Equivalent description of Hom-Lie algebroids

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.

The paper is organized as follows. In Section 2, we recall some basic notions. In Section 3, first, we study representations of Hom-Lie algebroids, and give some properties of Hom-Lie algebroids. Then, we prove that two known definitions of Hom-Lie algebroids can be transformed into each other(Theorem3.3, Theorem3.4).

Hom-Lie algebras and their representations
The notion of a Hom-Lie algebra was introduced in [3], see also [2,8] for more information.
A Hom-Lie algebra is called a regular Hom-Lie algebra if α is a linear automorphism.
(2) A subspace h ⊂ g is a Hom-Lie sub-algebra of (g, [ Representation and cohomology theories of Hom-Lie algebra are systematically introduced in [1,9]. See [10] for homology theories of Hom-Lie algebras.
The set of k-cochains on g with values in V , which we denote by C k (g; V ), is the set of skewsymmetric k-linear maps from g × · · · × g(k-times) to V : In [15], when β ∈ GL(V ), there is a series operators d s : where β −1 is the inverse of β, η ∈ C k (g; V ), and the author have the results: d s • d s = 0.
So, we proved the necessity of this Theorem. The sufficiency of this theorem is similar with Theorem 3.3.

Remark 3.5.
• For E is a vector bundle over M , we hope that we are able to get a pull-back diagram: E So, we assume that ϕ 2 = id.
• For given Hom-Lie algebriod E, we have operators d s , then, by operators d s , we can get the other definitions of Hom-Lie algebriod.