The Full Classification of Orthogonal Easy Quantum Groups

By Sven Raum, Moritz Weber. 2013

90 citations recorded by the source corpus.

We study easy quantum groups, a combinatorial class of orthogonal quantum groups introduced by Banica–Speicher in 2009. We show that there is a countable descending chain of easy quantum groups interpolating between Bichon’s free wreath product with the permutation group Sn and a semi-direct product of a permutation action of Sn on a free product. This reveals a series of new commutation relations interpolating between a free product construction and the tensor product. Furthermore, we prove a dichotomy result saying that every hyperoctahedral easy quantum group is either part of our new interpolating series of quantum groups or belongs to a class of semi-direct product quantum groups recently studied by the authors. This completes the classification of easy quantum groups. We also study combinatorial and operator algebraic aspects of the new interpolating series.

Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/98bcc8db4e63264a2789f245e086d22805ddb792

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Citations in the corpus

Free Probability and Random Matrices (2014, background): “So we are, for example, not addressing such exciting topics as free stochastic and Malliavin calculus [40, 108, 114], or the rectangular version of free probability [28], or the strong version of asymptotic freeness [48, 58, 89], or free monotone transport [83], or the relation with representation t”
Easy quantum groups and quantum subgroups of a semi-direct product quantum group (2013, methodology): “Indeed, in [RW13b] we classify by combinatorial arguments the remaining hyperoctahedral easy quantum groups which are not contained in C(Z∗n 2 ) ⋈C(Sn).”
ON THE PARTITION APPROACH TO SCHUR-WEYL DUALITY AND FREE QUANTUM GROUPS (2014, background): “Orthogonal easy quantum groups are completely classified (see [39]) and some partial results are known in the unitary case (in the forthcoming paper [41]).”