Two-parameter families of quantum symmetry groups

TL;DR

This paper introduces two-parameter families of quantum groups that generalize classical symmetry groups, exploring their structures and relations to free quantum groups and quantum isometry groups of free group duals.

Contribution

It defines new two-parameter quantum groups related to classical and free quantum groups, and investigates their properties and connections to quantum isometry groups.

Findings

H^+(p,0) is isomorphic to the quantum isometry group of the free group C*-algebra

The new quantum groups relate to free quantum groups studied earlier

H^+(p,0) can be viewed as a liberation of the classical isometry group of the p-dimensional torus

Abstract

We introduce and study natural two-parameter families of quantum groups motivated on one hand by the liberations of classical orthogonal groups and on the other by quantum isometry groups of the duals of the free groups. Specifically, for each pair (p,q) of non-negative integers we define and investigate quantum groups O^+(p,q), B^+(p,q), S^+(p,q) and H^+(p,q) corresponding to, respectively, orthogonal groups, bistochastic groups, symmetric groups and hyperoctahedral groups. In the first three cases the new quantum groups turn out to be related to the (dual free products of) free quantum groups studied earlier. For H^+(p,q) the situation is different: we show that H^+(p,0) is isomorphic to the quantum isometry group of the C*-algebra of the free group and it can be viewed as a liberation of the classical isometry group of the p-dimensional torus.