Categorical duality for Yetter-Drinfeld algebras (Q2339186)

Summary: We study tensor structures on \((\mathrm{Rep}\,G)\)-module categories defined by actions of a compact quantum group \(G\) on unital \(\mathrm C^*\)-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 \(G\)-\(\mathrm C^*\)-algebras is equivalent to the category of generating unitary tensor functors from \(\mathrm{Rep}\,G\) into \(\mathrm C^*\)-tensor categories. To illustrate this equivalence, we discuss coideals of quotient type in \(C(G)\), Hopf-Galois extensions and noncommutative Poisson boundaries.