Classification of non-Kac compact quantum groups of SU(n) type

Abstract

We classify up to isomorphism all non-Kac compact quantum groups with the same fusion rules and dimension function as $SU(n)$. For this we first prove, using categorical Poisson boundary, the following general result. Let $G$ be a coamenable compact quantum group and $K$ be its maximal quantum subgroup of Kac type. Then any dimension-preserving unitary fiber functor $RepG\to Hil{b}_f$ factors, uniquely up to isomorphism, through $RepK$. Equivalently, we have a canonical bijection $H^2(\hat{G};T)\cong H^2(\hat{K};T)$. Next, we classify autoequivalences of the representation categories of twisted $q$-deformations of compact simple Lie groups.

22 pages; v1: subsumes and strengthens the classification result from arXiv:1310.4407; v2: minor improvements, appendix corrected; v3: minor corrections, final version

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Cited by in corpus (5)

• Graded twisting of categories and quantum groups by group actions
• Poisson boundaries of monoidal categories
• Noncommutative Poisson boundaries and Furstenberg-Hamana boundaries of Drinfeld doubles
• Towards a classification of compact quantum groups of Lie type
• On rank one 2-representations of web categories