Hom-Lie Bialgebroid

In this paper, we introduce the notion of Hom-Lie bialgebroid and study the related theories. In particular, by use of Hom-big bracket, we have done a deep research on Hom-Lie bialgebra.The big bracket is exactly the graded Poisson bracket on cotangent bundle, which was the appropriate tool to study the theory of Lie bialgebra. Many generalizations are made for the big bracket and there are many applications. In this paper, we define the hom-analogue of the big bracket, i.e. the hom-big bracket, and provide a tool to study hom-structures. Since the Nijenhuis-Richardson bracket is a part of the big bracket, first we define the Hom-Nijenhuis-Richardson bracket, and show that the Hom-Nijenhuis-Richardson bracket gives rise to a graded hom-Lie algebra. The Hom-Nijenhuis-Richardson bracket has some good properties. On one hand, it can describe hom-Lie algebra structure. On the other hand, it can give rise to an example which is related to the representation of Hom-Lie algebra. This example is very important to study the Hom-Lie structure. Furthermore, the Hom-Nijenhuis-Richardson bracket can induce a cohomology operator, which is decided by Hom-big bracket and is different from the one existed. That is to say, the theory of cohomology of Hom-Lie algebra is not unique. Next we introduce the Hom-big bracket and show that gives rise to a graded Hom-Lie algebra. Furthermore, it also gives rise to a purely Hom-Poisson structure. As the application of the Hom-big bracket, we define Hom-Nijenhuis operators. In [42], a notion of Hom-Nijenhuis operator was given. However, the Hom-Nijenhuis operator defined here is different from the existing one. We can also to prove there is a one-to-one correspondence between Hom-Nijenhuis operator and trivial deformation. Of course, the trivial deformation is also different, the one here containing the twisted morphism. Similarly, the notion of a Hom-O-operator is also different from the one given in [43]. But we believe that the current definitions are more reasonable (see Remark 3.3.1 and 3.3.2). which justifies the usage of the Hom-big bracket. At last, we also clear the relations among Hom-O-operator, Hom-Nijenhuis operator and Hom-right-symmetry operator (see Lemma 3.3.2 and Proposition 3.3.2).We define Hom-Lie bialgebra using the Hom-big bracket. The Hom-Lie bialgebra defined here is the same as the one given in [47]. But this definition can not induce the theory of Manin triple. Moreover, there exists another definition of Hom-Lie bialgebra. see [43], which is dependent of the existence of the coadjoint-representation, which imposes a strong condition. To deal the problem, we introduce an improved definition:pruely Hom-Lie bialgebra. Under this new definition, we can realize the theory of Manin triple, without the dependence of coadjoint representation. By use of the definition of purely Hom-Lie bialgebra,we can give rise to a generalized Lie bracket of Hom-Lie bialgebra. Similarly as the classical case, Hom-Lie bialgebra (V. V*) is a compatible pair of Hom-Lie algebras (V,μ) and (V*,â–³). Moreover, the compatibility condition has three equivalent descriptions:the cohomology operator satisfies the property of derivation:â–³ is a 1-cocyle; V(?) V* has a Hom-Lie algebra structure. Next, we give the definition of Hom-Lie quasi-bialgebra and Hom-quasi Lie-bialgebra. Furthermore,we also give the equivalent descriptions of them using usual language.The definition of Hom-Lie algebroid was firstly introduced in [32]. The new definition of Hom-Lie algebroid has some relations to [32], but the are not the same. We write down the definition of differential operators, such as cohomology operator, interior product, Lie derivation and etc. Moreover, we write down a list of important equations, such as twisted Cartan formula. Moreover, we get the conclusion:(Aâ†'M,φ,[·,·]A,α,(?)A) is a Hom-Lie algebroid if and only if ((?)Τ(∧kA*),∧,(?)A(?),d) is a((?)A(?),(?)A(?))-differential graded commutative algebra. Finally, we give the definition of Hom-Courant algebroid, which is a generalization of Courant algebroid. The conclusion of Lie algebroid can naturally push into the Hom-case. Using these operators, we can give rise to Hom-Lie bialgebroid and Hom-Courant algebroid. Furthermore, we clear the relation between Hom-Lie bialgebroid and Hom-Courant algebroid:for a Hom-Lie bialgebroid (A, A*), there exists an natural Hom-Courant algebroid structure on A (?)A*.