The classification of left-invariant para-K\"ahler structures on simply connected four-dimensional Lie groups

TL;DR

This paper classifies all left-invariant para-K"ahler structures on four-dimensional simply connected Lie groups and explores their curvature properties, including flatness, Ricci flatness, and Ricci solitons.

Contribution

It provides a complete classification of these structures on four-dimensional Lie groups, a novel comprehensive analysis in this context.

Findings

Classification of para-K"ahler structures achieved

Identification of flat and Ricci flat cases

Existence of Ricci solitons in certain structures

Abstract

We give a complete classification of left invariant para-K\"ahler structures on four-dimensional simply connected Lie groups up to an automorphism. As an application we discuss some curvatures properties of the canonical connection associated to these structures as flat, Ricci flat and existence of Ricci solitons.