Liberation of orthogonal Lie groups

By T. Banica, Roland Speicher. 2008

208 citations recorded by the source corpus.

Abstract We show that under suitable assumptions, we have a one-to-one correspondence between classical groups and free quantum groups, in the compact orthogonal case. We classify the groups under correspondence, with the result that there are exactly 6 of them: O n , S n , H n , B n , S n ′ , B n ′ . We investigate the representation theory aspects of the correspondence, with the result that for O n , S n , H n , B n , this is compatible with the Bercovici–Pata bijection. Finally, we discuss some more general classification problems in the compact orthogonal case, notably with the construction of a new quantum group.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fe2dca16750941cdb9c744745691ea605fd4f558

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

On the classification of easy quantum groups (2012, methodology): “organized as follows. In Section 1 we give a short introduction into the concept of the classiﬁcation of easy quantum groups. For details on this, we refer to the initial paper by Banica and Speicher [BS09]. In Section 2 we focus on the classiﬁcation of the free easy quantum groups S+ n ⊆ G ⊆ O+ n f”
The Connes embedding property for quantum group von Neumann algebras (2014, background): “m to determine all intermediate quantum subgroups ON ≤ G ≤ O+ N. At this time, only one such intermediate quantum subgroup is known, namely the half-liberated orthogonal quantum groupON ≤ O∗ N ≤ O+ N [7]. Moreover, it is known that the inclusion ON ≤ O∗ N is maximal, meaning that there is no interme”
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].”
Quantum Isometries and Noncommutative Spheres (2009, methodology): “ consider the following universal algebras: C(O ∗ n) = C ∗ u11,...,unn uij = uij,uijuklust = ustukluij,u t = u−1 C(O+ n ) = C ∗ u11,...,unn uij = u∗ ij,u t = u−1 These algebras, introduced in [6], [24], are Hopf algebras in the sense of Woronowicz [25], [26]. We refer to the recent paper [7] for”
The Full Classification of Orthogonal Easy Quantum Groups (2013, methodology): “Examples of categories of partitions include the set of all partitions P , the set of all pair partitions P2 (all blocks have size two), the set of all noncrossing partitions NC (the lines may be drawn such that they do not cross) and NC2 ([BaSp09]).”
Unitary Easy Quantum Groups: the free case and the group case (2015, methodology): “the generators ◦• and ◦ ◦ , since they are always in a category, by deﬁnition. UNITARY EASY QUANTUM GROUPS 5 1.4. The non-colored case. When deﬁning orthogonaleasy quantum groups, Banica and Speicher [BS09] introduced categories of partitions whose points are not colored. These categories are in one”
Easy quantum groups and quantum subgroups of a semi-direct product quantum group (2013, methodology): “In order to describe corepresentation categories of compact quantum matrix groups combinatorially, Banica and Speicher introduced the notions of a category of partitions and of easy quantum groups [BS09].”
Examples of weakly amenable discrete quantum groups (2012, background): “ Remark 5.11. Let us also point out that the above results and the isomorphisms of [28, Thm 4.1] imply that the free bistochastic quantum groups B+ N and their symmetrized versions (B+ N) ′ (see e.g. [7] for the deﬁnitions) have the Haagerup property and are weakly amenable with Cowling-Haagerup con”
ON THE PARTITION APPROACH TO SCHUR-WEYL DUALITY AND FREE QUANTUM GROUPS (2014, background): “ by PA. A coloured partition is said to be noncrossing if the underlying uncoloured partition is noncrossing and the collection of all noncrossing partitions is denoted by NCA. Following the ideas of [10], we now have to associate linear maps to coloured partitions. This is done by simply forgetting”