Some Properties Of Linear Transformations On Hom-Lie Color Algebras And Perfect Lie Color Algebras

There are two parts in this thesis.Part 1 devotes to studying two linear transformations of perfect Lie color algebras over a field of characteristic different from 2,i.e.,triple derivations and triple homomorphisms.We prove that every triple derivation of a perfect Lie color algebra is a derivation,and every triple derivation of the derivation algebra is an inner derivation.Moreover,we give a necessary condition such that homomorphisms,anti-homomorphisms,and sums of homomorphisms and anti-homomorphisms are all triple homomorphisms.Part 2 studies six linear transformations of Hom-Lie color algebras,i.e.,derivations,quasi-derivations,generalized derivations,central derivations,centroids and quasi-centroids.First we prove that every central derivation algebra is a Hom-ideal of a derivation algebra and every centroid is an ideal of a derivation algebra.Then we give that the quasi-centroid of a Hom-Lie color algebras with zero center is Abel if and only if it is a Hom-Lie color algebra.In particular,we prove that the sums of quasi-derivation algebras and quasi-centroids are generalized derivation algebras.We also prove that quasi-derivation algebras can be embedded as derivations in a larger Hom-Lie color algebra.