On idempotent states on quantum groups

TL;DR

This paper explores the structure of idempotent states on compact quantum groups, showing their connection to group-like projections and finite quantum hypergroups, and introduces an order structure on these states.

Contribution

It establishes a canonical correspondence between idempotent states and Haar states on finite quantum hypergroups, and studies their order structure.

Findings

Idempotent states correspond to group-like projections in the dual.

Every finite quantum group’s idempotent state arises from a quantum hypergroup.

An order structure on idempotent states is introduced and analyzed.

Abstract

Idempotent states on a compact quantum group are shown to yield group-like projections in the multiplier algebra of the dual discrete quantum group. This allows to deduce that every idempotent state on a finite quantum group arises in a canonical way as the Haar state on a finite quantum hypergroup. A natural order structure on the set of idempotent states is also studied and some examples discussed.