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) = a ⊗ x + x ⊗ b + v(x ⊗ x) + w, with a,b ∈ R and v,w ∈ R ⊗_kR, where Δ : R → R ⊗_kR is the comultiplication map. In this paper, we show that after a change of variables one can in fact assume that Δ(x) = β^− 1 ⊗ x + x ⊗ 1 + w, with β is a grouplike element in R and w ∈ R ⊗_kR, when R ⊗_kR 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：

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霍普夫矿石延伸

棕色，奥黑根，章，和庄给一组条件上的构σ和σ -derivation δ一个的Hopf的ķ代数- [R ，用于当斜多项式扩展Ť = - [R [ X，σ，δ ]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σ和δ可以假设下发生的Δ（X）=一个⊗ X + X ⊗b + v（X ⊗ X）+瓦特，具有一个，b ∈ [R和v，瓦特∈ [R ⊗ _ķ - [R ，其中Δ：- [R → [R ⊗ _ķ - [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 ^{- 1} ⊗ X + X ⊗1 +瓦特，具有β处于类群元件ř和瓦特∈ [R ⊗ _ķ - [R，当[R ⊗ _ķ - [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。