On the Betti numbers of filiform Lie algebras over fields of characteristic two

Abstract

An $n$-dimensional Lie algebra $\mathfrak{g}$ over a field $\mathbb{F}$ of characteristic two is said to be of Vergne type if there is a basis $e_1,\dots ,e_n$ such that $[e_1,e_i]=e_{i+1}$ for all $2\le i\le n-1$ and $[e_i,e_j]=c_{i,j}{e}_{i+j}$ for some $c_{i,j}\in \mathbb{F}$ for all $i,j\ge 2$ with $i+j\le n$. We define the algebra ${\mathfrak{m}}_0$ by its nontrivial bracket relations: $[e_1,e_i]=e_{i+1},2\le i\le n-1$, and the algebra ${\mathfrak{m}}_2$: $[e_1,e_i]=e_{i+1},2\le i\le n-1$, $[e_2,e_j]=e_{j+2},3\le j\le n-2$. We show that, in contrast to the corresponding real and complex cases, ${\mathfrak{m}}_0(n)$ and ${\mathfrak{m}}_2(n)$ have the same Betti numbers. We also prove that for any Lie algebra of Vergne type of dimension at least $5$, there exists a non-isomorphic algebra of Vergne type with the same Betti numbers.

Cited by in corpus (1)

• On cohomology of filiform Lie superalgebras