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 $D\in W_n(K),$ $D\ne 0$ 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 $C_{\mathrm{Der}A}(D)$ is a "large" \ subalgebra in ${\mathrm{Der}}_{K}A$, namely $\rk_A C_{\Der A}(D) := \dim_R RC_{\Der A}(D)$ equals $\mathrm{tr}.{°}_{K}R,$ where $R$ is the field of fraction of the ring $A.

10 pages