# Hopf Ore Extensions

## TL;DR

This paper characterizes when skew polynomial extensions of Hopf algebras can be given a compatible Hopf structure, simplifying the conditions on the comultiplication map after a change of variables.

## Contribution

It shows that, under certain conditions, the comultiplication in skew polynomial extensions can be simplified, providing a complete characterization of such Hopf algebra extensions.

## Key findings

- Simplification of the comultiplication map after change of variables
- Complete characterization of Hopf structures on skew polynomial extensions
- Applicable to noetherian cocommutative Hopf algebras of finite Gelfand-Kirillov dimension

## Abstract

Brown, O'Hagan, Zhang, and Zhuang gave a set of conditions on an automorphism $\sigma$ and a $\sigma$-derivation $\delta$ of a Hopf $k$-algebra $R$ for when the skew polynomial extension $T=R[x, \sigma, \delta]$ of $R$ admits a Hopf algebra structure that is compatible with that of $R$. In fact, they gave a complete characterization of which $\sigma$ and $\delta$ can occur under the hypothesis that $\Delta(x)=a\otimes x +x\otimes b +v(x\otimes x) +w$, with $a, b\in R$ and $v, w\in R\otimes_k R$, where $\Delta: R\to R\otimes_k R$ is the comultiplication map. In this paper, we show that after a change of variables one can in fact assume that $\Delta(x)=\beta^{-1}\otimes x +x\otimes 1 +w$, with $\beta $ is a grouplike element in $R$ and $w\in R\otimes_k R,$ when $R\otimes_k R$ is a domain and $R$ is noetherian. In particular, this completely characterizes skew polynomial extensions of a Hopf algebra that admit a Hopf structure extending that of the ring of coefficients under these hypotheses. We show that the hypotheses hold for domains $R$ that are noetherian cocommutative Hopf algebras of finite Gelfand-Kirillov dimension.