Idempotent states on Sekine quantum groups

Abstract

Sekine quantum groups are a family of finite quantum groups. The main result of this article 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.

Keywords:

• Finite quantum groups
• Sekine quantum groups
• idempotent states
• random walks

Mathematics Subject Classification (2010) Primary:

• 20G42
• 15A24
• Secondary: 60B15

Acknowledgments

Part of this work was done during a visit to Seoul National University. The author would like to thank Hun Hee Lee and Xiao Xiong for their hospitality. He would also like to thank Adam Skalski and Uwe Franz for many useful comments and pointing out some mistakes in an earlier version of the article.

Additional information

Funding

The research was partially supported by the NCN (National Centre of Science) grant 2014/14/E/ST1/00525, the French “Investissements d’Avenir” program, project ISITE-BFC (contract ANR-15-IDEX-03) and NSFC No. 11431011.