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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.

Topological automorphism groups of compact quantum groups