The classification of tensor categories of two-colored noncrossing partitions

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.

Models of quantum permutations

“…Orthogonal easy quantum groups have been defined for the first time in [BS09] and they have been generalized in [TW18] and [TW17] to unitary easy quantum groups. This section is aimed for readers familiar with easy quantum groups and we refer to the references above for more details.…”

Categories of two-colored pair partitions part I: categories indexed by cyclic groups

“…For a more detailed introduction to two-colored partitions and their categories, confer [TW17a], and, more specifically for this article, for a treatment of twocolored partitions with neutral blocks, including more examples and illustrations, see [MW18].…”

“…Banica and Speicher showed that categories of onecolored partitions correspond to certain quantum subgroups of Wang's ([Wan95]) free orthogonal quantum group O + n , namely the orthogonal easy quantum groups (see[BS09],[Web16] and[Web17a]). Categories of two-colored partitions are in bijection with so-called unitary easy quantum groups (cf [TW17a]…”

“…This is an ongoing project of the authors. (2) In general, the complete classification of categories of two-colored partitions is still an open question (see also [TW17a]; see [RW16] for the complete classification in the orthogonal case). The question of all such categories with ⊗ ∈ C has recently been settled by Gromada in [Gro18].…”

Classification of globally colorized categories of partitions

“…This paper can be divided into two parts. The first one being purely combinatorial is a contribution to the classification of categories of two-colored partitions, by which it extends the work [TW18]. The second part applies the results on the theory of compact matrix quantum groups using techniques developed in [TW17].…”