Categorical Duality for Yetter--Drinfeld Algebras

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

2010 Mathematics Subject Classification: Primary 20G42; Secondary 18D10, 46L53, 57T05.

Keywords and Phrases: C^*-tensor category, quantum group, Yetter--Drinfeld algebra, Poisson boundary.

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