On split regular BiHom-Poisson color algebras


               <jats:p>The purpose of this paper is to introduce the class of split regular BiHom-Poisson color algebras, which can be considered as the natural extension of split regular BiHom-Poisson algebras and of split regular Poisson color algebras. Using the property of connections of roots for this kind of algebras, we prove that such a split regular BiHom-Poisson color algebra <jats:inline-formula>
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                        <jats:tex-math>L={\oplus }_{\left[\alpha ]\in \Lambda \text{/} \sim }{I}_{\left[\alpha ]}</jats:tex-math>
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                  </jats:inline-formula>. In particular, a necessary and sufficient condition for the simplicity of this algebra is determined, and it is shown that <jats:inline-formula>
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                  </jats:inline-formula> is the direct sum of the family of its simple (graded) ideals.</jats:p>

In particular, a necessary and sufficient condition for the simplicity of this algebra is determined, and it is shown that L is the direct sum of the family of its simple (graded) ideals.

Introduction
The interest in Poisson algebras has grown in the last few years, motivated especially by their applications in geometry and mathematical physics. For example, Poisson algebras play a fundamental role in deformation of commutative associative algebras [1]. Moreover, the cohomology group, deformation, tensor product and Γ-graded of Poisson algebras have been studied by many authors in [2][3][4][5]. A Hom-algebra is an algebra such that a linear homomorphism appears in the identities satisfied by its multiplication. This class of algebras appeared in the study of quasi-deformations of vector fields, in particular quasi-deformations of Witt and Virasoro algebras in [6]. So far, many authors have studied Hom-type algebras [7][8][9][10][11][12][13]. In particular, the notion of Hom-Lie color algebra was introduced in [8] and presented the methods to construct this color algebra, which can be viewed as an extension of a Hom-Lie algebra to a Γ-graded algebra, where Γ is any abelian group. Furthermore, a BiHom-algebra is an algebra in such a way that the identities defining the structure are twisted by two homomorphisms ϕ, ψ. A BiHom-Poisson color algebra has simultaneously a BiHom-Lie algebra structure and a BiHom-associative algebra structure, satisfying the BiHom-Leibniz identity.
The class of the split algebras is especially related to addition quantum numbers, graded contractions and deformations. For instance, for a physical system which displays a symmetry of L, it is interesting to know in detail the structure of the split decomposition because its roots can be seen as certain eigenvalues which are the additive quantum numbers characterizing the state of such system. Recently, the structure of different classes of split algebras such as split regular Hom-Poisson algebras, split regular Hom-Poisson color algebras, split regular BiHom-Lie superalgebras, split BiHom-Leibniz superalgebras and split Leibniz triple systems have been studied by using techniques of connections of roots (see for instance [14][15][16][17][18][19][20][21][22][23][24][25][26]).
Later, these techniques of connections become powerful to study not only split algebras but also graded algebras and algebras having multiplicative bases [27][28][29][30]. The purpose of this paper is to consider the decomposition and simplicity of split regular BiHom-Poisson color algebras by the techniques of connections of roots.

Preliminaries
First we recall the definitions of Lie color algebra, Poisson color algebra, Hom-Lie color algebra and Hom-Poisson color algebra. The following definition is well known from the theory of graded algebra.
be a Γ-graded -vector space. For a nonzero homogeneous element ∈ v L, denote by v the unique group element in Γ such that ∈ v L v , which will be called the homogeneous degree of v. We shall say that L is a Lie color algebra if it is endowed with a -bilinear map[⋅ ⋅] × → L L L , : , ,¯,¯, , (Jacobi identity), for all homogeneous elements ∈ v w t L , , .
Lie superalgebras are examples of Lie color algebras with = Γ 2 and ( ) = (− ) ε i j , 1 ij , for any ∈ i j , 2 . We also note that L 0 is a Lie algebra. , and a bi-character ε on Γ satisfying the following conditions: , two homomorphisms ϕ ψ , and a bi-character ε on Γ satisfying 1. (BiHom-Jacobi identity), for any ∈ x y z L , , , x y z,¯,¯denote the homogeneous degree of x y z , , . When ϕ ψ , furthermore are algebra automorphisms, it is said that L is a regular BiHom-Lie color algebra.
Definition 2.7. A BiHom-Poisson color algebra is a BiHom-Lie color algebra endowed with a BiHomassociative color product, that is, a bilinear product denoted by juxtaposition such that for all ∈ x y z L , , , and such that the BiHom-Leibniz color identity  ,¯,¯, . This gives the conclusion. □ Throughout this paper we will consider a regular BiHom-Poisson color algebra L being of arbitrary dimension and over an arbitrary base field . denotes the set of all non-negative integers and denotes the set of all integers. A subalgebra A of L is a graded subspace such that [ ] 0 and its only ideals are {0} and L. Let us introduce the class of split algebras in the framework of regular Hom-Poisson color algebras L. First, we recall that a Hom-Poisson color algebra ( [⋅ ⋅] ) L ϕ ε , , , , , over a base field , is called split with respect to a maximal Abelian subalgebra H of L, if L can be written as the direct sum , for any , a maximal Abelian (graded) subalgebra, of a regular BiHom-Poisson color algebra L. For a linear functional → α H : 0 , we define the root space of L (with respect to H) associated with α as the subspace We also say that Λ is the root system of L.
Note that when = = ϕ ψ Id, the split Poisson color algebras become examples of split regular BiHom-Poisson color algebras and when = ϕ ψ, the split regular Hom-Poisson color algebra become examples of split regular BiHom-Poisson color algebras. Hence, the present paper extends the results in [15].
From now on = ⊕ (⊕ ) ∈ L H L α α Λ denotes a split regular BiHom-Poisson color algebras. Also, and for an easier notation, the mappings | will be denoted by ϕ, ψ, − ϕ 1 , − ψ 1 , respectively. It is clear that the root space associated with the zero root L 0 satisfies ⊂ H L 0 . Conversely, given any Lemma 2.10. Let L be a split regular BiHom-Poisson color algebra. Then, for any α, ∈ ∪ { } β Λ 0 , the following assertions hold.
Now, let us show and apply equation (2.3).

For each
0 , we can apply BiHom-Jacobi identity and BiHom-skew-symmetry to get
We shall also say that { … } α α , , k 1 is a connection from α to β.
Our next goal is to show that the connection is an equivalence relation on Λ. For any ∈ α Λ, we denote by Our next goal is to associate an adequate ideal L Λα of L with any Λ α . For Λ α , ∈ α Λ, we define We denote by L Λα the following graded subspace of L,  Second, we will verify that By arguing as above, we have  By BiHom-associativity, we have

Hence, it just remains to check that H H
Proof. This is a direct consequence of Lemma 2.10-1,2. □ , and suppose there exist ∈ β Λ α and ∈ η Λ γ such that The following is divided into four situations to discuss.
Case 1: In a similar way, we get As above, BiHom-Leibniz color identity or BiHom-associativity identity gives us that this fact implies a contradiction either with equation (3.14) or equation (3.15). From here,   On the other hand, by BiHom-associativity, we get The proof is completed. □ Theorem 3.7. The following assertions hold.
1. For any ∈ α Λ, the subalgebra 2. If L is simple, then there exists a connection from α to β for any ∈ α β , Λ and α , taking into account Propositions 3.3 and 3.5, we have  By Propositions 3.3 and 3.6, we get   Let us introduce the concepts of root-multiplicativity and maximal length in the framework of split BiHom-Poisson color algebras, in a similar way to the ones for split BiHom-Lie algebras (see [15]   and so ∈ α Λ g 2 1 . From here, we have ∈ α Λ g 1 and ∈ α Λ g 2 1 , such that + ∈ − − + α ϕ α ψ Λ g g  α ϕ α ψ g g ,