On Lie algebras consisting of locally nilpotent derivations

TL;DR

This paper investigates the structure of finite-dimensional Lie algebras composed of locally nilpotent derivations over algebraically closed fields, proving their nilpotency and conjugation properties in specific cases.

Contribution

It establishes that finite-dimensional subalgebras of locally nilpotent derivations are nilpotent and characterizes their conjugation in the polynomial ring case.

Findings

Finite-dimensional subalgebras are nilpotent.

Subalgebras in $K[x,y]$ are conjugate to triangular Lie algebras.

Results extend understanding of derivation Lie algebras in algebraic geometry.

Abstract

Let $K$ be an algebraically closed field of characteristic zero and $A$ an integral $K$-domain. The Lie algebra $Der_{K}(A)$ 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 $Der_{K}(A)$ consisting of locally nilpotent derivations is nilpotent. In the case $A=K[x, y],$ it is also proved that any subalgebra of $Der_{K}(A)$ consisting of locally nilpotent derivations is conjugated by an automorphism of $K[x, y]$ with a subalgebra of the triangular Lie algebra.