Some restrictions on the Betti numbers of a nilpotent Lie algebra

Abstract

We have recently shown that a nilpotent Lie algebra $L$ of dimension $n \ge1 $ satisfies the inequality $\mathrm{dim} \ H_2(L,\mathbb{Z}) \leq \frac{1}{2}(n+m-2)(n-m-1)+1$, where $\mathrm{dim} \ L^2=m \ge 1$ and $H_2(L,\mathbb{Z})$ is the 2-nd integral homology Lie algebra of $L$. Our first main result correlates this bound with the $i$-th Betti number $\mathrm{dim} \ H^i(L,\mathbb{C}^\times)$ of $L$, where $H^i(L,\mathbb{C}^\times)$ denotes the $i$-th complex cohomology Lie algebra of $L$. Our second main result describes a more general restriction, which follows an idea of Ellis in [G. Ellis, The Schur multiplier of a pair of groups, Appl. Categ. Structures 6 (1998), 355--371].