Categorical duality for Yetter-Drinfeld algebras

Abstract

We study tensor structures on $(\mathrm{Rep}G)$-module categories defined by actions of a compact quantum group $G$ on unital C${}^{\ast }$-algebras. We show that having a tensor product which defines the module structure is equivalent to enriching the action of $G$ to the structure of a braided-commutative Yetter–Drinfeld algebra. This shows that the category of braided-commutative Yetter–Drinfeld $G$-C${}^{\ast }$-algebras is equivalent to the category of generating unitary tensor functors from $\mathrm{Rep}G$ into C${}^{\ast }$-tensor categories. To illustrate this equivalence, we discuss coideals of quotient type in $C(G)$, Hopf–Galois extensions and noncommutative Poisson boundaries.