Quantum automorphisms of twisted group algebras and free hypergeometric laws

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.