Some Study On Hom-Lie Algebras

This paper study coboundary operators and representations of Hom-Lie algebras, and properties of Hom-Lie-Rinehart algebras, Hom-Courant-Dorfman algebras, and also application of Hom-Lie algebras.We give the definition of (σ,τ)-differential graded commutative algebra, and give the coboundary operator of a Hom-Lie algebra with respect to the trivial representation, study the trivial representations of a Hom-Lie algebra. If (g, [.,., a) is Hom-Lie algebra, then (Ag*, a*, d) is an (a*, a*)-differential graded commutative algebra, where d is coboundary operator of g with respect to the trivial representation. Conversely, if (Ag*, a*, d) is an (a*,a*)-differential graded commutative algebra, then (g, [.,., a) is Hom-Lie algebra. We construct a regular Hom-Lie algebra base on gl(V), and study its properties. We prove that there is a series of coboundary operators ds of Hom-Lie algebras, and have some properties of ds, and also give the relation between cohomology groups with respect to ds. In case of regular Hom-Lie algebras, we give a special coboundary operator d and study the representation of regular Hom-Lie algebras, (g, [.,.,,a) is a regular Hom-Lie algebra, and p:g�'gl(Ⅴ)is a representation of (g, [.,.,,a) on the vector space V with respect to φ∈GL{V if and only if there exists:d:Ck{g;\V)�'Ck+l(g; V), and such that:(i) dod= 0;(ii) for and ζ∈Λkg*,η∈C1(g;V),(iii) (a-1)*d=do(a-1)* At the same time, this paper give the definition of an Omni-Hom-Lie algebra and the Dirac structure on Omni-Hom-Lie algebra. We study the property of Omni-Hom-Lie algebras. There is a one-to-one correspondence between Dirac structures of gl(V)(?) V and regular Horn-Lie algebra structures on subspace of V. We show that the underlying algebraic structure of the Omni-Hom-Lie algebras is a Hom-Leibniz algebra, and skewsymmetrization of the bracket operator of Omni-Hom-Lie algebras give rise to a Horn-Lie 2-algebra.Then, we study some examples and properties of Horn-Lie algebroids. We give the definition of a Horn-Lie-Rinehart algebra, which is the algebraic version of Horn-Lie algebroids, study the representation of a Hom-Lie-Rinehart algebra. Let R be a comnmtative K-algebra and E an R-module, a : E�' E, a’ is reversible and R-linear or α(fx)= fa,(x), x∈E,f∈ R. Then (E, [., .],α, R,ρ) is a Horn-Lie Rinehart algebra if and only if there is d : Ck(E; R)�' Ck+1(E; R) such that:(i) d o d = 0:(ii) (a-1)* od= do(a-1)*;(iii) for ζ∈Ck(E: R), η∈Cl(E: R),We give the definition of a Hom-Courant-Dorfman algebra, study formulas on Horn-Lie-Rinehart algebras and their dual: Lie derivative. Cartan formula and so on. We give one example of Hom-Courant-Dorfman algebras, and study properties of this example.Finally, we study application of Hom-Lie algebras on integrable system. This paper study differential system on Hom-Lie algebras as follow:It comes from one-parameter deformation of differential system:We study this svstem and get solutions of this system under different conditions.