Tannaka-Krein reconstruction and ergodic actions of easy quantum groups

TL;DR

This paper introduces a new reconstruction method for ergodic actions of compact quantum groups, characterizes certain algebraic structures, constructs examples from combinatorial data, and proves that quantum permutation groups cannot act ergodically on classical connected spaces.

Contribution

It provides an alternative reconstruction procedure, refines characterizations of Yetter-Drinfeld C*-algebras, and addresses a question about quantum permutation groups' actions.

Findings

New reconstruction method for ergodic actions.

Construction of ergodic actions from combinatorial data.

Quantum permutation group cannot act ergodically on classical connected spaces.

Abstract

We give a new alternative version of the reconstruction procedure for ergodic actions of compact quantum groups and we refine it to include characterizations of (braided commutative) Yetter-Drinfeld C*-algebras. We then use this to construct families of ergodic actions of easy quantum groups out of combinatorial data involving partitions and study them. Eventually, we use this categorical point of view to show that the quantum permutation group cannot act ergodically on a classical connected compact space, thereby answering a question of D. Goswami and H. Huang.