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

TL;DR

This paper investigates the Betti numbers of certain filiform Lie algebras over fields of characteristic two, revealing that different algebra structures can share identical Betti numbers, contrasting with real and complex cases.

Contribution

It demonstrates that in characteristic two, distinct Vergne-type Lie algebras can have the same Betti numbers, and constructs non-isomorphic algebras with identical Betti numbers.

Findings

$rak{m}_0(n)$ and $rak{m}_2(n)$ have the same Betti numbers in characteristic two.

Existence of non-isomorphic Vergne-type algebras with identical Betti numbers for dimensions ≥ 5.

Contrasts with real and complex cases where Betti numbers distinguish algebra structures.

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 $2\leq i \leq n-1$ 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: $[e_1,e_i]=e_{i+1}, 2\leq i \leq n-1$, 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.