Complex structures on nilpotent Lie algebras and descending central series

We study the algebraic constraints on the structure of nilpotent Lie algebra $\mathbb{g}$, which arise because of the presence of an integrable complex structure $J$. Particular attention is paid to non-abelian complex structures. Constructed various examples of positive graded Lie algebras with complex structures, in particular, we construct an infinite family $\mathfrak{D}(n)$ of such algebras that we have for their nil-index $s(\mathfrak{D}(n))$: $$s(\mathfrak{D}(n))=[ \frac{2}{3}\dim{\mathfrak{D}(n)} ].$$