关于余根基为余中心阿贝尔裂扩张的Hopf代数
On Hopf algebras whose coradical is a cocentral abelian cleft extension
AI总结：
本文针对余根基为余中心阿贝尔裂扩张所得半单Hopf代数Kₙ的Hopf代数族开展研究，刻画单Yetter-Drinfeld模、计算融合规则、确定部分有限维Nichols代数，还得到新的216维Hopf代数。
摘要：
本文是全面描述一类Hopf代数的第一步，这类Hopf代数的余根基同构于半单Hopf代数Kₙ（n为正奇数），而Kₙ通过余中心阿贝尔裂扩张得到。我们刻画了单Yetter-Drinfeld模，计算了融合规则，并确定了其中部分模的有限维Nichols代数。特别地，我们给出了K₃上单模上的有限维Nichols代数的刻画，其中包含一族依赖于三次单位根的12维Nichols代数B_ξ。此处B₁同构于著名的Fomin-Kirillov代数；作为分次代数，B_ξ与B_{ξ²}同构，但当ξ≠1时，B₁与B_ξ作为代数不同构。作为副产品，我们得到了新的216维Hopf代数。
英文摘要：
This paper is a first step toward the full description of a family of Hopf algebras whose coradical is isomorphic to a semisimple Hopf algebra K_{n}, n an odd positive integer, obtained by a cocentral abelian cleft extension. We describe the simple Yetter-Drinfeld modules, compute the fusion rules and determine the finite-dimensional Nichols algebras for some of them. In particular, we give the description of the finite-dimensional Nichols algebras over simple modules over K_{3}. This includes a family of 12-dimensional Nichols algebras $B_ξ$ depending on 3rd roots of unity. Here, $B_{1}$ is isomorphic to the well-known Fomin-Kirillov algebra, and $B_ξ \simeq B_{ξ^{2}} $ as graded algebras but $B_1$ is not isomorphic to $B_ξ $ as algebra for $ξ\neq 1$. As a byproduct we obtain new Hopf algebras of dimension 216.
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