Liberation of orthogonal Lie groups

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: $O_n,S_n,H_n,B_n,S_n^{\prime },B_n^{\prime }$. 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.

42 pages

References in corpus (5)

• Integration with respect to the Haar measure on unitary, orthogonal and symplectic group
• A noncommutative de Finetti theorem: Invariance under quantum permutations is equivalent to freeness with amalgamation
• Integration over compact quantum groups
• The hyperoctahedral quantum group
• Integration over quantum permutation groups

Cited by in corpus (5)

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• Classification of Globally Colorized Categories of Partitions
• Riesz transforms on compact quantum groups and strong solidity
• Principles of operator algebras
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