Cohomology of Lie algebras of polynomial vector fields on the line over fields of characteristic $2$

TL;DR

This paper computes the cohomology of certain Lie algebras of polynomial vector fields over fields of characteristic 2, providing a basis and an analog of Goncharova's theorem for this case.

Contribution

It introduces a basis for the cohomology of Lie algebras of polynomial vector fields over fields of characteristic 2, extending Goncharova's theorem to this setting.

Findings

Basis for cohomology with finite support constructed

Analog of Goncharova's theorem established for characteristic 2

Results applicable to derivations of polynomial rings

Abstract

For a field $\mathbb{F}$, let $L_k(\mathbb{F})$ be the Lie algebra of derivations $f(t)\frac{d}{dt}$ of the polynomial ring $\mathbb{F}[t]$, where $f(t)$ is a polynomial of degree $\geqslant k$. For any $k\geqslant -1$, we present a basis of the space of the cohomology with finite-dimensional support of the Lie algebra $L_k(\mathbb{F})$ with coefficients in the trivial module $\mathbb{F}$ for the case where ${\rm char}(\mathbb{F})=2$. The main result obtained is an analog of the famous Goncharova's Theorem for the case ${\rm char}(\mathbb{F})=0$ and $k\geqslant 1$.