Communications in Mathematical Physics

, Volume 341, Issue 3, pp 751–779 | Cite as

The Full Classification of Orthogonal Easy Quantum Groups

  • Sven Raum
  • Moritz Weber


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 S n and a semi-direct product of a permutation action of S n 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.


Quantum Group Compact Quantum Group Noncrossing Partition Completion Lemma Haagerup Property 
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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Faktultät Mathematik und InformatikWestfälische Wilhelmsuniversität MünsterMünsterGermany
  2. 2.Fachbereich MathematikSaarland UniversitySaarbrückenGermany

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