Finite-dimensional Nichols Algebras of Simple Yetter-Drinfeld Modules over the Suzuki Algebras

Abstract

In this paper, we continue to investigate finite-dimensional Nichols algebras of simple Yetter-Drinfeld modules over the Suzuki algebras

A_{Nn}^{\mu \lambda }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$A_{Nn}^{\mu\lambda}$$\end{document}

. It is finished for the case

A_{N2n}^{\mu \lambda }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$A_{N\,2n}^{\mu\lambda}$$\end{document}

. As for the case

A_{N2n+1}^{\mu \lambda }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$A_{N\,2n+1}^{\mu\lambda}$$\end{document}

, it boils down to the long-standing open problem: calculate dimensions of Nichols algebras of dihedral rack type

{\mathbb{D}}_{2n+1}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{D}_{2n+1}$$\end{document}

. It is interesting to see that the Suzuki algebras are set-theoretical. We pose some question or problems for our future research. In particular, we are curious about how to generalize the correspondence between braidings of rack type and group algebras to braidings of set-theoretical type.

Keywords

Nichols algebra / Suzuki algebra / Hopf algebra / Yetter-Drinfeld module

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