Centralizers of linear and locally nilpotent derivations

Abstract

UDC 512.715, 512.554.31

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero, $\mathbb{K}[x_1,\dots,x_n]$ be the polynomial algebra and $W_n(\mathbb{K})$ be the Lie algebra of all $\mathbb K$-derivations on $\mathbb{K}[x_1,\dots,x_n].$ For any derivation $D$ with linear components,  we describe the centralizer of $D$ in $W_n(\mathbb{K}),$ and  propose an algorithm for finding the generators of this centralizer regarded as a module over the ring of constants of the derivation $D$ in the case where $D$ is a basic Weitzenboeck derivation. In a more general case where a finitely generated integral domain $A$ over the field $\mathbb{K}$ is considered instead of the polynomial algebra $\mathbb{K}[x_1,\dots,x_n]$ and $D$ is a locally nilpotent derivation on $A,$ we prove that the centralizer ${\rm C}_{{\rm Der} A}(D)$ of $D$ in the Lie algebra ${\rm Der} A$ of all $\mathbb K$-differentiations on $A$ is a ``large'' subalgebra of ${\rm Der} A.$ Specifically, the rank of ${\rm C}_{{\rm Der} _A}(D)$ over $A$ is equal to the transcendence degree of the field of fractions $\mathrm{Frac}(A)$ over the field~$\mathbb K.$