Para-K\"ahler and pseudo-K\"ahler structures on Lie-Yamaguti algebras

TL;DR

This paper introduces para-Kähler and pseudo-Kähler structures on Lie-Yamaguti algebras, explores their properties, and connects them with pre-Lie-Yamaguti algebras and complex product structures.

Contribution

It develops the theory of para-Kähler and pseudo-Kähler structures on Lie-Yamaguti algebras and links these with complex product structures and Levi-Civita products.

Findings

Construction of semidirect product Lie-Yamaguti algebras

Definition of para-Kähler and pseudo-Kähler structures on Lie-Yamaguti algebras

Relation between Levi-Civita products and pre-Lie-Yamaguti algebras

Abstract

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-K\"ahler structures and pseudo-K\"ahler structures on Lie-Yamaguti algebras, and also gives the definition of complex product structures on Lie-Yamaguti algebras which was first introduced in [25]. 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.