Unitary quantum groups vs quantum reflection groups

TL;DR

This paper investigates the structure of intermediate quantum groups related to classical unitary and reflection groups, introducing new generation formulas and conjecturing a duality between their liberations.

Contribution

It provides new insights into the structure of intermediate quantum groups and proposes a conjecture on duality between their liberations, advancing understanding in quantum group theory.

Findings

Derived a generation formula for $H_N^{[ abla]}$ using crossed product methods

Solved specific questions about the structure of intermediate quantum groups

Conjectured a duality between liberations of $H_N$ and $U_N$

Abstract

We study the intermediate liberation problem for the real and complex unitary and reflection groups, namely $O_N,U_N,H_N,K_N$. For any of these groups $G_N$, the problem is that of understanding the structure of the intermediate quantum groups $G_N\subset G_N^\times\subset G_N^+$, in terms of the recently introduced notions of "soft" and "hard" liberation. We solve here some of these questions, our key ingredient being the generation formula $H_N^{[\infty]}=<H_N,T_N^+>$, coming via crossed product methods. Also, we conjecture the existence of a "contravariant duality" between the liberations of $H_N$ and of $U_N$, as a solution to the lack of a covariant duality between these liberations.