Examples of finite-dimensional Hopf algebras with the dual Chevalley property

Abstract

We present new Hopf algebras with the dual Chevalley property by determining all semisimple Hopf algebras Morita-equivalent to a group algebra over a finite group, for a list of groups supporting a non-trivial finite-dimensional Nichols algebra.

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

“…It is also effective to study finite-dimensional copointed Hopf algebras ( [16], [27], [22]). We note that there are very few classification results on finite-dimensional Hopf algebras whose coradical is neither a group algebra nor the dual of a group algebra, some exceptions being [19], [26], [11]. It should be mentioned that [11] constructed Hopf algebras with the Chevalley property over a semisimple Hopf algebra H that is Morita-equivalent to a group algebra KG (in the sense of H H YD KG KG YD as braided tensor categories).…”

“…We note that there are very few classification results on finite-dimensional Hopf algebras whose coradical is neither a group algebra nor the dual of a group algebra, some exceptions being [19], [26], [11]. It should be mentioned that [11] constructed Hopf algebras with the Chevalley property over a semisimple Hopf algebra H that is Morita-equivalent to a group algebra KG (in the sense of H H YD KG KG YD as braided tensor categories). It doesn't cover our case since H 8 can be obtained from a group algebra by a 2-pseudo-cocycle twist but not by a 2-cocycle twist [45].…”

“…The author would like to thank Prof. Wenxue Huang, Yunnan Li, and Rongchuan Xiong for useful discussions, and also, thank the referees for careful reading and helpful comments on the writing of this paper. Special thanks to Prof. Nicolás Andruskiewitsch for recommending the Revista de la Unión Matemática Argentina to the author and pointing out references [11] and [45].…”

Finite-dimensional Hopf algebras over the Hopf algebra $H_{b:1}$ of Kashina

“…The lifting method has been applied to classify some finite-dimensional pointed Hopf algebras such as [3], [4], [5], [6], [9], [13], [17], [20], etc., and copointed Hopf algebras [14], [26], etc. Nevertheless, there are a few classification results on finite-dimensional Hopf algebras whose coradical is neither a group algebra nor the dual of a group algebra, for instance, [7], [16], [19], [24], [25], [33], [34], etc. In fact, Shi began a program in [33] to classify the objects of finite-dimensional growth from a given nontrivial semisimple Hopf algebra A 0 = H 8 via some relevant Nichols algebras B(V ) derived from its semisimple Yetter-Drinfeld modules V ∈ A 0 A 0 YD.…”

“…(1) U 1 (n Except for the case (7), the remaining families of Hopf algebras contain non-trivial lifting relations.…”