COMPLEX PRODUCT STRUCTURES ON HOM-LIE ALGEBRAS

In this paper, we introduce the notion of complex product structures on hom-Lie algebras and show that a hom-Lie algebra carrying a complex product structure is a double hom-Lie algebra and it is also endowed with a hom-left symmetric product. Moreover, we show that a complex product structure on a hom-Lie algebra determines uniquely a left symmetric product such that the complex and the product structures are invariant with respect to it. Finally, we introduce the notion of hyper-para-Kahler hom-Lie algebras and we present an example of hyper-para-Kahler hom-Lie algebras.