A note on free quantum groups

TL;DR

This paper explores the free complexification operation for compact quantum groups, establishing a bijective correspondence between certain free orthogonal and free unitary quantum groups under specific conditions.

Contribution

It introduces a framework linking free orthogonal and free unitary quantum groups via free complexification, clarifying their structural relationship.

Findings

Established a one-to-one correspondence between free orthogonal and free unitary quantum groups.

Defined conditions under which the free complexification operation induces this correspondence.

Provided new insights into the structure of quantum groups at infinite level.

Abstract

We study the free complexification operation for compact quantum groups, $G\to G^c$. We prove that, with suitable definitions, this induces a one-to-one correspondence between free orthogonal quantum groups of infinite level, and free unitary quantum groups satisfying $G=G^c$.