Some restrictions on the Betti numbers of a nilpotent Lie algebra

We have recently shown that a nilpotent Lie algebra \(L\) of dimension \(n \geq 1\) satisfies the inequality \(\dim H_2(L,\mathbb{Z}) \leq \frac{1}{2}(n+m-2)(n-m-1)+1\), where \(\dim L^2=m \geq 1\) 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 \(\dim H^i(L,\mathbb{C}^\times)\) 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, Appl. Categ. Struct. 6, No. 3, 355–371].