Derivations of Lie Algebras of Vector Fields in Infinitely Many Variables

TL;DR

This paper characterizes all derivations of the Lie algebra of vector fields with infinitely many variables, showing they are all inner, and describes derivations of specific subalgebras.

Contribution

It provides a complete description of derivations of the Lie algebra of vector fields in infinitely many variables, establishing that all are inner and detailing derivations of subalgebras.

Findings

All derivations of $W_X( ext{F})$ are inner.

Derived the structure of derivations for subalgebras $W_ ext{infty}( ext{F})$ and $W_ ext{fin}( ext{F})$.

Extended the understanding of derivations in infinite-dimensional Lie algebras.

Abstract

Let $W_X(\mathbb{F})$ be the Lie algebra of all derivations of the polynomial algebra $\mathbb{F}[X]$ in infinitely many variables. We describe all derivations of $W_X(\mathbb{F})$ over a field of characteristic zero and prove that all such derivations are inner. We also consider the subalgebras $W_\infty(\mathbb{F})$ and $W_{\text{fin}}(\mathbb{F})$ of the algebra $W_X(\mathbb{F})$ and describe all of their derivations.