Cohomology of $\mathbb{N}$-graded Lie algebras of maximal class over $\mathbb{Z}_2$

TL;DR

This paper computes the cohomology of certain maximal class Lie algebras over the field _2, revealing isomorphisms in the infinite-dimensional case and providing Betti numbers in the finite-dimensional case.

Contribution

It provides a complete description of the cohomology rings of _2-graded Lie algebras of maximal class, highlighting differences from characteristic zero.

Findings

Cohomology rings of _2-graded Lie algebras are isomorphic in the infinite-dimensional case.

Explicit Betti numbers are computed for finite-dimensional cases.

The structure of cohomology over _2 differs from that over characteristic zero.

Abstract

We compute the cohomology with trivial coefficients of Lie algebras $\mathfrak{m}_0$ and $\mathfrak{m}_2$ of maximal class over the field $\mathbb{Z}_2$. In the infinite-dimensional case, we show that the cohomology rings $H^*(\mathfrak{m}_0)$ and $H^*(\mathfrak{m}_2)$ are isomorphic, in contrast with the case of the ground field of characteristic zero, and we obtain a complete description of them. In the finite-dimensional case, we find the first three Betti numbers of $\mathfrak{m}_0(n)$ and $\mathfrak{m}_2(n)$ over $\mathbb{Z}_2$.