【作者】 吴金勇；

【导师】 丁南庆； 刘公祥；

【作者基本信息】 南京大学 ， 基础数学， 2016， 博士

【摘要】 作为非交换代数群理论的一种自然形式,无限维Hopf代数的研究在近些年取得了实质性的进展.在众多方面,无限维Hopf代数和有限维Hopf代数一样都表现出了它的优美的性质.在代数群理论中,一个经典的结果是一维连通代数群只有两类：k+和k×.本论文就是在前人的工作基础上完成非交换情形的分类,即GK-维数为1的素正则Hopf代数的分类.具体来说,Lu-Wu-Zhang[24]定义了同调积分,并发起了GK维数为1的Hopf代数的分类工作.在此基础上.Brown-Zhang[12]部分分类了GK-维数为1的素正则Hopf代数.本论文构造了一类新的Hopf代数D(m,d,ζ),并完成了GK-维数为1的有限生成素正则Hopf代数的分类.进一步地,详细研究了该类新的Hopf代数的性质.更多还原

【Abstract】 As a natural form of non-commutative algebraic group theory, infinite dimensional Hopf algebras have been studied intensively and substantial progress has been made in classifying infinite dimensional noetherian Hopf algebras of low GK-dimension in recent years. Infinite dimensional Hopf algebras and finite dimensional Hopf algebras share exquisite properties in many aspects. It is well-known that there are only two connected algebraic groups of dimension one:k+ and k×. This fact makes us believe that there should be a complete classification of affine prime regular Hopf algebras of GK-dimension one. In this thesis, we finish the classification based on previous studies. Concretely, Lu-Wu-Zhang [24] introduced the notion of homological integral of Hopf algebras and initiated the classification of Hopf algebras of GK-dimension one. Brown and Zhang [12] made further efforts in this direction and classified all affine prime regular Hopf algebras H of GK-dimension one under a hypothesis. We construct a new class of Hopf algebras D(m,d,ζ) and finish the classification of affine prime regular Hopf algebras of GK-dimension one. Further, properties of these new Hopf algebras are studied detailed.更多还原