Quantum automorphisms of twisted group algebras and free hypergeometric laws

TL;DR

This paper establishes an isomorphism between quantum automorphism algebras of twisted and untwisted group algebras, leading to a new free probability relation connecting free hyperspherical and free hypergeometric laws.

Contribution

It proves a general isomorphism for quantum automorphism algebras of twisted group algebras and derives a novel free probability formula linking free hyperspherical and free hypergeometric laws.

Findings

Isomorphism between $A_{aut}(C_\sigma[G])$ and $A_{aut}(C[G])^\sigma$

Identification of subalgebras in $A_o(n)$ and $A_s(n^2)$

New free probability relation at $n=\infty$ between semicircle and free Poisson 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.