∈ Γ+ and x ∈ 𝒩. Assume that πl and πr are ergodic. We prove that π is locally rigid, that is, the orbit of π under O(Lp(𝒩)) is open in Hom (Γ, O(Lp(𝒩))). As a corollary, we obtain that, if moreover Γ is an ICC group, then the embedding g ↦ Ad (λ(g)) is locally rigid in O(Lp(𝒩(Γ))), where 𝒩(Γ) is the von Neumann algebra generated by the left regular representation λ of Γ.

scholarly journals GAUSSIAN FLUCTUATIONS OF REPRESENTATIONS OF WREATH PRODUCTS

2006 ◽  
Vol 09 (04) ◽  
pp. 529-546
Author(s):  
PIOTR ŚNIADY
Keyword(s):  
Finite Group ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Regular Representation ◽  
Wreath Products ◽  
Young Diagrams ◽  
Reducible Representation ◽  
Irreducible Components ◽  
Left Regular Representation ◽  
Left Regular

We study the asymptotics of the reducible representations of the wreath products G≀Sq = Gq ⋊ Sq for large q, where G is a fixed finite group and Sq is the symmetric group in q elements; in particular for G = ℤ/2ℤ we recover the hyperoctahedral groups. We decompose such a reducible representation of G≀Sq as a sum of irreducible components (or, equivalently, as a collection of tuples of Young diagrams) and we ask what is the character of a randomly chosen component (or, what are the shapes of Young diagrams in a randomly chosen tuple). Our main result is that for a large class of representations, the fluctuations of characters (and fluctuations of the shape of the Young diagrams) are asymptotically Gaussian. The considered class consists of the representations for which the characters asymptotically almost factorize and it includes, among others, the left regular representation therefore we prove the analogue of Kerov's central limit theorem for wreath products.

scholarly journals Random products of automorphisms of Heisenberg nilmanifolds and Weil’s representation

2010 ◽  
Vol 31 (5) ◽  
pp. 1277-1286 ◽  
Author(s):  
BACHIR BEKKA ◽  
JEAN-ROMAIN HEU
Keyword(s):  
Probability Measure ◽  
Spectral Gap ◽  
Regular Representation ◽  
Metaplectic Representation ◽  
Matrix Coefficients ◽  
Left Regular Representation ◽  
Left Regular ◽  
Random Products ◽  
Image Position ◽  
Dense Set

AbstractForn≥1, letHbe the (2n+1)-dimensional real Heisenberg group, and let Λ be a lattice inH. Let Γ be the group of automorphisms of the corresponding nilmanifold Λ∖HandUthe associated unitary representation of Γ onL2(Λ∖H) . Denote byTthe maximal torus factor associated to Λ∖H. Using Weil’s representation (also known as the metaplectic representation), we show that a dense set of matrix coefficients of the restriction ofUto the orthogonal complement ofL2(T) inL2(Λ∖H) belong toℓ4n+2+ε(Γ) for every ε>0 . We give the following application to random walks on Λ∖Hdefined by a probability measureμon Aut (Λ∖H) . Denoting by Γ(μ) the subgroup of Aut (Λ∖H) generated by the support ofμand byU0andV0the restrictions ofUto, respectively, the subspaces ofL2(Λ∖H) andL2(T) with zero mean, we prove the following inequality:whereλis the left regular representation of Γ(μ) onℓ2(Γ(μ)) . In particular, the action of Γ(μ) on Λ∖Hhas a spectral gap if and only if the corresponding action of Γ(μ) onThas a spectral gap.

scholarly journals LOCALLY COMPACT QUANTUM GROUPS IN THE UNIVERSAL SETTING

2001 ◽  
Vol 12 (03) ◽  
pp. 289-338 ◽  
Author(s):  
JOHAN KUSTERMANS
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Group Structure ◽  
Locally Compact ◽  
Bijective Correspondence ◽  
Locally Compact Quantum Groups ◽  
Algebraic Quantum ◽  
Compact Quantum Groups ◽  
Group A ◽  
Fine Tune

In this paper we associate to every reduced C *-algebraic quantum group (A, Δ) (as defined in [11]) a universal C *-algebraic quantum group (Au, Δu). We fine tune a proof of Kirchberg to show that every *-representation of a modified L 1-space is generated by a unitary corepresentation. By taking the universal enveloping C *-algebra of a dense sub *-algebra of A we arrive at the C *-algebra Au. We show that this C *-algebra Au carries a quantum group structure which is a rich as its reduced companion. We also establish a bijective correspondence between quantum group morphisms and certain co-actions.

scholarly journals Automorphisms of Full II1 Factors, II

1978 ◽  
Vol 21 (3) ◽  
pp. 325-328 ◽  
Author(s):  
John Phillips
Keyword(s):  
Regular Representation ◽  
Conjugacy Classes ◽  
Type Ii ◽  
Nonabelian Group ◽  
Left Regular Representation ◽  
Left Regular ◽  
Free Generators

The purpose of this note is to continue the author's study of the automorphisms of certain factors of type II1 Namely, those factors arising from the left regular representation of a free nonabelian group. Our main result shows that the outer conjugacy classes of automorphisms of such a factor are not countably separated. This had previously been shown only when the number of free generators was assumed to be infinite.

scholarly journals Classification of globally colorized categories of partitions

2018 ◽  
Vol 21 (04) ◽  
pp. 1850029 ◽  
Author(s):  
Daniel Gromada
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Tensor Category ◽  
Set Partitions

Set partitions closed under certain operations form a tensor category. They give rise to certain subgroups of the free orthogonal quantum group [Formula: see text], the so-called easy quantum groups, introduced by Banica and Speicher in 2009. This correspondence was generalized to two-colored set partitions, which, in addition, assign a black or white color to each point of a set. Globally colorized categories of partitions are those categories that are invariant with respect to arbitrary permutations of colors. This paper presents a classification of globally colorized categories. In addition, we show that the corresponding unitary quantum groups can be constructed from the orthogonal ones using tensor complexification.

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