Easy quantum groups and quantum subgroups of a semi-direct product quantum group

By Sven Raum, Moritz Weber. 2013

46 citations recorded by the source corpus.

We consider compact matrix quantum groups whose fundamental corepresentation matrix has entries which are partial isometries with central support. We show that such quantum groups have a simple representation as semi-direct product quantum groups of a group dual quantum group by an action of a permutation group. This general result allows us to completely classify easy quantum groups with the above property by certain reflection groups. We give four applications of our result. First, there are uncountably many easy quantum groups. Second, there are non-easy quantum groups between the free orthogonal quantum group and the permutation group. Third, we study operator algebraic properties of the hyperoctahedral series. Finally, we prove a generalised de Finetti theorem for easy quantum groups in the scope of this article.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ceb54b3c22f9ad1029267394c92a9774c72ad980

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Roland Speicher
2. Moritz Weber
3. Sven Raum

In Geometry and Topology

1. Moritz Weber
2. Sven Raum
3. Roland Speicher
4. Amaury Freslon

In Mathematical Physics

1. Moritz Weber
2. Sven Raum
3. Roland Speicher
4. Amaury Freslon
5. Pierre Tarrago

Citations in the corpus

• Unitary Easy Quantum Groups: the free case and the group case (2015, unknown): “sic combinatorial feature. In this sense, general theorems about compact matrix quantum groups could be found by first investigating the easy quantum groups. An example for that method can be found in [RW15a], where any quantum group Sn ⊆G ⊆O+ n with u2 ij being central projections can be understood ”
• Free Probability and Random Matrices (2014, background): “This has triggered quite some investigations on more general quantum symmetries and its relations to de Finetti theorems, see [4, 5, 32, 52].”
• The Full Classification of Orthogonal Easy Quantum Groups (2013, methodology): “It began in [BaSp09], and was continued in [BaCuSp10], [We13],[RaWe13a] and [RaWe13b].”