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Ergodic coactions with large multiplicity and monoidal equivalence of quantum groups. (English) Zbl 1122.46046
New examples of ergodic coactions of compact quantum groups in which the multiplicity of an irreducible corepresentation can be strictly larger than the dimension of the latter are constructed. These examples are obtained using a bijective correspondence between certain ergodic coactions on \(C^*\)-algebras and unitary fiber functors on the representation category of a compact quantum group. A complete classification of the ergodic coactions of full quantum multiplicity of the universal orthogonal groups \(A_O (F)\) and the universal unitary groups \(A_U (F)\) is obtained, as well as a computation of the \(2\)-cohomology of their duals. It is shown that the associated \(C^*\)-algebras and von Neumann algebras can be defined by generators and relations.

MSC:
46L55 Noncommutative dynamical systems
46L53 Noncommutative probability and statistics
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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