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Quantum automorphisms of twisted group algebras and free hypergeometric laws

Authors: Teodor Banica, Julien Bichon and Stephen Curran
Journal: Proc. Amer. Math. Soc. 139 (2011), 3961-3971
MSC (2010): Primary 46L65; Secondary 16W30, 46L54
Published electronically: March 15, 2011
MathSciNet review: 2823042
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Abstract: We prove that we have an isomorphism of type $ A_{aut}(\mathbb{C}_\sigma[G])\simeq A_{aut}(\mathbb{C}[G])^\sigma$, for any finite group $ G$, and any $ 2$-cocycle $ \sigma$ on $ G$. In the particular case $ G=\mathbb{Z}_n^2$, this leads to a Haar measure-preserving identification between the subalgebra of $ A_o(n)$ generated by the variables $ u_{ij}^2$ and the subalgebra of $ A_s(n^2)$ generated by the variables $ X_{ij}=\sum_{a,b=1}^np_{ia,jb}$. Since $ u_{ij}$ is ``free hyperspherical'' and $ X_{ij}$ is ``free hypergeometric'', we obtain in this way a new free probability formula, which at $ n=\infty$ corresponds to the well-known relation between the semicircle law and the free Poisson law.

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Additional Information

Teodor Banica
Affiliation: Department of Mathematics, Cergy-Pontoise University, 95000 Cergy-Pontoise, France

Julien Bichon
Affiliation: Department of Mathematics, Clermont-Ferrand University, Campus des Cezeaux, 63177 Aubiere Cedex, France

Stephen Curran
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Address at time of publication: Department of Mathematics, University of California, Los Angeles, California 90095

Keywords: Quantum automorphism group, free hypergeometric law
Received by editor(s): February 16, 2010
Received by editor(s) in revised form: September 13, 2010
Published electronically: March 15, 2011
Communicated by: Marius Junge
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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