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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Quantum automorphisms of twisted group algebras and free hypergeometric laws
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by Teodor Banica, Julien Bichon and Stephen Curran PDF
Proc. Amer. Math. Soc. 139 (2011), 3961-3971 Request permission

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
  • Email: teodor.banica@u-cergy.fr
  • Julien Bichon
  • Affiliation: Department of Mathematics, Clermont-Ferrand University, Campus des Cezeaux, 63177 Aubiere Cedex, France
  • MR Author ID: 633469
  • Email: bichon@math.univ-bpclermont.fr
  • 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
  • Email: curransr@math.berkeley.edu, curransr@math.ucla.edu
  • 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
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 139 (2011), 3961-3971
  • MSC (2010): Primary 46L65; Secondary 16W30, 46L54
  • DOI: https://doi.org/10.1090/S0002-9939-2011-10877-3
  • MathSciNet review: 2823042