The intertwiner spaces of non-easy group-theoretical quantum groups

Abstract

In 2015, Raum and Weber gave a definition of group-theoretical quantum groups, a class of compact matrix quantum groups with a certain presentation as semi-direct product quantum groups, and studied the case of easy quantum groups. In this article we determine the intertwiner spaces of non-easy group-theoretical quantum groups. We generalise group-theoretical categories of partitions and use a fiber functor to map partitions to linear maps which is slightly different from the one for easy quantum groups.We show that this construction provides the intertwiner spaces of group-theoretical quantum groups in general.

Group-theoretical graph categories

“…3, we introduce compact matrix quantum groups and their connections with categories of partitions and graphs. In particular, we recall the important results from [7,10]. In Sect.…”

“…The definition of a category of partitions was modified in [7] in order to generalize the above-described correspondence to all S V -invariant normal subgroups A Z * V 2 . We define the following operations that are essentially based on the group multiplication in Z * V 2 Considering P = ker(a, b) and Q = ker(c, d), we call the partition ker(ac, bd) a connected tensor product of P and Q.…”

“…In this section, we generalize the group-theoretical categories of partitions and skew categories of partitions from [7,9] (summarized here in Sects. 1.3 and 1.4) by introducing the following concepts.…”

“…A natural question is then, whether there is an alternative approach to model the representation categories. This was solved in [7] by introducing skew categories of partitions.…”

“…In Sect. 1, we introduce categories of partitions and graphs and recall the important combinatorial results from [9] and [7]. In Sect.…”