Authors: Ivan Kaygorodov ; Samuel A. Lopes ¹; Farukh Mashurov ²

• 1 Universidade do Porto = University of Porto
• 2 Kazakh-British Technical University

We connect the theorems of Rentschler [18] and Dixmier [10] onlocally nilpotent derivations and automorphisms of the polynomial ring A0and of the Weyl algebra A1, both over a field of characteristic zero, byestablishing the same type of results for the family of algebrasAh = hx, y | yx − xy = h(x)i,where h is an arbitrary polynomial in x. In the second part of the paper weconsider a field F of prime characteristic and study F[t]-comodule algebrastructures on Ah. We also compute the Makar-Limanov invariant of absolute constants of Ah over a field of arbitrary characteristic and show howthis subalgebra determines the automorphism group of Ah.

• 13N15 - Derivations and commutative rings
• 16S10 - Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
• 16S32 - Rings of differential operators (associative algebraic aspects)
• 16W20 - Automorphisms and endomorphisms