On the classification of left-invariant para-K\"{a}hler structures on four-dimensional Lie groups

TL;DR

This paper classifies left-invariant para-K"{a}hler structures on four-dimensional Lie groups by leveraging symplectic Lie algebra classifications, providing explicit forms of compatible structures and exploring related Sasaki para-contact structures.

Contribution

It offers a new classification approach based on symplectic Lie algebras, expanding previous work on para-K"{a}hler structures on four-dimensional Lie algebras.

Findings

Explicit forms of compatible para-complex structures

Classification of Sasaki para-contact structures

Connection to symplectic Lie algebra classifications

Abstract

A first classification of para-K\"{a}hler structures on four-dimensional Lie algebras was obtained by D. Calvaruzo in 2015. In this paper, we propose another classification based on the classification of symplectic Lie algebras. For each four-dimensional symplectic Lie algebra, compatible para-complex structures and the corresponding pseudo-Riemannian metrics are found in explicit form. This leads to the classification of Sasaki para-contact structures on five-dimensional contact Lie algebras with a nontrivial center.