Classification of Globally Colorized Categories of Partitions

TL;DR

This paper classifies globally colorized categories of partitions, which are linked to easy quantum groups, and shows how the associated unitary quantum groups can be derived from orthogonal ones via tensor complexification.

Contribution

It provides a complete classification of globally colorized categories of partitions and introduces a method to construct their quantum groups from orthogonal quantum groups.

Findings

Classification of all globally colorized categories of partitions

Construction of associated unitary quantum groups from orthogonal ones

Extension of the theory to two-colored set partitions

Abstract

Set partitions closed under certain operations form a tensor category. They give rise to certain subgroups of the free orthogonal quantum group $O_n^+$, the so called easy quantum groups, introduced by Banica and Speicher in 2009. This correspondence was generalized to two-colored set partitions, which, in addition, assign a black or white color to each point of a set. Globally colorized categories of partitions are those categories that are invariant with respect to arbitrary permutations of colors. This article presents a classification of globally colorized categories. In addition, we show that the corresponding unitary quantum groups can be constructed from the orthogonal ones using tensor complexification.