Integration over Compact Quantum Groups

Abstract

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.

Graphs having no quantum symmetry

“…For quantum groups such methods are worked out in [7], but their relation with the present results is very unclear.…”

“…The motivation comes from certain combinatorial aspects of subfactors, free probability, and statistical mechanical models. See [3], [5], [7].…”

Free wreath product quantum groups: The monoidal category, approximation properties and free probability

“…This tool has mainly two advantages : on one hand it allows us sometimes to get some interesting formulae for the Haar state on the matrix entries of a corepresentation, and on the other hand it yields some asymptotic results on the joint law of a finite set of elements when the dimension of the quantum group goes to infinity. Let us first sum up the pattern of this method coming from [BC05]: let G = (A, (u ij ) 1≤i,j≤n ) be a matrix compact quantum group acting on V ⊗k = X i ⊗k 1≤i≤n…”

“…It was mainly developped in the framework of compact quantum groups and permutation quantum groups by Banica and Collins (see [BC05], [BC07]). This tool has mainly two advantages : on one hand it allows us sometimes to get some interesting formulae for the Haar state on the matrix entries of a corepresentation, and on the other hand it yields some asymptotic results on the joint law of a finite set of elements when the dimension of the quantum group goes to infinity.…”

“…Once again the main strategy to get some results on the stochastic level is to restrict to the associated C * -tensor category through the Tannak-Krein duality. See Banica and Collins [BC05], [BC07], Banica and Speicher in [BS09], Banica, Curran and Speicher in [BCS11], Kostler and Speicher in [KS09] for more information on the subject.…”