Centralizers of linear and locally nilpotent derivations

TL;DR

This paper characterizes the centralizers of linear and locally nilpotent derivations in polynomial and finitely generated domains, providing algorithms and structural insights into their algebraic properties.

Contribution

It offers a description of the centralizer of linear derivations and proves the size of the centralizer for locally nilpotent derivations in finitely generated domains.

Findings

Centralizer of linear derivations described explicitly.

Algorithm provided for generators of the centralizer in specific cases.

Size of the centralizer for locally nilpotent derivations established.

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.