Hopf Ore Extensions

Abstract

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.