Isomorphism problems and groups of automorphisms for Ore extensions $K[x][y; δ]$ (zero characteristic)

Abstract

Let $Ł(f) = K[x][y; f\frac{d}{dx} ]$ be an Ore extension of a polynomial algebra $K[x]$ over a field $K$ of characteristic zero where $f\in K[x]$. For a given polynomial $f$, the automorphism group of the algebra $Ł(f) $ is explicitly described. The polynomial case $Ł(0) = K[x,y]$ and the case of the Weyl algebra $A_1= K[x][y; \frac{d}{dx} ]$ were done done by Jung (1942) and van der Kulk (1953), and Dixmier (1968), respectively. In 1997, Alev and Dumas proved that the algebras $Ł(f)$ and $Ł(g)$ are isomorphic iff $g(x) = łf(αx+β)$ for some $ł, α\in K\backslash \{ 0\}$ and $β\in K$. In 2015, Benkart, Lopes and Ondrus gave a complete description of the set of automorphism groups of algebras $Ł(f)$. In this paper we complete the picture, i.e. {\em given} the polynomial $f$ we have the explicit description of the automorphism group of $Ł(f)$.