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

TL;DR

This paper investigates bases of the free module of derivations over polynomial rings, showing that any n-dimensional Lie algebra over an algebraically closed field can be realized as a subalgebra with a basis that forms an R-basis of the derivation module.

Contribution

It establishes that every n-dimensional Lie algebra over an algebraically closed field of characteristic zero can be embedded as a subalgebra with a basis that is also an R-basis of the derivation module.

Findings

Existence of subalgebras isomorphic to any given n-dimensional Lie algebra.

Such subalgebras have bases that serve as R-bases of the derivation module.

The result applies to polynomial rings over algebraically closed fields of characteristic zero.

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 $W_n(\mathbb{K})$.