Locally nilpotent Lie algebras of derivations of integral domains

TL;DR

This paper studies the structure of locally nilpotent subalgebras of derivations of integral domains over a field of characteristic zero, revealing their ideal series and classifying maximal cases with rank three.

Contribution

It characterizes the ideal structure of locally nilpotent subalgebras of derivations and classifies maximal such subalgebras with rank three.

Findings

Locally nilpotent subalgebras have a specific ideal series with abelian quotients.

Every such subalgebra with rank n has a chain of ideals with increasing rank.

Maximal locally nilpotent subalgebras of rank 3 are fully described.

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 $rk_{R}L=3.$