On nilpotent Lie algebras of derivations of fraction fields

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 $W(A)=RDe{r}_{\mathbb{K}}A,$ the Lie algebra of $\mathbb{K}$-derivations of $R$ obtained from $De{r}_{\mathbb{K}}A$ via multiplication by elements of $R.$ If $L\subseteq W(A)$ is a subalgebra of $W(A)$ denote by $r{k}_{R}L$ 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 $r{k}_{R}L\le 3$. 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 $DerF[x_1,x_2,x_3],$ 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.

arXiv commentCorrected typos. Revised formulation of Theorem 1, results unchanged

References in corpus (1)

• Lie algebras of triangular polynomial derivations and an isomorphism criterion for their Lie factor algebras