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.
Categories: math.RA