Ergodic Coactions with Large Multiplicity and Monoidal Equivalence of Quantum Groups

By Julien Bichon, An De Rijdt, S. Vaes. 2005

157 citations recorded by the source corpus.

We construct 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. 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. We classify these unitary fiber functors on the universal orthogonal and unitary quantum groups. The associated C*-algebras and von Neumann algebras can be defined by generators and relations, but are not yet well understood.

Subfields: Atomic and Molecular Physics, and Optics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d438b6db0f18d53949843dfde7ef0554944989b2

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In Atomic and Molecular Physics, and Optics

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Citations in the corpus

CCAP for Universal Discrete Quantum Groups (2013, background): “satisfying obvious coassociativity conditions, with (O(Gr,Gr),∆rr) = (O(Gr),∆r) and the coactions ∆rs and ∆ s rs being ergodic [5].”
The Baum–Connes conjecture for free orthogonal quantum groups (2009, methodology): “In order to explain this, we shall discuss the notion of monoidal equivalence for compact quantum groups introduced by Bichon, de Rijdt and Vaes [8].”
Examples of weakly amenable discrete quantum groups (2012, methodology): “We will use the notion of monoidal equivalence and we refer the reader to [7] for the relevant definitions and properties.”
INVARIANTS OF THE HALF-LIBERATED ORTHOGONAL GROUP (2009, background): “This shows also that O n is not monoidally equivalent, in the sense of [10], to any known compact quantum group so far, in particular it is the first original example of a compact quantum group with exponential growth as considered in [7].”
Classification results for easy quantum groups (2009, background): “One interesting question would be to understand the twisting and deformation of the easy quantum groups, say with the objective of extending our formalism to the S(2) 6= id case, via monoidal equivalence [15].”
Reduced operator algebras of trace-preserving quantum automorphism groups (2012, unknown): “In [13], the notion of monoidal equivalence for compact quantum groups was introduced, and in [32], Vaes and Vergnioux showed that exactness of reduced C∗-algebras of compact quantum groups is preserved by monoidal equivalence.”
The Property of Rapid Decay for Discrete Quantum Groups (2020, background): “Moreover the dual of Ao(Q) is unimodular if and only if Q is a multiple of a unitary matrix, and up to an isomorphism one can then assume that Q = IN or Q = ( 0 −Ik Ik 0 ) [5].”