Left-invariant almost para-K\"{a}hler structures on six-dimensional nilpotent Lie groups

TL;DR

This paper classifies six-dimensional nilpotent Lie groups with left-invariant para-K"{a}hler structures, providing explicit structures and analyzing their curvature, showing they are Ricci-flat and nilpotent.

Contribution

It provides a complete classification of six-dimensional nilpotent Lie groups admitting para-K"{a}hler structures and explicitly constructs these structures.

Findings

All such Lie groups admit nilpotent para-complex structures.

The associated para-K"{a}hler metrics are Ricci-flat.

Explicit expressions for the structures are derived.

Abstract

In this paper, we consider left-invariant para-complex structures on six-dimensional nilpotent Lie groups. A complete list of six-dimensional nilpotent Lie groups that admit para-K\"{a}hler structures is obtained, explicit expressions for para-complex structures are found, and curvature properties of associated para-K\"{a}hler metrics are investigated. It is shown that paracomplex structures are nilpotent and the corresponding para-K\"{a}hler metrics are Ricci-flat.