Duals of finite quantum permutation groups

TL;DR

This paper explores the axioms and properties of duals of finite quantum permutation groups, simplifying their theory and discussing potential extensions to infinite cases and quantum group actions on infinite graphs.

Contribution

It formulates axioms for duals of finite quantum permutation groups and analyzes their simplified structure, also considering extensions to infinite quantum groups.

Findings

Axioms for duals of finite quantum permutation groups are established.

The theory of such quantum groups simplifies in the dual setting.

Potential extensions to infinite quantum permutation groups are discussed.

Abstract

We work out axioms for the duals $G\subset U_N^+$ of the finite quantum permutation groups, $F\subset S_N^+$ with $|F|<\infty$, and we discuss how the basic theory of such quantum permutation groups partly simplifies in the dual setting. We discuss as well some potential extensions to the infinite case, in connection with the well-known question of axiomatizing the discrete quantum group actions on the infinite graphs.