The structure of connected (graded) Hopf algebras revisited

Let H be a connected graded Hopf algebra over a field of characteristic zero and K an arbitrary graded Hopf subalgebra of H. We show that there is a family of homogeneous elements of H indexed by a totally order set that satisfy several desirable conditions, which reveal interesting connections betw...

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Bibliographic Details
Published inJournal of algebra Vol. 610; pp. 684 - 702
Main Authors Li, C.-C., Zhou, G.-S.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.11.2022
Subjects
Connected Hopf algebra
Gelfand-Kirillov dimension
Hopf Ore extension
Lyndon word
Connected Hopf algebra
16W50
Lyndon word
68R15
16T05
Hopf Ore extension
16P90
16S15
Gelfand-Kirillov dimension
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Summary:Let H be a connected graded Hopf algebra over a field of characteristic zero and K an arbitrary graded Hopf subalgebra of H. We show that there is a family of homogeneous elements of H indexed by a totally order set that satisfy several desirable conditions, which reveal interesting connections between H and K. In particular, the set on non-decreasing products of these elements give a basis for H as a left and right graded K-module. As one of its consequences, we see that H is a graded iterated Hopf Ore extension of K of derivation type provided that H is of finite Gelfand-Kirillov dimension. The main tool of this work is Lyndon words, along the idea developed by Lu, Shen and the second-named author in [24].
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2022.07.031