On noncommutative bases of the free module $W_n(\mathbb{K})$

Abstract

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and $R=\mathbb{K}[x_1,x_2,...x_n]$ the polynomial ring in $n$ variables over $\mathbb{K}.$ We study bases of the free $R$-module $W_n(\mathbb{K})$ of all $\mathbb{K}$-derivations of the ring $R$, such that their linear span over $\mathbb{K}$ is a subalgebra of the Lie algebra $W_n(\mathbb{K})$. We proved that for any Lie algebra $L$ of dimension $n$ over $\mathbb{K}$ there exists a subalgebra $\bar{L}$ of $W_n(\mathbb{K})$ which is isomorphic to $L$ and such that every $\mathbb{K}$-basis of $\bar{L}$ is an $R$-basis of the $R$-module $W_n(\mathbb{K})$.