Hom-Big Brackets: Theory and Applications

In this paper, we introduce the notion of hom-big brackets, which is a generalization of Kosmann-Schwarzbach's big brackets. We show that it gives rise to a graded hom-Lie algebra. Thus, it is a useful tool to study hom-structures. In particular, we use it to describe hom-Lie bialgebras and hom-Nijenhuis operators.


Introduction
The notion of hom-Lie algebras was introduced by Hartwig, Larsson and Silvestrov in [10] as 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 [10]. Because of their close relation to discrete and deformed vector fields and differential calculus [10,20,21], hom-Lie algebras were widely studied recently [1,24,25,31,32].
The big bracket {·, ·} on ∧ • (V ⊕ V * ) is exactly the graded Poisson bracket on T * V [1]. See [11,17,23] for more details. It was already clear that the big bracket was the appropriate tool to study the theory of Lie bialgebras. Many generalizations are made for the big bracket and there are many applications, e.g., in the theory of strong homotopy bialgebras [18], in the theory of Poisson geometry and Lie algebroids [12,14,16,27], in the theory of deformations of Courant algebroids [2,13], and etc.
The purpose of this paper is to 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 [26] on the direct sum ⊕ k Hom(∧ k V, V ) is a part of the big bracket, first we define the hom-Nijenhuis-Richardson bracket [·, ·] α , where α ∈ GL(V ), 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 structures, namely for µ ∈ Hom(∧ 2 V, V ), [µ, µ] α = 0 if and only if µ satisfies the hom-Jacobi identity. On the other hand, for A, B ∈ Hom(V, V ), we have This bracket is exactly the one introduced in [30], which plays an important role in the representation theory. Then we introduce the hom-big bracket and show that it gives rise to a graded hom-Lie algebra. Moreover, it also gives rise to a purely hom-Poisson structure introduced in [22].
As the first application, we define hom-Lie bialgebras using the hom-big bracket. A Lie bialgebra [8] is the Lie-theoretic case of a bialgebra: it is a set with a Lie algebra structure and a Lie coalgebra one which are compatible. Lie bialgebras are the infinitesimal objects of Poisson-Lie groups. Both Lie bialgebras and Poisson-Lie groups are considered as semiclassical limits of quantum groups. The solutions of the classical Yang-Baxter equations provide examples of Lie bialgebras. The hom-analogue of the Yang-Baxter equation and quantum groups are studied in [31,32]. Furthermore, hom-analogues of a Lie bialgebra are studied in two approaches recently [29,33]. The hom-Lie bialgebra defined here is the same as the one given in [33]. As a byproduct, we give the definitions of a hom-Lie quasi-bialgebra and a hom-quasi-Lie bialgebra. We hope that they are connected with hom-quantum groups [32]. They also provide a way to study the hom-analogue of Drinfeld twists.
As the second application, we define hom-Nijenhuis operators using the hom-big bracket. For a Lie algebra (l, [·, ·] l ), a Nijenhuis operator is a linear map N : l −→ l satisfying which gives a trivial deformation of Lie algebra l and plays an important role in the study of integrability of Hamilton equations [7,15]. In general, a 1-parameter infinitesimal deformation is controlled by a 2-cocycle ω : ∧ 2 l −→ l (see [26] for more details). In [4], the authors identified the role that Nijenhuis operators play in the theory of contractions and deformations of both Lie algebras and Leibniz (Loday) algebras. Nijenhuis operators on algebras other than Lie algebras, including for L ∞ -algebras, Poisson structures and Courant algebroids, can be found in [2,3,5,9,13,15]. In [28], a notion of a hom-Nijenhuis operator was given. However, the hom-Nijenhuis operator defined here is different from the existing one. Similarly, the notion of a hom-O-operator is also different from the one given in [29]. But we believe that the current definitions are more reasonable (see Remarks 6.3 and 6.10) and this justifies the usage of the hom-big bracket.
The paper is organized as follows. In Section 2, we recall notions of hom-Lie algebras, representations of hom-Lie algebras, hom-right-symmetric algebras, big brackets, Lie bialgebras and Nijenhuis operators. In Section 3, we give the definition of the hom-Nijenhuis-Richardson bracket [·, ·] α and show that the composition gives rise to a hom-right-symmetric algebra structure (Theorem 3.5). Consequently, [·, ·] α satisfies the hom-Jacobi identity. Then we obtain a new cohomology of a hom-Lie algebra via the hom-Nijenhuis-Richardson bracket, see (3.8).
In Section 4, we give the definition of the hom-big bracket and show that it gives rise to a graded hom-Lie algebra (Theorem 4.3). In particular, it is consistent with the hom-Nijenhuis-Richardson bracket. In Section 5, we define a hom-Lie bialgebra using the hom-big bracket and describe it using the usual algebraic language. We also give the definitions of a hom-Lie quasibialgebra and a hom-quasi-Lie bialgebra. In Section 6, we define a hom-Nijenhuis operator and a hom-O-operator using the hom-big bracket and study their properties.
(1) A (multiplicative) hom-Lie algebra is a triple (V, [·, ·], α) consisting of a vector space V , a skew-symmetric bilinear map (bracket) [·, ·] : ∧ 2 V −→ V and a linear map α : V → V preserving the bracket, such that the following hom-Jacobi identity with respect to α is satisfied (2) A hom-Lie algebra is called a regular hom-Lie algebra if α is an algebra automorphism.
Definition 2.2. A representation of the hom-Lie algebra (V, [·, ·], α) on the vector space W with respect to β ∈ gl(W ) is a linear map ρ : V −→ gl(W ), such that for all x, y ∈ V , the following equalities are satisfied

Definition 2.3 ([24]
). A hom-right-symmetric algebra is a triple (V, * , γ) consisting of a linear space V , a bilinear map * : V ⊗V → V and a linear map γ : V → V preserving the multiplication such that the following equality is satisfied Given a hom-right-symmetric algebra (V, * , γ), define [·, ·] :
Using the big bracket, a Nijenhuis operator N : V −→ V (viewed as an element in V * ⊗ V ) on a Lie algebra (V, µ) can be described by

The hom-Nijenhuis-Richardson bracket
The Nijenhuis-Richardson bracket is a graded Lie algebra structure on the space of alternating multilinear forms of a vector space to itself, introduced by Nijenhuis and Richardson [26]. In this section, we introduce the notion of hom-Nijenhuis-Richardson brackets and study their properties.
Let V be a vector space. For any k ≥ 0, denote by where the composition • is given by where Ad α is the adjoint map, i.e., Ad α N = αN α −1 .
In [30], the authors showed that (gl(V ), [·, ·] α , Ad α ) is a hom-Lie algebra, which plays important roles in the representation theory of hom-Lie algebras. More precisely, any representation of a hom-Lie algebra g on V can be realized as a homomorphism from g to the hom-Lie algebra (gl(V ), [·, ·] α , Ad α ).
The Jacobi identity can be described by the Nijenhuis-Richardson bracket. Similarly, the hom-Jacobi identity can be described by the hom-Nijenhuis-Richardson bracket.
is a graded hom-Lie algebra, i.e., we have By Lemma 3.4 and Corollary 3.6, for any hom-Lie algebra (V, µ, α), there is a coboundary operator d : This formula can be easily generalized for any representation. More precisely, for any representation ρ of the hom-Lie algebra (V, µ, α) on W with respect to β, define d : Theorem 3.7. With above notations, d 2 = 0. Thus, we have a well-defined cohomology.
Proof . By straightforward computation.
Remark 3.8. The coboundary operator d given above is different from the one given in [30]. It turns out that cohomology theories are not unique for hom-Lie algebras.

The hom-big bracket
In this section, we introduce the notion of hom-big brackets. Let α : V → V be an invertible linear map, α −1 its inverse and α * : V * → V * its dual map. α induces a linear map from ∧ p+1 V to ∧ p+1 V , for which we use the same notation, by In particular, we have . Thus, we use the same notation. For an invertible linear map α ∈ GL(V ), on the graded vector space ∧ • (V ⊕V * ), we define the hom-big bracket: which is uniquely determined by the following properties: (i) for all x, y ∈ V , {x, y} α = 0; (ii) for all ξ, η ∈ V * , {ξ, η} α = 0; (iii) For all x ∈ V and ξ ∈ V * , {x, ξ} α = ξ(α −1 x); (iv) {·, ·} α satisfies the following graded-commutative relation: In the classical case, the big bracket gives rise to a graded Lie algebra structure. Similarly, the hom-big bracket also induces a graded hom-Lie algebra structure, which is the main result in this section. To prove the theorem, we need some preparations. For all Ξ ∈ ∧ q+1 V * , x, x 1 , . . . , x q ∈ V , define the interior product i α We can get the following formulas by straightforward computations.
At the end of this section, we show that the hom-big bracket is consistent with the hom-Nijenhuis-Richardson bracket. (4.10) Proof . By (4.8), we have which implies that (4.10) holds.

Hom-Lie bialgebras
The big bracket is a very useful tool to study bialgebra structures. In this section, we follow the classical approach to define a hom-Lie bialgebra using the hom-big bracket, which turns out to be the same as the one given in [33]. Furthermore, using the hom-big bracket, it is very easy to give the notions of hom-Lie quasi-bialgebras and hom-quasi-Lie bialgebras.
First we describe hom-Lie algebras and hom-Lie coalgebras using the hom-big bracket.
Proposition 5.1. Let V be a vector space and µ : ∧ 2 V → V a skew-symmetric bilinear map satisfying Ad α µ = µ. Then we have Furthermore, {µ, µ} α = 0 if and only if µ satisfies the hom-Jacobi identity with respect to α.
Proof . It is obvious that ( Remark 5.5. The definition of a hom-Lie bialgebra given above is the same as the one given in [33]. However, to obtain the Manin triple theory, we need to follow the approach given in [29].
At the end of this section, we give the notions of a hom-Lie quasi-bialgebra and a hom-quasi-Lie bialgebra.

Hom-Nijenhuis operators and hom-O-operators
In this section, we give the notion of a hom-Nijenhuis operator using the hom-big bracket. We show that a hom-Nijenhuis operator gives rise to a trivial deformation. Furthermore, a new definition of a Hom-O-operator is given. The next proposition characterizes a hom-Nijenhuis operator using the usual algebraic formula. To be simple, we write µ(x, y) by [x, y] in the sequel. Therefore, we have By (6.2) and (6.3), we have By (5.1), (6.2)-(6.4), we have Therefore, N is a hom-Nijenhuis operator if and only if (6.1) holds.
Definition 6.4. A deformation is said to be trivial, if there exists a linear operator N : The condition (6.8) is equivalent to Therefore, N is a hom-Nijenhuis operator. Thus, a trivial deformation gives rise to a hom-Nijenhuis operator. The converse is also true.
Theorem 6.5. Let N be a hom-Nijenhuis operator. Then a deformation can be obtained by putting Furthermore, this deformation is trivial.
Proof . Obviously, ω = dN . Therefore, (6.6) holds naturally. By Ad α N = N and Ad α µ = µ, we can deduce that Ad α ω = ω. Finally, we need to check the hom-Jacobi identity for ω, which follows from the Nijenhuis condition (6.1). We omit details. Therefore, ω generates a trivial deformation. Furthermore, also by (6.1), it is straightforward to see that (6.8) holds. Thus, this deformation is trivial.
As in the classical case, any polynomial of a Nijenhuis operator is still a Nijenhuis operator. The following formula can be obtained by straightforward computations. Lemma 6.6. Let N be a hom-Nijenhuis operator acting on a hom-Lie algebra (V, [·, ·], α). Then for all i, j ∈ N, there holds Theorem 6.7. Let N be a hom-Nijenhuis operator acting on a hom-Lie algebra (V, [·, ·], α).
Then for any polynomial P (z) = Therefore, P (N ) is a Nijenhuis operator.
At the end of this section, we introduce a new definition of a hom-O-operator, which is a generalization of an O-operator introduced by Kupershmidt in [19].
Definition 6.8. Let (V, [·, ·], α) be a hom-Lie algebra and ρ : V −→ gl(W ) a representation of (V, [·, ·], α) on W with respect to β ∈ GL(W ). A linear map T : W → V is called a hom-Ooperator if T satisfies Lemma 6.9. With the above notations, a linear map T : W → V is a hom-O-operator if and only if 0 T 0 0 is a hom-Nijenhuis operator for the semidirect product hom-Lie algebra V ρ W .
Proof . By straightforward computations. Remark 6.10. As in the case of hom-Nijenhuis operators, the above definition of a hom-Ooperator is different from the one given in [29]. Now our principle is that T is a hom-O-operator if and only if 0 T 0 0 is a hom-Nijenhuis operator as the above lemma shows. Since the present definition of a hom-Nijenhuis operator is different from the one given in [28] (see Remark 6.3), it is reasonable that the definition of a hom-O-operator is also different from the old one. Recently, some applications of hom-O-operators were given in [6].
As in the classical case, a hom-O-operator can give rise to a hom-right-symmetric algebra.