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AbstractWe connect the theorems of Rentschler [18] and Dixmier [10] on locally nilpotent derivations and automorphisms of the polynomial ring A0 and of the Weyl algebra A1, both over a field of characteristic zero, by establishing the same type of results for the family of algebras Ah=〈x,y|yx−xy=h(x)〉,{A_h} = \left\langle {x,y|yx - xy = h\left( x \right)} \right\rangle ,, where h is an arbitrary polynomial in x. In the second part of the paper we consider a field 𝔽 of prime characteristic and study 𝔽[t]-comodule algebra structures on Ah. We also compute the Makar-Limanov invariant of absolute constants of Ah over a field of arbitrary characteristic and show how this subalgebra determines the automorphism group of Ah.
Communications in Mathematics – de Gruyter
Published: Jun 1, 2021
Keywords: Derivations; iterative higher derivations; rings of differential operators; Weyl algebra; 13N15; 16W20; 16S10; 16S32
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