Representation categories of compact matrix quantum groups

• Darstellungskategorien kompakter Matrixquantengruppen

Maaßen, Laura; Hiß, Gerhard (Thesis advisor); Weber, Moritz (Thesis advisor); Freslon, Amaury (Thesis advisor)

Aachen : RWTH Aachen University (2021)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2021

Abstract

One key result obtained from the investigation of compact matrix quantum groups is a Tannaka-Krein type duality, by which any compact matrix quantum group can be fully recovered from its representation category. Following this idea, easy quantum groups are defined through a combinatorial description of their representation categories. In this thesis, we study the representation categories of so-called group-theoretical quantum groups and show that they can be described by a combinatorial calculus similar to that used for easy quantum groups. Furthermore, we analyse the structure of abstract tensor categories that interpolate the representation categories of easy quantum groups. This thesis thus concerns research problems at the intersection of the theory of compact quantum groups, combinatorics and category theory with links to group theory. The first part of this thesis concerns group-theoretical quantum groups. We define an analogue of orthogonal group-theoretical quantum groups in the unitary setting and show that their description as semi-direct product quantum groups can be generalised. We describe their representation categories, both in the easy and the non-easy case. For this purpose, we introduce modified versions of categories of partitions, which model the 'group-theoretical structure' of the diagonal subgroups of group-theoretical quantum groups. Moreover, we define a modified fiber functor linked with the classical fiber functor via Moebius inversion. Subsequently, we show that the application of the Tannaka-Krein duality yields the desired description of the representation categories of group-theoretical quantum groups. Next, we restrict our attention to the orthogonal case. Although it is known that uncountably many orthogonal group-theoretical easy quantum groups exist, almost no concrete examples have been studied. We compute various examples with small generators, including in particular a new series of easy quantum groups between the hyperoctahedral series and higher hyperoctahedral series. We conclude our analysis of orthogonal group-theoretical quantum groups by an improved version of a de Finetti theorem by Raum and Weber. In the second part of this thesis, we study interpolating partition categories in the framework of Deligne's interpolation categories. Interpolating partition categories are the categorial abstraction of categories of partitions together with a complex interpolation parameter. We explain that their semisimplifications interpolate the representation categories of easy quantum groups. Next, we show that the semisimplicity of an interpolating partition category is encoded in the determinants of certain Gram matrices. We compute the set of interpolation parameters yielding semisimple interpolating partition categories for all group-theoretical easy quantum groups. Moreover, we parametrise the indecomposable objects in all interpolating partition categories by an explicitly constructible system of finite groups and exhibit their Grothendieck rings as filtered deformations. We apply these results to orthogonal easy groups and free orthogonal easy quantum groups.