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@PHDTHESIS{Maaen:822228,
      author       = {Maaßen, Laura},
      othercontributors = {Hiß, Gerhard and Weber, Moritz and Freslon, Amaury},
      title        = {{R}epresentation categories of compact matrix quantum
                      groups},
      school       = {RWTH Aachen University},
      type         = {Dissertation},
      address      = {Aachen},
      publisher    = {RWTH Aachen University},
      reportid     = {RWTH-2021-06610},
      pages        = {1 Online-Ressource : Illustrationen},
      year         = {2021},
      note         = {Veröffentlicht auf dem Publikationsserver der RWTH Aachen
                      University; 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.},
      cin          = {114710 / 110000},
      ddc          = {510},
      cid          = {$I:(DE-82)114710_20140620$ / $I:(DE-82)110000_20140620$},
      typ          = {PUB:(DE-HGF)11},
      doi          = {10.18154/RWTH-2021-06610},
      url          = {https://publications.rwth-aachen.de/record/822228},
}