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 [Formula: see text]. With this definition, the study of para-Kähler left Leibniz algebras mirrors that of para-Kähler Lie algebras. Specifically, we show that para-Kähler left Leibniz algebras can be constructed from two copies of dialgebras, which generalize left-symmetric algebras, and we refer to them as pre-para-Kähler left Leibniz dialgebras. To provide large families of para-Kähler left Leibniz algebras, we introduce new classes of pre-para-Kähler left Leibniz dialgebras, including flat pseudo-Euclidean left Leibniz algebras and cyclic pre-symplectic left Leibniz algebras, thereby demonstrating the generality and versatility of these structures within the para-Kähler framework.