Noetherian pointed Hopf algebras are affine

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 $k$ be a field. In this paper, we introduce the notions of $\textit{reduction order}$ and $\textit{reduction-factorization}$ on words, and use them to show that any right or left Noetherian pointed Hopf algebra over $k$ 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 $H$ over $k$, we construct a well-ordered set $X$ such that: (1) $H$ is generated, as an algebra, by the subset $X_I$ of irreducible letters (with respect to the reduction order); and (2) $X_I$ is finite whenever $H$ is right Noetherian.