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

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.