Teodor Banica

Representations of compact quantum groups and subfactors

Abstract We associate Popa systems (= standard invariants of subfactors) to the finite dimensional representations of compact quantum groups. We characterise the systems arising in this way: these are the ones which can be “represented” on finite dimensional Hilbert spaces. This is proved by a universal construction. We explicitely compute (in terms of some free products) the operation of going from representations of compact quantum groups to Popa systems and the back via the universal construction. We prove a Kesten type result for the co-amenability of compact quantum groups, which allows us to compare it with the amenability of subfactors.

Free Bessel Laws

We introduce and study a remarkable family of real probability measures that we call free Bessel laws. These are related to the free Poisson law. Our study includes: definition and basic properties, analytic aspects (supports, atoms, densities), combinatorial aspects (functional transforms, moments, partitions), and a discussion of the relation with random matrices and quantum groups.

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.

Integration over quantum permutation groups

Abstract We find a combinatorial formula for the Haar measure of quantum permutation groups. This leads to a dynamic formula for laws of diagonal coefficients, explaining the Poisson/free Poisson convergence result for characters.

Quantum Isometries and Noncommutative Spheres

We introduce and study two new examples of noncommutative spheres: the half-liberated sphere, and the free sphere. Together with the usual sphere, these two spheres have the property that the corresponding quantum isometry group is “easy”, in the representation theory sense. We present as well some general comments on the axiomatization problem, and on the “untwisted” and “non-easy” case.

De Finetti theorems for easy quantum groups

We study sequences of noncommutative random variables which are invariant under “quantum transformations” coming from an orthogonal quantum group satisfying the “easiness” condition axiomatized in our previous paper. For 10 easy quantum groups, we obtain de Finetti type theorems characterizing the joint distribution of any infinite quantum invariant sequence. In particular, we give a new and unified proof of the classical results of de Finetti and Freedman for the easy groups Sn, On, which is based on the combinatorial theory of cumulants. We also recover the free de Finetti theorem of Kostler and Speicher, and the characterization of operator-valued free semicircular families due to Curran. We consider also finite sequences, and prove an approximation result in the spirit of Diaconis and Freedman.

QUANTUM GROUPS ACTING ON 4 POINTS

Abstract We classify the compact quantum groups acting on 4 points. These are the quantum subgroups of the quantum permutation group 𝓠4. Our main tool is a new presentation for the algebra C(𝓠4), corresponding to an isomorphism of type 𝓠4 ≃ SO –1(3). The quantum subgroups of 𝓠4 are subject to a McKay type correspondence, that we describe at the level of algebraic invariants.

HOPF IMAGES AND INNER FAITHFUL REPRESENTATIONS

We develop a general theory of Hopf image of a Hopf algebra representation, with the associated concept of inner faithful representation, modelled on the notion of faithful representation of a discrete group. We study several examples, including group algebras, enveloping algebras of Lie algebras, pointed Hopf algebras, function algebras, twistings and cotwistings, and we present a Tannaka duality formulation of the notion of Hopf image.

Quantum Groups and Fuss--Catalan Algebras

Abstract: The categories of representations of compact quantum groups of automorphisms of certain inclusions of finite dimensional ℂ*-algebras are shown to be isomorphic to the categories of Fuss–Catalan diagrams.

Fusion rules for quantum reflection groups

We find the fusion rules for the quantum analogues of the complex reflection groups