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Actions of the additive group Ga on certain noncommutative deformations of the plane

Actions of the additive group Ga on certain noncommutative deformations of the plane 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Communications in Mathematics de Gruyter

Actions of the additive group Ga on certain noncommutative deformations of the plane

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Publisher
de Gruyter
Copyright
© 2021 Ivan Kaygorodov et al., published by Sciendo
eISSN
2336-1298
DOI
10.2478/cm-2021-0024
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

Communications in Mathematicsde Gruyter

Published: Jun 1, 2021

Keywords: Derivations; iterative higher derivations; rings of differential operators; Weyl algebra; 13N15; 16W20; 16S10; 16S32

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