On para-Kahler and hyper-para-Kahler Lie algebras

In this study, we consider Lie algebras that admit para-Kahler and hyper-para-Kahler structures. We provide new characterizations of these Lie algebras and develop many methods for building large classes of examples. Previously, Bai considered para-Kahler Lie algebras as left symmetric bialgebras. We reconsider this viewpoint and make improvements in order to obtain some new results. The study of para-Kahler and hyper-para-Kahler is intimately linked to the study of left symmetric algebras, particularly those that admit invariant symplectic forms. In this study, we provide many new classes of left symmetric algebras and complete descriptions of all the associative algebras that admit an invariant symplectic form. We also describe all four-dimensional hyper-para-Kahler Lie algebras. (C) 2015 Elsevier Inc. All rights reserved.