Fixed Point Algebras for Easy Quantum Groups

TL;DR

This paper investigates fixed point algebras arising from actions of easy quantum groups on Cuntz algebras, establishing conditions under which these algebras are Kirchberg and computing their K-groups.

Contribution

It proves that free easy quantum groups satisfy certain conditions making their fixed point algebras Kirchberg and provides explicit K-group calculations for these algebras.

Findings

Fixed point algebras are Kirchberg algebras under certain conditions.

K-groups of fixed point algebras are explicitly computed.

Examples include quantum permutation and orthogonal quantum groups.

Abstract

Compact matrix quantum groups act naturally on Cuntz algebras. The first author isolated certain conditions under which the fixed point algebras under this action are Kirchberg algebras. Hence they are completely determined by their $K$-groups. Building on prior work by the second author, we prove that free easy quantum groups satisfy these conditions and we compute the $K$-groups of their fixed point algebras in a general form. We then turn to examples such as the quantum permutation group $S_n^+$, the free orthogonal quantum group $O_n^+$ and the quantum reflection groups $H_n^{s+}$. Our fixed point-algebra construction provides concrete examples of free actions of free orthogonal easy quantum groups, which are related to Hopf-Galois extensions.