The structure of connected (graded) Hopf algebras

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 cons...

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Bibliographic Details
Published inAdvances in mathematics (New York. 1965) Vol. 372; p. 107292
Main Authors Zhou, G.-S., Shen, Y., Lu, D.-M.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 07.10.2020
Subjects
Connected Hopf algebra
Gelfand-Kirillov dimension
Iterated Hopf Ore extension
Lyndon word
Connected Hopf algebra
16W50
16E65
Lyndon word
68R15
16T05
Iterated Hopf Ore extension
16P90
16S15
Gelfand-Kirillov dimension
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Summary: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.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2020.107292