In this paper, first we show that (g, [·, ·], α) is a Hom-Lie algebra if and only if ( g * , α * , d) is an (α * , α * ) differential graded-commutative algebra. Then, we revisit representations of Hom-Lie algebras and show that there are a series of coboundary operators. We also introduce the notion of an omni-Hom-Lie algebra associated to a vector space and an invertible linear map. We show that regular Hom-Lie algebra structures on a vector space can be characterized by Dirac structures in the corresponding omni-Hom-Lie algebra. The underlying algebraic structure of the omni-Hom-Lie algebra is a Hom-Leibniz algebra, or a Hom-Lie 2-algebra.

In this paper, we study representations of Hom-Lie algebroids, give some properties of Hom-Lie algebroids and discuss equivalent statements of Hom-Lie algebroids. Then, we prove that two known definitions of Hom-Lie algebroids can be transformed into each other under some conditions. IntroductionThe notion of Hom-Lie algebras was introduced by Hartwig, Larsson, and Silvestrov in [3] as a part of a study of deformations of the Witt and the Virasoro algebras. In a Hom-Lie algebra, the Jacobi identity is twisted by a linear map, called the Hom-Jacobi identity. Some q-deformations of the Witt and the Virasoro algebras have the structure of a Hom-Lie algebra [3,4]. Because of close relations to discrete and deformed vector fields and differential calculus [3, 5, 6], more people pay special attention to this algebraic structure. For a party of k-cochains on Hom-Lie algebras, name k-Hom-cochains, there is a series of coboundary operators [11]; for regular Hom-Lie algebras, [12] gives a new coboundary operator on k-cochains, and there are many works have been done by the special coboundary operator [12,13]. In [15], there is a series of coboundary operators, and the author generalizes the result " If k is a Lie algebra, ρ : k −→ gl(V ) is a representation if and only if there is a degree-1 operator D on Λk * ⊗ V satisfying D 2 = 0, andwhere d k : ∧ k g * −→ ∧ k+1 g * is the coboundary operator associated to the trivial representation."Geometric generalizations of Hom-Lie algebras are given in [7][13]. In [7], C. Laurent-Gengoux and J. Teles proved that there is a one-to-one correspondence between Hom-Gerstenhaber algebras and Hom-Lie algebroids; in [14], base on Hom-Lie algebroids from [7], the authors study representation of Hom-Lie algebroids. In [13], the authors make small modifications to the definition of Hom-Lie algebroids, and give a new definition of Hom-Lie algebroids, base on the new definition of Hom-Lie algebroids, definitions of Hom-Lie bialgebroids and Hom-Courant algebroids are given.In this article, we first study representations of Hom-Lie algebroids, give equivalent statements of Hom-Lie algebroids and prove that different definitions of Hom-Lie algebroids are given by the same Hom-Lie algebras and their representations. 0 Keyword: Hom-Lie algebroids; Hom-Lie algebras; representations 0 MSC: 17B99,58H05 0

In this paper, the definition of Hom-Lie groups is given and one conntected component of Lie group GL(V ), which is not a subgroup of GL(V ), is a Hom-Lie group. More, we proved that there is a one-to-one relationship between Hom-Lie groups and Hom-Lie algebras (gl(V ), [•, •] β , Ad β ). Next, we also proved that if there is a Hom-Lie group homomorphism, then, there is a morphism between their Hom-Lie algebras. Last, as an application, we use these results on Toda lattice equation.

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