Para-Kahler and pseudo-Kahler 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-Kahler structures and pseudo-Kahler 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.