Lie algebras of derivations with large abelian ideals

Abstract

Let \(\mathbb K\) be a field of characteristic zero, \(A=\mathbb{K}[x_{1},\ldots ,x_{n}]\) the polynomial ring and \(R=\mathbb{K}(x_{1},\dots,x_{n})\) the field of rational functions. The Lie algebra \({\widetilde W}_{n}(\mathbb{K}):=\operatorname{Der}_{\mathbb{K}}R\) of all \(\mathbb{K}\)-derivation on \(R\) is a vector space (of dimension n) over \(R\) and every its subalgebra \(L\) has rank \(\operatorname{rk}_{R}L=\dim_{R}RL\). We study subalgebras \(L\) of rank m over \(R\) of the Lie algebra \(\widetilde{W}_{n}(\mathbb{K})\) with an abelian ideal \(I\subset L\) of the same rank \(m\) over \(R\). Let \(F\) be the field of constants of \(L\) in \(R\). It is proved that there exist a basis \(D_1, \ldots, D_m\) of \(FI\) over \(F\), elements \(a_1, \ldots, a_k\in R\) such that \(D_i(a_j)=\delta_{ij}\), \(i=1, \ldots, m\), \(j=1,\ldots, k\), and every element \(D\in FL\) is of the form \(D=\sum_{i=1}^{m}f_i(a_1, \ldots, a_k)D_i\) for some \(f_i\in F[t_1, \ldots t_k]\), \(\deg f_i\leq 1\). As a consequence it is proved that \(L\) is isomorphic to a subalgebra (of a very special type) of the general affine Lie algebra \({\rm aff}_{m}(F)\).