Abelian para-Kähler structures on Lie algebras

Para-Kähler Lie algebras which decompose as the sum of two abelian Lagrangian subalgebras are studied. We propose several constructions and provide an inductive description of such Lie algebras. The curvatures of the para-Kähler metric are computed and sufficient conditions to ensure flatness or Ricci-flatness are given. The Lie algebras for which the para-Kähler metric is Einstein and non-Ricci-flat are completely characterized.