Left-invariant para-Kähler structures on six-dimensional nilpotent Lie groups

Left-invariant para-complex structures on six-dimensional nilpotent Lie groups are considered. A complete list of six-dimensional nilpotent Lie groups that admit para-Kähler structures is obtained, explicit expressions for para-complex structures are found, and curvature properties of associated para-Kähler metrics are investigated. It is shown that paracomplex structures are nilpotent and the corresponding para-Kähler metrics are Ricci-flat.