Categorically Morita Equivalent Compact Quantum Groups

Abstract

We give a dynamical characterization of categorical Morita equivalence between compact quantum groups. More precisely, by a Tannaka-Krein type duality, a unital ${\mathrm{C}}^{\ast }$-algebra endowed with commuting actions of two compact quantum groups corresponds to a bimodule category over their representation categories. We show that this bimodule category is invertible if and only if the actions are free, with finite dimensional fixed point algebras, which are in duality as Frobenius algebras in an appropriate sense. This extends the well-known characterization of monoidal equivalence in terms of bi-Hopf-Galois objects.