Hopf Ore Extensions

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.

9 pages