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 $s(\mathfrak{ D}(n))$: $$ s(\mathfrak{ D}(n))=[ \frac{ 2}{ 3}\dim{ \mathfrak{ D}(n)} ]. $$
Categories: math.RA