Noetherian pointed Hopf algebras are affine
Huan Jia, Yinhuo Zhang
TL;DR
Question: Are Noetherian Hopf algebras affine? This paper addresses it for pointed Hopf algebras by introducing a reduction framework on words: reduction order, reduction-factorization, and prime words. Using skew-triangular comultiplications, it shows every word can be expressed modulo a defining ideal as a product of irreducible letters, and that right/left Noetherian pointed Hopf algebras are finitely generated as algebras. Consequently, any Noetherian pointed Hopf algebra is affine; furthermore, Hopf subalgebras of such algebras are affine and connected Noetherian Hopf algebras are affine. The results extend and generalize prior work over algebraically closed fields of characteristic zero and illustrate a PBW-type, combinatorial method for affineness.
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.