Abstract
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.
1. Introduction
The notion of Hom-Lie algebras was introduced by Hartwig, Larsson, and Silvestrov in [1] 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 [5]; for regular Hom-Lie algebras, [6] gives a new coboundary operator on -cochains, and there are many works have been done by the special coboundary operator [6, 7]. In [8], 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 [9], C. Laurent-Gengoux and J. Teles proved that there is a one-to-one correspondence between Hom-Gerstenhaber algebras and Hom-Lie algebroids; in [10], based on Hom-Lie algebroids from [9], the authors study representation of Hom-Lie algebroids. In [7], 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. Preliminaries
2.1. Hom-Lie Algebras and Their Representations
The notion of a Hom-Lie algebra was introduced in [1]; see also [11, 12] for more information.
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 .
Representation and cohomology theories of Hom-Lie algebra are systematically introduced in [13, 14]. See [15] for homology theories of Hom-Lie algebras.
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 [8], 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
Now, we introduce two kinds of definitions of Hom-Lie algebroids; they are from [7, 9]. For more about Hom-Lie algebroids, please see [7, 9].
Definition 3 (see [9]). 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 [9];(d), when , then , and Hom-Lie algebroid is just a Lie algebroid;(e). It follows from
Definition 4 (see [7]). 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 [7].
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.
Now, we revisited representations of Hom-Lie algebroids, respectively, based on Definitions 3 and 4.
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), Then, for any , we have Sept 5, for , by (36), (42), and (i), we have So, we haveBy (39) and (50), is a Hom-Lie algebra.
By (44), we have .
By (35) and (42), is a representation of Hom-Lie algebra on with respect to .
The proof is completed.
Theorem 8. 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 define by Definition 4 if and only if there is a series of operators , such that(1);(2)for any , we have (3); ;(4)for , we have ;(5)for , we have
Proof. For necessity, with above Propositions which we proved, we just need to prove and .
For Hom-Lie algebriod and , by the definition of , we have For Hom-Lie algebriod and , we have So, we proved the necessity of this theorem. The sufficiency of this theorem is similar to Theorem 7.
Remark 9. (i) For is a vector bundle over , we hope that we are able to get a pull-back diagram: So, we assume that .
(ii) For given Hom-Lie algebriod , we have operators , then, by operators , we can get the other definitions of Hom-Lie algebriod.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study is supported by the NSF of China (no. 11771382) and the Science and Technology Project (GJJ161029) of Department of Education, Jiangxi Province.