Homogeneous quantum groups and their easiness level

TL;DR

This paper introduces the concept of easiness level for homogeneous quantum groups, extending the classification beyond easy groups and analyzing maximality properties at higher levels.

Contribution

It defines the easiness level for homogeneous quantum groups and investigates their properties, especially maximality, at levels beyond the easy case.

Findings

Easiness level p=1 corresponds to easy groups.

Maximality of certain liberation inclusions persists at p=2.

Construction of the easiness level p for homogeneous quantum groups.

Abstract

Given a closed subgroup $G\subset U_N^+$ which is homogeneous, in the sense that we have $S_N\subset G\subset U_N^+$, the corresponding Tannakian category $C$ must satisfy $span(\mathcal{NC}_2)\subset C\subset span(P)$. Based on this observation, we construct a certain integer $p\in\mathbb N\cup\{\infty\}$, that we call "easiness level" of $G$. The value $p=1$ corresponds to the case where $G$ is easy, and we explore here, with some theory and examples, the case $p>1$. As a main application, we show that $S_N\subset S_N^+$ and other liberation inclusions, known to be maximal in the easy setting, remain maximal at the easiness level $p=2$ as well.