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

Abstract

Let Λ ( f ) = K [ x ] [ y ; f d d x ] \Lambda (f) = K[x][y; f\frac {d}{dx} ] be an Ore extension of a polynomial algebra K [ x ] K[x] over a field K K of characteristic zero where f ∈ K [ x ] f\in K[x] . For a given polynomial f f , the automorphism group of the algebra Λ ( f ) \Lambda (f) is explicitly described. The polynomial case Λ ( 0 ) = K [ x , y ] \Lambda (0) = K[x,y] and the case of the Weyl algebra A 1 = K [ x ] [ y ; d d x ] A_1= K[x][y; \frac {d}{dx} ] were done by Jung [J. Reine Angew. Math. 184 (1942), pp. 161–174] and van der Kulk [Nieuw Arch. Wisk. (3) 1 (1953), pp. 33–41], and Dixmier [Bul. Soc. Math. France 96 (1968), pp. 209–242], respectively. Alev and Dumas [Comm. Algebra 25 (1997), pp. 1655–1672] proved that the algebras Λ ( f ) \Lambda (f) and Λ ( g ) \Lambda (g) are isomorphic iff g ( x ) = λ f ( α x + β ) g(x) = \lambda f(\alpha x+\beta ) for some λ , α ∈ K ∖ { 0 } \lambda , \alpha \in K\backslash \{ 0\} and β ∈ K \beta \in K . Benkart, Lopes and Ondrus [Trans. Amer. Math. Soc. 367 (2015), pp. 1993–2021] gave a complete description of the set of automorphism groups of algebras Λ ( f ) \Lambda (f) . In this paper we complete the picture, i.e. given the polynomial f f we have the explicit description of the automorphism group of Λ ( f ) \Lambda (f) . The key concepts in finding the automorphism groups are the eigenform, the eigenroot and the eigengroup of a polynomial (introduced in the paper; they are of independent interest).