A New Characterisation of Idempotent States on Finite and Compact Quantum Groups

TL;DR

This paper establishes a correspondence between idempotent states and pre-subgroups in finite quantum groups, revealing lattice isomorphisms with coidalgebras, and extends these results to compact quantum groups under certain conditions.

Contribution

It introduces a new characterization of idempotent states on finite and compact quantum groups, linking them to pre-subgroups and coidalgebras, and demonstrates lattice isomorphisms.

Findings

Idempotent states correspond to pre-subgroups in finite quantum groups.

Lattices of idempotent states and coidalgebras are isomorphic in finite quantum groups.

Lattices of idempotent states and expected coidalgebras are isomorphic in compact quantum groups.

Abstract

We show that idempotent states on finite quantum groups correspond to pre-subgroups in the sense of Baaj, Blanchard, and Skandalis. It follows that the lattices formed by the idempotent states on a finite quantum group and by its coidalgebras are isomorphic. We show furthermore that these lattices are also isomorphic for compact quantum groups, if one restricts to expected coidalgebras.