On left-invariant semi-Kähler structures on six-dimensional nilpotent nonsymplectic Lie groups

It is known that there exist 34 classes of six-dimensional nilpotent Lie groups many of which admit left-invariant symplectic and complex structures. Among them, there are three classes of groups on which there are no left-invariant symplectic structures but there exist complex structures. The aim of the work is to determine new left-invariant geometric structures on these three six-dimensional Lie groups, compensating in a sense for the absence of symplectic structures, as well as to study their geometric properties. We study Lie groups Gi that have the following Lie algebras with nonzero Lie brackets: g1: [e1, e2] = e4, [e2, e3] = e5, [e1, e4] = e6, [e3, e5] = - e6, g2: [e1, e2] = e4, [e1, e4] = e5, [e2, e4] = e6, g3: [e1, e2] = e6, [e3, e4] = e6. It is shown that on these Lie algebras there exist non-degenerate 2-forms ю for which the property (oAdw = 0 holds. Such forms ю are called semi-Kahler. For each group Gi, families of semi-Kahler 2-forms ю, compatible complex and para-complex structures, and corresponding pseudo-Riemannian metrics are obtained.