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Towards a Classification of Compact Quantum Groups of Lie Type

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Operator Algebras and Applications

Part of the book series: Abel Symposia ((ABEL,volume 12))

Abstract

This is a survey of recent results on classification of compact quantum groups of Lie type, by which we mean quantum groups with the same fusion rules and dimensions of representations as for a compact connected Lie group G. The classification is based on a categorical duality for quantum group actions recently developed by De Commer and the authors in the spirit of Woronowicz’s Tannaka–Krein duality theorem. The duality establishes a correspondence between the actions of a compact quantum group H on unital C∗-algebras and the module categories over its representation category \(\mathop{\mathrm{Rep}}\nolimits H\). This is further refined to a correspondence between the braided-commutative Yetter–Drinfeld H-algebras and the tensor functors from \(\mathop{\mathrm{Rep}}\nolimits H\). Combined with the more analytical theory of Poisson boundaries, this leads to a classification of dimension-preserving fiber functors on the representation category of any coamenable compact quantum group in terms of its maximal Kac quantum subgroup, which is the maximal torus for the q-deformation of G if q ≠ 1. Together with earlier results on autoequivalences of the categories \(\mathop{\mathrm{Rep}}\nolimits G_{q}\), this allows us to classify up to isomorphism a large class of quantum groups of G-type for compact connected simple Lie groups G. In the case of G = SU(n) this class exhausts all non-Kac quantum groups.

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Acknowledgements

S.N. was supported by the European Research Council under the EuropeanUnion’s Seventh Framework Programme (FP/2007–2013)/ERC Grant Agreementno. 307663.

M.Y. was supported by JSPS KAKENHI Grant Number 25800058, and partially by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).

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Authors and Affiliations

  1. Department of Mathematics, University of Oslo, 1053, Blindern, NO-0316, Oslo, Norway

    Sergey Neshveyev

  2. Department of Mathematics, Ochanomizu University, Otsuka 2-1-1, Bunkyo, 112-8610, Tokyo, Japan

    Makoto Yamashita

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  1. Sergey Neshveyev
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Correspondence to Sergey Neshveyev .

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Editors and Affiliations

  1. Department of Science and Technology, University of the Faroe Islands, Tórshavn, Faroe Islands

    Toke M. Carlsen

  2. Department of Mathematics, University of Oslo, Oslo, Norway

    Nadia S. Larsen

  3. Department of Mathematics, University of Oslo, Oslo, Norway

    Sergey Neshveyev

  4. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Christian Skau

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Neshveyev, S., Yamashita, M. (2016). Towards a Classification of Compact Quantum Groups of Lie Type. In: Carlsen, T.M., Larsen, N.S., Neshveyev, S., Skau, C. (eds) Operator Algebras and Applications. Abel Symposia, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-39286-8_11

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