Unitary easy quantum groups: geometric aspects

TL;DR

This paper explores the classification of unitary easy quantum groups with geometric properties, introducing associated noncommutative structures and axioms to understand their organization and examples.

Contribution

It develops a formalism linking quantum groups with noncommutative geometric objects, providing a framework for classification and conjecturing restrictions on known examples.

Findings

Main examples fit the developed formalism

Associated structures include noncommutative spheres, tori, and reflection groups

Axioms help restrict and understand the classification of quantum groups

Abstract

We discuss the classification problem for the unitary easy quantum groups, under strong axioms, of noncommutative geometric nature. Our main results concern the intermediate easy quantum groups $O_N\subset G\subset U_N^+$. To any such quantum group we associate its Schur-Weyl twist $\bar{G}$, two noncommutative spheres $S,\bar{S}$, a noncommutative torus $T$, and a quantum reflection group $K$. Studying $(S,\bar{S},T,K,G,\bar{G})$ leads then to some natural axioms, which can be used in order to investigate $G$ itself. We prove that the main examples are covered by our formalism, and we conjecture that in what concerns the case $U_N\subset G\subset U_N^+$, our axioms should restrict the list of known examples.