Categorical duality for Yetter-Drinfeld algebras

Abstract

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.

24 pages; v3: the last section was split off as a separate manuscript arxiv:1405.6574, as it no longer relies on this paper, a converse to Tomatsu's result on Poisson boundaries added; v4: minor corrections, to appear in Doc Math

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• Noncommutative Poisson boundaries and Furstenberg-Hamana boundaries of Drinfeld doubles
• Martin boundaries of the duals of free unitary quantum groups
• Towards a classification of compact quantum groups of Lie type
• Probabilistic boundaries of finite extensions of quantum groups
• Convergence to the boundary for random walks on discrete quantum groups and monoidal categories