The structure of connected (graded) Hopf algebras
The structure of connected (graded) Hopf algebras
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DOI：
10.1016/j.aim.2020.107292
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发表时间：
2019-04
期刊：
影响因子：
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通讯作者：
G.-S. Zhou;Y. Shen;D. Lu
中科院分区：
文献类型：
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作者：
G.-S. Zhou;Y. Shen;D. Lu
本文通过证明在特征为0的域上存在一族齐次生成元和指数集上的全序满足一定条件，建立了连通分次Hopf代数的结构定理.结构定理的方法是建设性的，基于林登词的组合性质和标准括号的话。作为一个令人惊讶的结果的结构定理，我们表明，连接的分次Hopf代数有限Gelfand-Kirillov维数在一个领域的特征为0都是迭代的Hopf Ore扩展的基本领域。此外，在不假设Hopf代数具有有限的Gelfand-Kirillov维数（或仿射性）或基域是代数闭的情况下，作为结构定理的推论，观察到了特征为0的域上连通Hopf代数的一些关键事实.
In this paper, we establish a structure theorem for connected graded Hopf algebras over a field of characteristic 0 by claiming the existence of a family of homogeneous generators and a total order on the index set that satisfy some desirable conditions. The approach to the structure theorem is constructive, based on the combinatorial properties of Lyndon words and the standard bracketing on words. As a surprising consequence of the structure theorem, we show that connected graded Hopf algebras of finite Gelfand-Kirillov dimension over a field of characteristic 0 are all iterated Hopf Ore extensions of the base field. In addition, some keystone facts of connected Hopf algebras over a field of characteristic 0 are observed as corollaries of the structure theorem, without the assumptions of having finite Gelfand-Kirillov dimension (or affineness) on Hopf algebras or of that the base field is algebraically closed.