Quantum automorphisms of twisted group algebras and free hypergeometric laws

Abstract

We prove that we have an isomorphism of type A a u t ( C σ [ G ] ) ≃ A a u t ( C [ G ] ) σ A_{aut}(\mathbb C_\sigma [G])\simeq A_{aut}(\mathbb C[G])^\sigma , for any finite group G G , and any 2 2 -cocycle σ \sigma on G G . In the particular case G = Z n 2 G=\mathbb Z_n^2 , this leads to a Haar measure-preserving identification between the subalgebra of A o ( n ) A_o(n) generated by the variables u i j 2 u_{ij}^2 and the subalgebra of A s ( n 2 ) A_s(n^2) generated by the variables X i j = ∑ a , b = 1 n p i a , j b X_{ij}=\sum _{a,b=1}^np_{ia,jb} . Since u i j u_{ij} is “free hyperspherical” and X i j X_{ij} is “free hypergeometric”, we obtain in this way a new free probability formula, which at n = ∞ n=\infty corresponds to the well-known relation between the semicircle law and the free Poisson law.