Topological automorphism groups of compact quantum groups
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We study the topological structure of the automorphism groups of compact quantum groups showing that, in parallel to a classical result due to Iwasawa, the connected component of identity of the automorphism group and of the “inner” automorphism group coincide. For compact matrix quantum groups, which can be thought of as quantum analogues of compact Lie groups, we prove that the inner automorphism group is a compact Lie group and the outer automorphism group is discrete. Applications of this to the study of group actions on compact quantum groups are highlighted. We end by providing examples of compact matrix quantum groups with infinitely-generated fusion rings, in stark contrast with the classical situation. Along the way we study the invariant theory of finite group actions on free Laurent rings and show that the rings of invariants are, in general, not finitely generated.
KeywordsAutomorphism group Outer automorphism group Fusion ring Compact quantum group Dynamical system
Mathematics Subject Classification46L52 46L55 46L85 16T05
This work was initiated at the 7th ECM satellite conference “Compact Quantum Groups” at Greifswald, Germany and the authors are grateful to the organizers, Uwe Franz, Malte Gerhold, Adam Skalski and Moritz Weber, for the invitation. The authors would also like to thank Yuki Arano and Makoto Yamashita for interesting discussions. The first author was partially supported by NSF grant DMS-1565226. The second author is supported by a DST Inspire Faculty Fellowship. We would like to thank the anonymous referees for numerous suggestions leading to the improvement of the paper.
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