MATHEMATICS

On left-invariant semi-kähler structures on six-dimensional nilpotent nonsymplectic lie groups

N. K. Smolentsev, K. V. Chernova

Kemerovo State University, Kemerovo, Russian Federation

Abstract: 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 $G_i$ that have the following Lie algebras with nonzero Lie brackets:
$ \mathbf{g}_1: [e_1, e_2] = e_4, [e_2, e_3] = e_5, [e_1, e_4] = e_6, [e_3, e_5] = - e_6$,
$ \mathbf{g}_2: [e_1, e_2] = e_4, [e_1, e_4] = e_5, [e_2, e_4] = e_6$,
$ \mathbf{g}_3: [e_1, e_2] = e_6, [e_3, e_4] = e_6$.
It is shown that on these Lie algebras there exist non-degenerate $2$-forms $\omega$ for which the property $\omega\wedge d\omega=0$ holds. Such forms $\omega$ are called semi-Kähler. For each group $G_i$, families of semi-Kahler $2$-forms $\omega$, compatible complex and para-complex structures, and corresponding pseudo-Riemannian metrics are obtained.

Keywords: six-dimensional nilpotent Lie algebras, left-invariant semi-Kähler structures, para-complex structures, almost para-semi-Kähler structures.

UDC: 514.76

MSC: 53C15, 53C30, 53C25, 22E25

Received: 16.05.2024
Accepted: June 9, 2025

DOI: 10.17223/19988621/95/4