# On the annihilators of rational functions in the Lie algebra of derivations of k[x, y]

## TL;DR

This paper studies the structure of derivations in the Lie algebra of all polynomial derivations in two variables that annihilate a given rational function, revealing they form a rank 1 free module and describing their maximal abelian subalgebras.

## Contribution

It characterizes the annihilators of rational functions in the Lie algebra of derivations as rank 1 free modules and describes their maximal abelian subalgebras.

## Key findings

- A_{W_2}(u) is a free rank 1 submodule of W_2.
- Descriptions of maximal abelian subalgebras and centralizers.
- Structural insights into derivations annihilating rational functions.

## Abstract

Let k be an algebraically closed field of zero characteristic. The Lie algebra W_2 of all k-derivations of the polynomial ring k[x, y] naturally acts on the polynomial ring k[x, y] and also on the field of rational functions k(x, y). For a fixed non-constant rational function u from k(x,y) we consider the set A_{W_2}(u) of all derivations D from W_2 such that D(u)=0. We prove that A_{W_2}(u) is a free submodule of rank 1 of the k[x,y]-module W_2. A description of the maximal abelian subalgebras as well of the centralizers of elements in the Lie algebra A_{W_2}(u) has been obtained.