量子$p$ -群的本原变形
Primitive Deformations of Quantum $p$-groups
摘要：
对于有限维Hopf代数，关于其特征$0$（例如在$\mathbb{C}$上）的分类研究已经有几十年的历史，取得了许多丰硕的成果，但其正特征的结构仍然是一个谜.本文在素特征为$p$的代数闭域$\mathbf{k}$上，引入了本原变形的概念，给出了一种结构化的方法来分类几乎连续生成的有限维连通Hopf代数，即这些连通Hopf代数是$p^{n+1}$维的，其本原空间是$n$维的交换限制李代数.我们举例说明这种技术的情况下$n=2$。结合我们在arXiv：1309.0286中的结果，我们给出了特征为$p>2$的$\mathbf{k}$上的$p^3$维连通Hopf代数的完全分类.
英文摘要：
For finite-dimensional Hopf algebras, their classification in characteristic $0$ (e.g. over $\mathbb{C}$) has been investigated for decades with many fruitful results, but their structures in positive characteristic have remained elusive. In this paper, working over an algebraically closed field $\mathbf{k}$ of prime characteristic $p$, we introduce the concept, called Primitive Deformation, to provide a structured technique to classify certain finite-dimensional connected Hopf algebras which are almost primitively generated; that is, these connected Hopf algebras are $p^{n+1}$-dimensional, whose primitive spaces are abelian restricted Lie algebras of dimension $n$. We illustrate this technique for the case $n=2$. Together with our preceding results in arXiv:1309.0286, we provide a complete classification of $p^3$-dimensional connected Hopf algebras over $\mathbf{k}$ of characteristic $p>2$.
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