Finite dimensional Hopf algebras over Kac-Paljutkin algebra $H_8$

Abstract

Let $H_8$ be the neither commutative nor cocommutative semisimple eight dimensional Hopf algebra, which is also called Kac-Paljutkin algebra \cite{MR0208401}. All simple Yetter-Drinfel'd modules over $H_8$ are given. As for simple objects and direct sums of two simple objects in ${}_{H_8}^{H_8}\mathcal{YD}$, we calculated dimensions for the corresponding Nichols algebras, except four semisimple cases which are generally difficult. Under the assumption that the four undetermined Nichols algebras are all infinite dimensional, we determine all the finite dimensional Nichols algebras over $H_8$. It turns out that the already known finite dimensional Nichols algebras are all diagonal type. In fact, they are Cartan types $A_1$, $A_2$, $A_2\times A_2$, $A_1\times \cdots \times A_1$, and $A_1\times \cdots \times A_1\times A_2$. By the way, we calculate Gelfand-Kirillov dimensions for some Nichols algebras. As an application, we obtain five families of new finite dimensional Hopf algebras over $H_8$ according to the lifting method.

36 pages. The article is being updated, comments are welcome

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• On some classification of finite-dimensional Hopf algebras over the Hopf algebra $H_{b:1}^{\ast }$ of Kashina
• Finite dimensional Nichols algebras over Suzuki algebra II: over simple Yetter-Drinfeld modules of $A_{N2n+1}^{\mu \lambda }$
• Multinomial expansion and Nichols algebras associated to non-degenerate involutive solutions of the Yang-Baxter equation
• The Nichols algebra $\mathfrak{B}(V_{abe})$ and a class of combinatorial numbers
• Automorphism group of Suzuki's Hopf algebra
• On Hopf algebras whose coradical is a cocentral abelian cleft extension