ИСТИНА

An almost complex structure $J$ on a Lie algebra $\mathfrak{g}$ ($J:\mathfrak{g} \to \mathfrak{g}$ satisfying $J^2=-1$) is called integrable (Nijenhuis tensor $N(J)$ vanishes) if $$ N(J)=[JX,JY]-[X,Y]-J[JX,Y]-J[X,JY]=0, \; \forall X,Y \in \mathfrak{g}. $$ An integrable almost complex structure on the tangent Lie algebra $\mathfrak{g}$ of a real simply connected Lie group $G$ defines a left invariant complex structure on $G$. If $G$ is nilpotent and $\Gamma \subset G$ is a cocompact lattice, $J$ defines a complex structure on corresponding nilmanifold $G/\Gamma$. We plan to discuss the algebraic constraints on the structure of nilpotent Lie algebra $\mathfrak{g}$ which arise because of the presence of an integrable almost complex structure $J$ on $\mathfrak{g}$.