Nilpotent Lie algebras of derivations with the center of small corank

Abstract

Let $\mathbb K$ be a field of characteristic zero, $A$ be an integral domain over $\mathbb K$ with the field of fractions $R=Frac(A),$ and $Der_{\mathbb K}A$ be the Lie algebra of all $\mathbb K$-derivations on $A$. Let $W(A):=RDer_{\mathbb K} A$ and $L$ be a nilpotent subalgebra of rank $n$ over $R$ of the Lie algebra $W(A).$ We prove that if the center $Z=Z(L)$ is of rank $\geq n-2$ over $R$ and $F=F(L)$ is the field of constants for $L$ in $R,$ then the Lie algebra $FL$ is contained in a locally nilpotent subalgebra of $W(A)$ of rank $n$ over $R$ with a natural basis over the field $R.$ It is also proved that the Lie algebra $FL$ can be isomorphically embedded (as an abstract Lie algebra) into the triangular Lie algebra $u_n(F)$, which was studied early by other authors.