Research Article | Open Access
Equivalent Description of Hom-Lie Algebroids
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 notion of Hom-Lie algebras was introduced by Hartwig, Larsson, and Silvestrov in  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 -deformations of the Witt and the Virasoro algebras have the structure of a Hom-Lie algebra [1, 2]. Because of close relations to discrete and deformed vector fields and differential calculus [1, 3, 4], more people pay special attention to this algebraic structure. For a party of -cochains on Hom-Lie algebras, called -Hom-cochains, there is a series of coboundary operators ; for regular Hom-Lie algebras,  gives a new coboundary operator on -cochains, and there are many works have been done by the special coboundary operator [6, 7]. In , there is a series of coboundary operators, and the author generalizes the result “If is a Lie algebra, is a representation if and only if there is a degree- operator on satisfying , and where is the coboundary operator associated with the trivial representation.”
Geometric generalizations of Hom-Lie algebras are given in [7, 9]. In , C. Laurent-Gengoux and J. Teles proved that there is a one-to-one correspondence between Hom-Gerstenhaber algebras and Hom-Lie algebroids; in , based on Hom-Lie algebroids from , the authors study representation of Hom-Lie algebroids. In , the authors make small modifications to the definition of Hom-Lie algebroids and give a new definition of Hom-Lie algebroids; based 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.
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 (Theorem 7, Theorem 8).
2.1. Hom-Lie Algebras and Their Representations
Definition 1. A Hom-Lie algebra is a triple consisting of a vector space , a skew-symmetric bilinear map (bracket) and a linear transformation satisfying , and the following Hom-Jacobi identity:A Hom-Lie algebra is called a regular Hom-Lie algebra if is a linear automorphism.
A subspace is a Hom-Lie subalgebra of if and is closed under the bracket operation , i.e., for all , .
A morphism from the Hom-Lie algebra to the Hom-Lie algebra is a linear map such that and .
Definition 2. A representation of the Hom-Lie algebra on a vector space with respect to is a linear map , such that, for all , the following equalities are satisfied:
Let be a Hom-Lie algebra, be a vector space, and be a representation of on the vector space with respect to .
The set of -cochains on with values in , which we denote by , is the set of skew-symmetric -linear maps from (-times) to :
In , when , there is a series of operators which is given by where is the inverse of , , and the authors have the results: .
2.2. Hom-Lie Algebroids
Definition 3 (see ). A Hom-Lie algebroid is a quintuple , where is a vector bundle over a manifold is a smooth map, is a bilinear map, called bracket, is a vector bundle morphism, called anchor, and is a linear endomorphism of , for such that (1);(2)the triple is a Hom-Lie algebra;(3)the following Hom-Leibniz identity holds: (4) is a representation of Hom-Lie algebra on with respect to .
In fact, according to Definition 3, for , we have the following properties:(a);(b), defined by is a morphism of ;(c). According to in Definition, where is defined in Definition 3.1 of ;(d), when , then , and Hom-Lie algebroid is just a Lie algebroid;(e). It follows from
Definition 4 (see ). A Hom-Lie algebroid is a quintuple , where is a vector bundle over a manifold is a smooth map, is a bilinear map, called bracket, is a bundle map, called anchor, and is a linear endomorphism of , for such that (1);(2)the triple is a Hom-Lie algebra;(3)the following Hom-Leibniz identity holds: (4) is a representation of Hom-Lie algebra on with respect to .
From Definition 4, for , we have(a);(b), defined by is a morphism of ;(c);(d);(e).
When and are invertible, Hom-Lie bialgebroids and Hom-Courant algebroids are given in .
3. Representations of Hom-Lie Algebroids
In this section, we assume that map is involution; i.e., .
Let be a Hom-Lie algebroid, whence is a representation of on , where is or . We define , , by setting where ,
We define map by When , we have .
Let . Then, is a subset of . Let acts on , we have Actually, when , for , we have so , and we have Let and ; we have that is a subset of . So, if , we have .
At the same time, can induce a map , which is defined by Then, we have
Proposition 5. With above notations, for , , we have
Proof. First let , ; we have So, when , we have By induction on , assume that when , we have For any , then ; we have The proof is completed.
Proposition 6. With above notations, we have
Proof. With straightforward computations, for any , we have At the same time, we have We complete this proof.
Theorem 7. Let be a vector bundle over manifold is a smooth map and , and is a linear endomorphism of , i.e., for . Then is a Hom-Lie algebriod defined by Definition 3 if and only if there is a series of operators , such that (i);(ii)for any , we have (iii); ;(iv)for , we have: ;(v)for , we have
Proof. For necessity, with above propositions which we proved, we just need to prove (iv) and (v). For Hom-Lie algebriod and , by the definition of , we have For , we have So, we proved the necessity of this theorem. Now, we prove the adequacy of this theorem.
Sept 1, we define byThen, by (iv), for , , we haveOn the other hand, we have By (ii), for , we have Then, we haveThe definition of is reasonable.
By (iii), we have We find the result:Sept 2, for any , we define bySo, by (36), we have By , (35), (iii), and (36), we have We have the following:For any , by (i), (29), and (36), we have We get Sept 3, by (36), (30), (33), and (v), we have So, for , we haveSept 4, by (iii), we have For , by (ii), we have By (36),