Complex, symplectic and Kaehler structures on four dimensional Lie groups

TL;DR

This paper classifies complex, symplectic, and Kaehler structures on four-dimensional solvable Lie groups, providing explicit descriptions and cohomology computations to deepen understanding of their geometric properties.

Contribution

It systematically characterizes invariant complex, symplectic, and Kaehler structures on four-dimensional solvable Lie groups and computes their real cohomology.

Findings

Explicit classification of invariant structures

Identification of conditions for Kaehler structures

Cohomology computations of Lie algebras

Abstract

In this work we deal with left invariant complex and symplectic structures on simply connected four dimensional solvable real Lie groups. We search the general form of such structures, when they exist and we make use of this information to determine all left invariant Kaehler structures. Finally, as an appendix we compute explicitly the real cohomology of the corresponding Lie algebras.