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

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 $⩾k$. For any $k⩾-1$, 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 $\mathrm{char}(\mathbb{F})=2$. The main result obtained is an analog of the famous Goncharova's Theorem for the case $\mathrm{char}(\mathbb{F})=0$ and $k⩾1$.

arXiv comment10 pages, accepted for publication in the "Journal of Algebra"