Almost contact Hom-Lie algebras and Sasakian Hom-Lie algebras

In this paper, we consider Horn-Lie groups and introduce left invariant almost contact structures on them (almost contact Horn-Lie algebras). On such Horn-Lie groups, we construct the almost contact metrics and the contact forms. We give the notion of normal almost contact Hom-Lie algebras and describe K-contact and Sasakian structures on Hom-Lie algebras. Also, we study some of their properties. In addition, it is shown that any Sasakian Hom-Lie algebra is a K-contact Hom-Lie algebra. Finally, we present examples of Sasakian Horn-Lie algebras and in particular, we show that the skew symmetric matrix so(3) carries a Sasakian structure.