Para-Kähler and pseudo-Kähler structures on Lie–Yamaguti algebras

For a pre-Lie–Yamaguti algebra $A$, by using its sub-adjacent Lie–Yamaguti algebra $A^c$, we are able to construct a semidirect product Lie–Yamaguti algebra via a representation of $A^c$. The investigation of such semidirect Lie–Yamaguti algebras leads us to the notions of para-Kähler structures and pseudo-Kähler structures on Lie–Yamaguti algebras, and also gives the definition of complex product structures on Lie–Yamaguti algebras. Furthermore, a Levi–Civita product with respect to a pseudo-Riemannian Lie–Yamaguti algebra is introduced and we explore its relation with pre-Lie–Yamaguti algebras.