Quantum groups and generalized circular elements
Michael Brannan, Kay Kirkpatrick
TL;DR
This paper demonstrates that generators of certain free orthogonal quantum groups behave like free families of generalized circular and semicircular elements in large dimensions, and these quantum groups serve as symmetries of free Araki-Woods factors.
Contribution
It establishes a connection between free orthogonal quantum groups and free probability distributions, revealing their role as symmetries in free Araki-Woods factors.
Findings
Generators modeled by free circular and semicircular elements
Quantum groups act as distributional symmetries
Results hold in the large quantum dimension limit
Abstract
We show that with respect to the Haar state, the joint distributions of the generators of Van Daele and Wang's free orthogonal quantum groups are modeled by free families of generalized circular elements and semicircular elements in the large (quantum) dimension limit. We also show that this class of quantum groups acts naturally as distributional symmetries of almost-periodic free Araki-Woods factors.
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