On Lie algebras consisting of locally nilpotent derivations

Abstract

Let $K$ be an algebraically closed field of characteristic zero and $A$ an integral $K$-domain. The Lie algebra $De{r}_K(A)$ of all $K$-derivations of $A$ contains the set $LND(A)$ of all locally nilpotent derivations. The structure of $LND(A)$ is of great interest, and the question about properties of Lie algebras contained in $LND(A)$ is still open. An answer to it in the finite dimensional case is given. It is proved that any finite dimensional (over $K$) subalgebra of $De{r}_K(A)$ consisting of locally nilpotent derivations is nilpotent. In the case $A=K[x,y],$ it is also proved that any subalgebra of $De{r}_K(A)$ consisting of locally nilpotent derivations is conjugated by an automorphism of $K[x,y]$ with a subalgebra of the triangular Lie algebra.