Finite dimensional Nichols algebras over Suzuki algebra I: simple Yetter-Drinfeld modules of $A_{N\,2n}^{\mu\lambda}$

TL;DR

This paper classifies simple Yetter-Drinfeld modules over Suzuki algebras and investigates their finite-dimensional Nichols algebras, revealing new types and dimensions, and establishing conditions for infinite-dimensionality.

Contribution

It provides a complete classification of simple Yetter-Drinfeld modules over Suzuki algebras and analyzes the structure and dimensions of associated Nichols algebras, including new realizations and infinite-dimensional criteria.

Findings

Finite-dimensional Nichols algebras of diagonal type are classified by Cartan and super types.

Existence of 64, 4m, and m^2-dimensional Nichols algebras over Suzuki algebras.

Conditions under which certain Nichols algebras are infinite-dimensional.

Abstract

The Suzuki algebra $A_{Nn}^{\mu \lambda}$ was introduced by Suzuki Satoshi in 1998, which is a class of cosemisimple Hopf algebras. It is not categorically Morita-equivalent to a group algebra in general. In this paper, the author gives a complete set of simple Yetter-Drinfeld modules over the Suzuki algebra $A_{N\,2n}^{\mu\lambda}$ and investigates the Nichols algebras over those simple Yetter-Drinfeld modules. The involved finite dimensional Nichols algebras of diagonal type are of Cartan type $A_1$, $A_1\times A_1$, $A_2$, $A_2\times A_2$, Super type ${\bf A}_{2}(q;I_2)$ and the Nichols algebra ufo(8). There are $64$, $4m$ and $m^2$-dimensional Nichols algebras of non-diagonal type over $A_{N\,2n}^{\mu \lambda}$. The $64$-dimensional Nichols algebras are of dihedral rack type $\Bbb{D}_4$. The $4m$ and $m^2$-dimensional Nichols algebras $\mathfrak{B}(V_{abe})$ discovered first by Andruskiewitsch and Giraldi can be realized in the category of Yetter-Drinfeld modules over $A_{Nn}^{\mu \lambda}$. By using a result of Masuoka, we prove that $\dim\mathfrak{B}(V_{abe})=\infty$ under the condition $b^2=(ae)^{-1}$, $b\in\Bbb{G}_{m}$ for $m\geq 5$.