Noetherian pointed Hopf algebras are affine

Abstract

Let $k$ be a field. In this paper, we introduce the notions of reduction order and reduction-factorization 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 $H$ over $k$, we construct a well-ordered set such that: (1) $H$ is generated, as an algebra, by the subset of irreducible letters (with respect to the reduction order); and (2) $H$ is finite whenever it is right/left Noetherian.