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.
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Acknowledgments
The author gratefully acknowledges her advisor Jason Bell for his constant encouragement and advice. The author also thanks Ken Brown for useful comments and thanks the referee for suggesting an improvement to the proofs of Lemmas 2.1 and 2.2.
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Presented by: Kenneth Goodearl
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The author acknowledges support from the National Sciences and Engineering Research Council of Canada.
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Huang, H. Hopf Ore Extensions. Algebr Represent Theor 23, 1477–1486 (2020). https://doi.org/10.1007/s10468-019-09901-8
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DOI: https://doi.org/10.1007/s10468-019-09901-8