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.
Categories: math.RA
