The inversion formulae for automorphisms of polynomial algebras and differential operators in prime characteristic

Abstract

Let $K$ be an {\em arbitrary} field of characteristic $p>0$, let $A$ be one of the following algebras: $P_n:=K[x_1,...,x_n]$ is a polynomial algebra, $\CD (P_n)$ is the ring of differential operators on $P_n$, $\CD (P_n)\t P_m$, the $n$'th {\em Weyl} algebra $A_n$, the $n$'th {\em Weyl} algebra $A_n\t P_m$ with polynomial coefficients $P_m$, the power series algebra $K[[x_1,...,x_n]]$, $T_{k_1,...,k_n}$ is the subalgebra of $\CD (P_n)$ generated by $P_n$ and the higher derivations $\der_i^{[j]}$, $0\le j<p^{k_i}$, $i=1,...,n$ (where $k_1,...,k_n\in \mathbb{N}$), $T_{k_1, ..., k_n}\t P_m$, an {\em arbitrary central simple} (countably generated) algebra over an {\em arbitrary} field. {\em The inversion formula} for automorphisms of the algebra $A$ is found {\em explicitly}.

29 pages

Cited by in corpus (3)

• An analogue of the Conjecture of Dixmier is true for the algebra of polynomial integro-differential operators
• Generators and defining relations for ring of differential operators on smooth affine algebraic variety in prime characteristic
• The group of order preserving automorphisms of the ring of differential operators on Laurent polynomial algebra in prime characteristic