Finite dimensional Nichols algebras over Suzuki algebra I: simple Yetter-Drinfeld modules of $A_{N2n}^{\mu \lambda }$

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 $A_{N2n}^{\mu \lambda }$ 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 ${\mathbf{A}}_2(q;I_2)$ and the Nichols algebra ufo(8). There are $64$, $4m$ and $m^2$-dimensional Nichols algebras of non-diagonal type over $A_{N2n}^{\mu \lambda }$. The $64$-dimensional Nichols algebras are of dihedral rack type ${\mathbb{D}}_4$. 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 $b^2=(ae{)}^{-1}$, $b\in {\mathbb{G}}_m$ for $m\ge 5$.

arXiv comment28 Pages, Comments are welcome

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Cited by in corpus (2)

• Multinomial expansion and Nichols algebras associated to non-degenerate involutive solutions of the Yang-Baxter equation
• Automorphism group of Suzuki's Hopf algebra