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Hopf Ore Extensions
Algebras and Representation Theory ( IF 0.5 ) Pub Date : 2019-05-22 , DOI: 10.1007/s10468-019-09901-8
Hongdi Huang

Brown, O’Hagan, Zhang, and Zhuang gave a set of conditions on an automorphism σ and a σ-derivation δ of a Hopf k-algebra R for when the skew polynomial extension T = R[x,σ,δ] of R admits a Hopf algebra structure that is compatible with that of R. In fact, they gave a complete characterization of which σ and δ can occur under the hypothesis that Δ(x) = ax + xb + v(xx) + w, with a,bR and v,wRkR, where Δ : RRkR is the comultiplication map. In this paper, we show that after a change of variables one can in fact assume that Δ(x) = β− 1x + x ⊗ 1 + w, with β is a grouplike element in R and wRkR, when RkR 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.

中文翻译:

霍普夫矿石延伸

棕色,奥黑根,章,和庄给一组条件上的构σσ -derivation δ一个的Hopf的ķ代数- [R ,用于当斜多项式扩展Ť = - [R [ Xσδ ]的ř接受与R相容的Hopf代数结构。事实上,他们给其中一个完整的表征σδ可以假设下发生的Δ(X)=一个X + Xb + vXX)+瓦特,具有一个b[Rv瓦特[Rķ - [R ,其中Δ:- [R[Rķ - [R是余乘法地图。在本文中,我们表明,变量的变化之后的一个实际上可以假定Δ(X)= β - 1X + X ⊗1 +瓦特,具有β处于类群元件ř瓦特[Rķ - [R,当[Rķ - [R是域和- [R是诺特。特别是,这完全表征了Hopf代数的斜多项式扩展,这种扩展允许在这些假设下Hopf结构扩展系数环的扩展。我们证明了对于域R的假设是有限Gelfand-Kirillov维数的noetherian协交换Hopf代数。
更新日期:2019-05-22
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