Idempotent states on Sekine quantum groups

TL;DR

This paper classifies all idempotent states on Sekine quantum groups, revealing new non-Haar idempotent states, and analyzes their order structure and convergence properties of convolution powers.

Contribution

It provides a complete computation of all idempotent states on Sekine quantum groups, including a new class of non-Haar idempotent states, advancing understanding of their algebraic structure.

Findings

Identified all idempotent states on Sekine quantum groups

Discovered a new class of non-Haar idempotent states

Established conditions for convergence of convolution powers

Abstract

Sekine quantum groups are a family of finite quantum groups. The main result of this paper is to compute all the idempotent states on Sekine quantum groups, which completes the work of Franz and Skalski. This is achieved by solving a complicated system of equations using linear algebra and basic number theory. From this we discover a new class of non-Haar idempotent states. The order structure of the idempotent states on Sekine quantum groups is also discussed. Finally we give a sufficient condition for the convolution powers of states on Sekine quantum group to converge.