Categories of Two-Colored Pair Partitions, Part I: Categories Indexed by Cyclic Groups

TL;DR

This paper classifies categories of two-colored pair partitions indexed by cyclic groups, revealing many new half-liberation procedures and connecting combinatorial structures to quantum groups.

Contribution

It introduces a new classification of partition categories indexed by cyclic groups and explores their connection to half-liberated quantum groups.

Findings

Categories are indexed by cyclic groups.

Many new half-liberation procedures are identified.

Connections to easy quantum groups are established.

Abstract

We classify certain categories of partitions of finite sets subject to specific rules on the colorization of points and the sizes of blocks. More precisely, we consider pair partitions such that each block contains exactly one white and one black point when rotated to one line; however crossings are allowed. There are two families of such categories, the first of which is indexed by cyclic groups and is covered in the present article; the second family will be the content of a follow-up article. Via a Tannaka-Krein result, the categories in the two families correspond to easy quantum groups interpolating the classical unitary group and Wang's free unitary quantum group. In fact, they are all half-liberated in some sense and our results imply that there are many more half-liberation procedures than previously expected. However, we focus on a purely combinatorial approach leaving quantum group aspects aside.