Study Of Finite Dimensional Hom-lie Algebras

In this paper, we firstly study the dimensions problem of(multiplicative)simple Hom-Lie algebras. We give the necessary and sufficient conditions for two multiplicative simple Hom-Lie algebras to be isomorphic by studying the corresponding semisimple Lie algebras. The correspondence between Weyl group and some special multiplicative simple Hom-Lie algebras is also obtained. We obtain some results on nilpotency and solvability. Furthermore we classify four dimensional nilpotent Hom-Lie algebras and also some of five dimension. Under the two conditions that twist operator belongs to centroid and Hom-Lie algebra is quadratic, we get some nontrivial ideals and decompose the Hom-Lie algebra into direct sum of ideals. The sufficient and necessary condition for adg to be a set of derivations is also presented.