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

Felix V. Weinstein

Published 2016-05-30Version 1

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 build a basis of the space of continuous cohomology 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 result obtained is similar to the famous Goncharova's Theorem for the case ${\rm char}(\mathbb{F})=0$.