On nilpotent Lie algebras of derivations of fraction fields

A. P. Petravchuk

Algebra and discrete mathematics, Tome 22 (2016) no. 1, pp. 116-128

Résumé

Let $\mathbb K$ be an arbitrary field of characteristic zero and $A$ an integral $\mathbb K$-domain. Denote by $R$ the fraction field of $A$ and by $W(A)=R\operatorname{Der}_{\mathbb K}A$, the Lie algebra of $\mathbb K$-derivations on $R$ obtained from $\operatorname{Der}_{\mathbb K}A$ via multiplication by elements of $R$. If $L\subseteq W(A)$ is a subalgebra of $W(A)$ denote by $\operatorname{rk}_{R}L$ 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 $\operatorname{rk}_{R}L\leq 3$. 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 $\operatorname{Der} F[x_{1}, x_{2}, x_{3}]$, 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 $f\in F[x_{2}, x_{3}]$, $g\in F[x_3]$, $c\in F$.

Détail

Keywords: Lie algebra, vector field, nilpotent algebra, derivation.