Topological automorphism groups of compact quantum groups

Abstract

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 with the construction of a compact matrix quantum group whose fusion ring is not finitely generated, unlike the classical case.

Automorphisms of compact quantum groups

“…Remark In a very recent work, the second author along with Chirvasitu have proved that indeed the outer automorphism group of any compact matrix quantum group is discrete. However, in the same paper, a counterexample is given to show that the representation ring of a compact matrix quantum group, need not be finitely generated as a ring, in a sharp departure from the classical case.…”