Product structures and complex structures of Hom-Lie-Yamaguti algebras

Abstract

In this paper, we mainly discuss linear deformations of a Hom-Lie-Yamaguti algebra and introduce the notion of a Hom-Nijenhuis operator. We introduce the notion of a product structure on a Hom-Lie-Yamaguti algebra, which is a Hom-Nijenhuis operator $E$ satisfying $E^2= Id$. There is a product structure on a Hom-Lie-Yamaguti algebra if and only if the Hom-Lie-Yamaguti algebra is the direct sum of two subalgebras (as vector spaces). At the same time, we also introduce the notion of a complex structure on a Hom-Lie-Yamaguti algebra. Finally, we add a compatibility condition between a product structure and a complex structure to introduce the notion of a complex product structure on a Hom-Lie-Yamaguti algebra.