Unitary Quantum Groups Research Articles

Unitary Easy Quantum Groups: The Free Case and the Group Case

Easy quantum groups form a class of compact matrix quantum groups which are completely determined by set partitions. They have been defined in 2009 by Banica and Speicher in the orthogonal case (as quantum subgroups of Wang’s free orthogonal quantum group |$O_n^+$|⁠). We extend their definition to the unitary case (as quantum subgroups of Wang’s free unitary quantum group |$U_n^+$|⁠) using colored partitions. In the free case (i.e., restricting to noncrossing partitions), the corresponding categories of partitions have recently been classified by the authors by purely combinatorial means. We show how the quantum groups associated with them can be constructed from other known examples using generalizations of Banica’s free complexification. For doing so, we introduce new kinds of products between quantum groups.

Examples of factors which have no Cartan subalgebras

We consider some conditions similar to Ozawa’s condition (AO) and prove that if a non-injective factor satisfies such a condition and has the W ∗ C B A P \rm W^*CBAP , then it has no Cartan subalgebras. As a corollary, we prove that I I 1 \rm II_1 factors of universal orthogonal and unitary discrete quantum groups have no Cartan subalgebras. We also prove that continuous cores of type I I I 1 \rm III_1 factors with such a condition are semisolid as a I I ∞ \rm II_\infty factor.

Strong Asymptotic Freeness for Free Orthogonal Quantum Groups

AbstractIt is known that the normalized standard generators of the free orthogonal quantum groupO+Nconverge in distribution to a free semicircular system as N → ∞. In this note, we substantially improve this convergence result by proving that, in addition to distributional convergence, the operator normof any non-commutative polynomial in the normalized standard generators ofO+Nconverges asN→ ∞ to the operator norm of the corresponding non-commutative polynomial in a standard free semicircular system. Analogous strong convergence results are obtained for the generators of free unitary quantum groups. As applications of these results, we obtain a matrix-coefficient version of our strong convergence theorem, and we recover a well-knownL2-L∞norm equivalence for noncommutative polynomials in free semicircular systems.

On the representation theory of partition (easy) quantum groups

Abstract Compact matrix quantum groups are strongly determined by their intertwiner spaces, due to a result by S. L. Woronowicz. In the case of easy quantum groups (also called partition quantum groups), the intertwiner spaces are given by the combinatorics of partitions, see the initial work of T. Banica and R. Speicher. The philosophy is that all quantum algebraic properties of these objects should be visible in their combinatorial data. We show that this is the case for their fusion rules (i.e. for their representation theory). As a byproduct, we obtain a unified approach to the fusion rules of the quantum permutation group S N + ${S_{N}^{+}}$ , the free orthogonal quantum group O N + ${O_{N}^{+}}$ as well as the hyperoctahedral quantum group H N + ${H_{N}^{+}}$ . We then extend our work to unitary easy quantum groups and link it with a “freeness conjecture” of T. Banica and R. Vergnioux.

CCAP for Universal Discrete Quantum Groups

We show that the discrete duals of the free orthogonal quantum groups have the Haagerup property and the completely contractive approximation property. Analogous results hold for the free unitary quantum groups and the quantum automorphism groups of finite-dimensional C*-algebras. The proof relies on the monoidal equivalence between free orthogonal quantum groups and SUq(2) quantum groups, on the construction of a sufficient supply of bounded central functionals for SUq(2) quantum groups, and on the free product techniques of Ricard and Xu. Our results generalize previous work in the Kac setting due to Brannan on the Haagerup property, and due to the second author on the CCAP.

Examples of weakly amenable discrete quantum groups

We prove that the free orthogonal and free unitary quantum groups FON+ and FUN+ are weakly amenable and that their Cowling–Haagerup constant is equal to 1. This is achieved by estimating the completely bounded norm of the projections on the coefficients of irreducible representations of their compact duals. An argument of monoidal equivalence then allows us to extend this result to quantum automorphism groups of finite spaces and even yields some examples of weakly amenable non-unimodular discrete quantum groups with the Haagerup property.

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

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 $$\gamma $$ -element and that $$\gamma = 1$$ . 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.

Quantum Symmetries and Strong Haagerup Inequalities

In this paper, we consider families of operators $\{x_r\}_{r \in \Lambda}$ in a tracial C$^\ast$-probability space $(\mathcal A, \phi)$, whose joint $\ast$-distribution is invariant under free complexification and the action of the hyperoctahedral quantum groups $\{H_n^+\}_{n \in \N}$. We prove a strong form of Haagerup's inequality for the non-self-adjoint operator algebra $\mathcal B$ generated by $\{x_r\}_{r \in \Lambda}$, which generalizes the strong Haagerup inequalities for $\ast$-free R-diagonal families obtained by Kemp-Speicher \cite{KeSp}. As an application of our result, we show that $\mathcal B$ always has the metric approximation property (MAP). We also apply our techniques to study the reduced C$^\ast$-algebra of the free unitary quantum group $U_n^+$. We show that the non-self-adjoint subalgebra $\mathcal B_n$ generated by the matrix elements of the fundamental corepresentation of $U_n^+$ has the MAP. Additionally, we prove a strong Haagerup inequality for $\mathcal B_n$, which improves on the estimates given by Vergnioux's property RD \cite{Ve}.

Free dynamical quantum groups and the dynamical quantum group \mathrmSU\mathrmdynQ(2)

We introduce dynamical analogues of the free orthogonal and free unitary quantum groups, which are no longer Hopf algebras but Hopf algebroids or quantum groupoids. These objects are constructed on the purely algebraic level and on the level of universal C*-algebras. As an example, we recover the dynamical SU_q(2) studied by Koelink and Rosengren, and construct a refinement that includes several interesting limit cases.

Approximation properties for free orthogonal and free unitary quantum groups

We show that the reduced von Neumann algebras of the free orthogonal and free unitary quantum groups have the Haagerup approximation property. Using this result and a Haagerup-type inequality for these quantum groups due to Vergnioux (J. Operator Theory 57, 303-324 (2007)), we also show that the reduced C$^\ast$-algebras of the free orthogonal and free unitary quantum groups have the metric approximation property.

Integration over Compact Quantum Groups

We find a combinatorial formula for the Haar functional of the orthogonal and unitary quantum groups. As an application, we consider diagonal coefficients of the fundamental representation, and we investigate their spectral measures.

Ergodic Coactions with Large Multiplicity and Monoidal Equivalence of Quantum Groups

We construct new examples of ergodic coactions of compact quantum groups, in which the multiplicity of an irreducible corepresentation can be strictly larger than the dimension of the latter. These examples are obtained using a bijective correspondence between certain ergodic coactions on C*-algebras and unitary fiber functors on the representation category of a compact quantum group. We classify these unitary fiber functors on the universal orthogonal and unitary quantum groups. The associated C*-algebras and von Neumann algebras can be defined by generators and relations, but are not yet well understood.

Decomposition of tensor product representations of the unitary matrix quantum group SUq(2)

Representations of SUq(2) are labeled by a phase playing the role of a Casimir operator. The coproduct SUq(2) is used to form direct product representations which can be reduced to irreducible ones. Direct product of two representations with given phases can be reduced to a representation with any given phase. Coefficients necessary for this procedure are calculated. The calculation can also be performed in the Fourier transformed space which replaces the phase by an integer.

Random sets, q-distributions and quantum groups

We have shown that the non-extensivity of classical set theory is related to unitary quantum groups. Using this non-extensivity property, we define a q-distribution, a binomial q-distribution and a Poisson q-distribution.

Solving the strongly coupled 2D gravity: 1. Unitary truncation and quantum group structure

Strongly compled gravity theories with Virasoro central charges equal to 7, 13, and 19 are shown to enjoy striking properties: at these values, the subset of chiral operators with real Virasoro-weights, acting on a subspaceHphys, is shown to be closed by fusion and braiding, and to leave this subspace invariant. Moreover, the representation of the Virasoro algebra becomes unitary when it is restricted toHphys. Strongly coupled 2D gravity withCgrav=7, 13, or 19 may thus be naturally truncated obtaining a consistent conformal theory (this result is similar to the truncation that occurs forC=1−6(p−p′)2/pp′ withp andp′ integers, where only a finite number of primary fields remains, as is well known in rational theories). The proof of this unitary truncation theorem, already summarized in a recent letter, is fully described here.

A Set of Orthogonal Polynomials That Generalize the Racah Coefficients or $6 - j$ Symbols

A very general set of orthogonal polynomials with five free parameters is given explicitly, the orthogonality relation is proved and the three term recurrence relation is found.