Noetherian pointed Hopf algebras are affine
Huan Jia, Yinhuo Zhang

TL;DR
This paper proves that all right or left Noetherian pointed Hopf algebras over a field are affine, using reduction orders and factorization techniques, partially answering a longstanding question in Hopf algebra theory.
Contribution
It introduces reduction order and reduction-factorization concepts to establish the affineness of Noetherian pointed Hopf algebras, advancing understanding of their algebraic structure.
Findings
Any right or left Noetherian pointed Hopf algebra over a field is affine.
Constructs a well-ordered generating set for such Hopf algebras.
Shows the finiteness of irreducible generators when the algebra is right Noetherian.
Abstract
Let be a field. In this paper, we introduce the notions of and on words, and use them to show that any right or left Noetherian pointed Hopf algebra over is affine. This result offers a partial affirmative answer to the classical affineness question for Noetherian Hopf algebras posed by Wu and Zhang \cite{WZ2003}. For a pointed Hopf algebra over , we construct a well-ordered set such that: (1) is generated, as an algebra, by the subset of irreducible letters (with respect to the reduction order); and (2) is finite whenever is right Noetherian.
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Taxonomy
TopicsAdvanced Algebra and Logic · Algebraic structures and combinatorial models · Advanced Combinatorial Mathematics
