In this paper, we introduce the notions of pseudo-Riemannian, para-Hermitian and para- K ̈ ahler structures on hom-Lie algebras. In addition, we present the characterization of these structures. Also, we provide an example including these structures. We then intro- duce the phase space of a hom-Lie algebra and using the hom-left symmetric product, we show that a para-K ̈ ahler hom-Lie algebra gives a phase space and conversely, we can construct a para-K ̈ ahler hom-Lie algebra using a phase space.
