Locally nilpotent Lie algebras of derivations of integral domains

Abstract

Let $\mathbb K$ be a field of characteristic zero and $A$ an integral domain over $\mathbb K.$ The Lie algebra $\Der_{\mathbb K} A$ of all $\mathbb K$-derivations of $A$ carries very important information about the algebra $A.$ This Lie algebra is embedded into the Lie algebra $R\Der_{\mathbb K} A\subseteq \Der_{\mathbb K}R$, where $R={\rm Frac}(A)$ is the fraction field of $A.$ The rank $rk_{R}L$ of a subalgebra $L$ of $R\Der_{\mathbb K} A$ is defined as dimension $\dim_R RL.$ We prove that every locally nilpotent subalgebra $L$ of $R\Der_{\mathbb K} A$ with $rk_{R}L=n$ has a series of ideals $0=L_0\subset L_1\subset L_2\dots \subset L_n=L$ such that $\rank_R L_i=i$ and all the quotient Lie algebras $L_{i+1}/L_{i}, i=0, \ldots , n-1,$ are abelian. We also describe all maximal (with respect to inclusion) locally nilpotent subalgebras $L$ of the Lie algebra $R\Der_{\mathbb K} A$ with $rk_{R}L=3.$