On nilpotent Lie algebras of derivations of fraction fields

TL;DR

This paper classifies nilpotent Lie algebras of derivations of fraction fields of commutative algebras, showing they embed into a specific triangular Lie algebra in three variables, providing a structural understanding of such derivations.

Contribution

It proves that nilpotent subalgebras of derivations with rank at most 3 embed into a finite-dimensional subalgebra of a triangular Lie algebra, offering a new structural characterization.

Findings

Nilpotent Lie algebras of derivations with rank ≤ 3 are embeddable into a triangular Lie algebra.

Provides a classification of nilpotent vector field Lie algebras with polynomial coefficients in three variables.

Establishes a connection between derivations of fraction fields and triangular Lie algebra structures.

Abstract

Let $K$ be an arbitrary field of characteristic zero and $A$ a commutative associative $ K$-algebra which is an integral domain. Denote by $R$ the fraction field of $A$ and by $W(A)=RDer_{\mathbb K}A,$ the Lie algebra of $\mathbb K$-derivations of $R$ obtained from $Der_{\mathbb K}A$ via multiplication by elements of $R.$ If $L\subseteq W(A)$ is a subalgebra of $W(A)$ denote by $rk_{R}L$ the dimension of the vector space $RL$ over the field $R$ and by $F=R^{L}$ the field of constants of $L$ in $R.$ Let $L$ be a nilpotent subalgebra $L\subseteq W(A)$ with $rk_{R}L\leq 3$. It is proven that the Lie algebra $FL$ (as a Lie algebra over the field $F$) is isomorphic to a finite dimensional subalgebra of the triangular Lie subalgebra $u_{3}(F)$ of the Lie algebra $Der F[x_{1}, x_{2}, x_{3}], $ where $u_{3}(F)=\{f(x_{2}, x_{3})\frac{\partial}{\partial x_{1}}+g(x_{3})\frac{\partial}{\partial x_{2}}+c\frac{\partial}{\partial x_{3}}\}$ with $f\in F[x_{2}, x_{3}], g\in F[x_3]$, $c\in F.$ In particular, a characterization of nilpotent Lie algebras of vector fields with polynomial coefficients in three variables is obtained.