Quantum groups based on spatial partitions

TL;DR

This paper introduces a new class of compact matrix quantum groups based on spatial partitions, expanding the framework of easy quantum groups and revealing novel quantum subgroups and categorical structures.

Contribution

It extends the easy quantum groups framework to three-dimensional partitions, enabling the discovery of new quantum subgroups and categorical interpretations.

Findings

Identified new quantum subgroups of $O_n^+$ not containing $S_n$

Developed new quantum group products from partition categories

Provided quantum group interpretations for complex partition categories

Abstract

We define new compact matrix quantum groups whose intertwiner spaces are dual to tensor categories of three-dimensional set partitions -- which we call spatial partitions. This extends substantially Banica and Speicher's approach of the so called easy quantum groups: It enables us to find new examples of quantum subgroups of Wang's free orthogonal quantum group $O_n^+$ which do not contain the symmetric group $S_n$; we may define new kinds of products of quantum groups coming from new products of categories of partitions; and we give a quantum group interpretation of certain categories of partitions which do neither contain the pair partition nor the identity partition.