On nilpotent Lie algebras of derivations of fraction fields

Abstract

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 \(W(A)=RDer_{\mathbb K}A,\) the Lie algebra of  \(\mathbb K\)-derivations on \(R\) 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 \(f\in F[x_{2}, x_{3}], g\in F[x_3]\), \(c\in F.\)