Para-Kahler Left Leibniz Algebras: Definition, Characterization, Constructions, and New Classes of Dialgebras

In this paper, we extend the concept of para-Kähler structures from Lie algebras to left Leibniz algebras. First, we show that the Levi-Civita product, initially defined for pseudo-Euclidean Lie algebras, can be generalized to any pseudo-Euclidean algebra. A para-Kähler left Leibniz algebra is defined as a para-Kähler vector space [Formula: see text] equipped with a left Leibniz product [Formula: see text], such that the Levi-Civita product associated to [Formula: see text] commutes with [Formul