The classification of tensor categories of two-colored noncrossing partitions

TL;DR

This paper classifies all categories of two-colored noncrossing partitions, a combinatorial framework used in the study of orthogonal and unitary easy quantum groups, identifying ten series and two additional categories.

Contribution

It provides a complete combinatorial classification of two-colored noncrossing partition categories, extending the understanding of quantum group symmetries.

Findings

Ten series of categories indexed by natural numbers

Two additional unique categories identified

Parameters specify colorization on global and local levels

Abstract

Our basic objects are partitions of finite sets of points into disjoint subsets. We investigate sets of partitions which are closed under taking tensor products, composition and involution, and which contain certain base partitions. These so called categories of partitions are exactly the tensor categories being used in the theory of Banica and Speicher's orthogonal easy quantum groups. In our approach, we additionally allow a coloring of the points. This serves as the basis for the introduction of unitary easy quantum groups, which is done in a separate article. The present article however is purely combinatorial. We find all categories of two-colored noncrossing partitions. For doing so, we extract certain parameters with values in the natural numbers specifying the colorization of the categories on a global as well as on a local level. It turns out that there are ten series of categories, each indexed by one or two parameters from the natural numbers, plus two additional categories. This is just the beginning of the classification of categories of two-colored partitions and we point out open problems at the end of the article.