Centralizers of linear and locally nilpotent derivations

Abstract

Let $K$ be an algebraically closed field of characteristic zero, $A = K[x_1,\dots,x_n]$ the polynomial ring, $R = K(x_1,\dots,x_n)$ the field of rational functions, and let $W_n(K) = \Der_{K}A$ be the Lie algebra of all $K$-derivations on $A$. If $D \in W_n(K),$ $D\not =0$ is linear (i.e. of the form $D = \sum_{i,j=1}^n a_{ij}x_j \frac{\partial}{\partial x_i}$) we give a description of the centralizer of $D$ in $W_n(K)$ and point out an algorithm for finding generators of $C_{W_n(K)}(D)$ as a module over the ring of constants in case when $D$ is the basic Weitzenboeck derivation. In more general case when the ring $A$ is a finitely generated domain over $K$ and $D$ is a locally nilpotent derivation on $A,$ we prove that the centralizer $C_{{\rm Der}A}(D)$ is a "large" \ subalgebra in ${\rm Der}_{K} A$, namely $\rk_A C_{\Der A}(D) := \dim_R RC_{\Der A}(D)$ equals ${\rm tr}.\deg_{K}R,$ where $R$ is the field of fraction of the ring $A.