ON THE STRUCTURE OF SPLIT INVOLUTIVE HOM-LIE COLOR ALGEBRAS

In this paper we study the structure of arbitrary split involutive regular Hom-Lie color algebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split involutive regular Hom-Lie color algebra L is of the form L = U ⊕ ∑ [α]∈Π/∼ I[α], with U a subspace of the involutive abelian subalgebra H and any I[α], a well-described involutive ideal of L, satisfying [I[α], I[β]] = 0 if [α] 6= [β]. Under certain conditions, in the case of L being of maximal length, the simplicity of the algebra is characterized and it is shown that L is the direct sum of the family of its minimal involutive ideals, each one being a simple split involutive regular Hom-Lie color algebra. Finally, an example will be provided to characterise the inner structure of split involutive Hom-Lie color algebras.


Introduction
The notion of Lie color algebras was introduced as generalized Lie algebras in 1960 by Ree in [11].In 1979, Scheunert investigated the Lie color algebras from a purely mathematical point of view (see [14]).So far, many results for this kind of algebras have been considered in the frameworks of enveloping algebras cohomology, representations, and related problems (see [5,10,12,13,15,16,19]).
In 2012, Yuan [20] introduced the notion of a Hom-Lie color algebra which can be considered as an extension of Hom-Lie superalgebras to Λ-graded algebras, where Λ is any additive abelian group.The pioneering works in these subjects are [1,2,17].
As is well-known, 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 Lie algebra 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 a system.Determining the structure of split algebras will become more and more meaningful in the area of research in mathematical physics.Recently, in [6,9,8,3,21], the structure of arbitrary split Lie algebras, arbitrary 62 VALIOLLAH KHALILI split involutive Lie algebras, arbitrary split Lie color algebras, arbitrary split regular Hom-Lie algebras, and arbitrary split involutive regular Hom-Lie algebras have been determined by the techniques of connection of roots.
Our goal in this work is to study the structure of arbitrary split involutive regular Hom-Lie color algebras by the techniques of connection of roots.The results of this article are based on some works in [3,7,21].
Throughout this paper, split involutive regular Hom-Lie color algebras L are considered of arbitrary dimension and over an arbitrary base field F, with characteristic zero.We also consider an additive abelian group Λ with identity zero.
To close this introduction, we briefly outline the contents of the paper.In Section 2, we begin by recalling the necessary background on split involutive regular Hom-Lie color algebras.Section 3 develops techniques of connections of roots for split involutive regular Hom-Lie color algebras.We also show that such an arbitrary split involutive regular Hom-Lie color algebra L with a root system Π is of the form L = U ⊕ [α]∈Π/∼ I [α] , with U a subspace of the involutive abelian subalgebra H and any I [α] , a well-described involutive ideal of L, satisfying [ In Section 4, we show that under certain conditions, in the case of L being of maximal length, the simplicity of the algebra is characterized and it is shown that L is the direct sum of the family of its minimal involutive ideals, each one being a simple split involutive regular Hom-Lie color algebra.Finally, Section 5 provides a concrete example which characterizes the inner structure of split involutive Hom-Lie color algebras.

Preliminaries
Let us begin with some definitions concerning graded algebraic structures.For a detailed discussion of this subject, we refer the reader to the literature [14].Let Λ be any additive abelian group.A vector space V is said to be Λ-graded if there is a family {V λ } λ∈Λ of vector subspaces such that V = λ∈Λ V λ .An element v ∈ V is said to be homogeneous of degree λ if v ∈ V λ , λ ∈ Λ, and in this case, λ is called the color of v.As usual, we denote by |v| the color of an element v ∈ V .Thus, each homogeneous element v in V determines a unique group element |v| ∈ Λ by v ∈ V |v| .Fortunately, we can almost always drop the symbol "| |", since confusion rarely occurs.
Let V = λ∈Λ V λ and W = λ∈Λ W λ be two Λ-graded vector spaces.A linear mapping f : If in addition f is homogeneous of degree zero, namely, f (V λ ) ⊂ W λ holds for any λ ∈ Λ, then we call f even.
An algebra A is said to be Λ-graded if its underlying vector space is Λ-graded, i.e., A = λ∈Λ A λ , and if A λ A µ ⊂ A λ+µ , for λ, µ ∈ Λ.A subalgebra of A is said to be graded if it is graded as a subspace of A.
Let B be another Λ-graded algebra.A homomorphism ϕ : A −→ B of Λ-graded algebras is a homomorphism of the algebra A into the algebra B, which is an even mapping.
Definition 2.1 ([14]).Let Λ be an abelian group.A map : The definition above implies that in particular, the following relations hold: Throughout this paper, if x and y are homogeneous elements of a Λ-graded vector space and |x| and |y|, which are in Λ, denote respectively their degrees, then for convenience we write (x, y) instead of (|x|, |y|).It is worth mentioning that, unless otherwise stated, in the sequel all the graded spaces are over the same abelian group Λ and the bi-character is the same for all structures.
Let L be a Hom-Lie color algebra over the base field F, and let − : F −→ F be an involutive automorphism which we call a conjugation on F. An involution is a conjugate-linear map * : and (φ(x)) * = φ(x * ), for all x, y ∈ L. Definition 2.3.A regular Hom-Lie color algebra endowed with an involution is said to be an involutive regular Hom-Lie color algebra.An involutive subset of an involutive algebra is a subset globally invariant by the involution.
The usual regularity conditions will be understood in the involutive graded sense.For instance, a subalgebra A of L is an involutive graded space A = λ∈Λ A λ of L such that [A, A] ⊂ A and φ(A) = A.An involutive graded subspace I = λ∈Λ I λ of L is called an ideal if [I, L] ⊂ I and φ(I) = I.We say that L is simple if [L, L] = 0 and its only (involutive graded) ideals are (0) and L.
From now on, (L, * ) denotes an involutive regular Hom-Lie color algebra.We introduce the concept of split involutive regular Hom-Lie color algebra in an analogous way.We begin by considering a maximal involutive abelian graded subalgebra H = λ∈Λ H λ among the involutive abelian graded subalgebras of L. Note that H is necessarily a maximal involutive abelian subalgebra of L (see [7,Lemma 2.4]).Let us introduce the class of split algebras in the framework of involutive split regular Hom-Lie color algebras.We denote by H = λ∈Λ H λ a maximal involutive abelian subalgebra of (L, * ).For a linear functional that is α(h * ) = α(h) for any h ∈ H 0 , we define the root space of L (with respect to H) associated to α as the subspace We also say that Π is the root system of L.
Proof.It is a directly consequence of Lemma 2.8 in [7].Definition 2.8.A root system Π of a split involutive regular Hom-Lie color algebra L is called symmetric if Π = −Π.

Connections of roots and decompositions
In the following, L denotes a split involutive regular Hom-Lie color algebra with a symmetric root system Π and L = H ⊕ ( α∈Π L α ), the corresponding root space decomposition.We begin by developing the techniques of connections of roots in the same setting as [3].Definition 3.1.Let α, β be two nonzero roots in Π.We say that α is connected to β, denoted by α ∼ β, if there exists a family satisfying the following conditions: Note that the case k = 1 in Definition 3.1 is equivalent to the fact that β = εαφ z for some z ∈ Z and ε ∈ {±1}.
We also have that αφ in case k ≥ 2, with ε ∈ {±1}.Then for any r ∈ N such that r ≥ m, there exists a connection {α 1 , α2 , . . ., αk } from α to β such that α1 = εβφ −r in case k = 1 or Proof.The assertions are proved in [ Next, we define Finally, we denote by I [α] the direct sum of the two graded subspaces above, that is, Proposition 3.4.For any α ∈ Π, the graded subspace Proof.First, we are going to check that . By the fact that L 0 = H and (3.1), it is clear that [I 0,[α] , I 0,[α] ] = 0, and we have Let us consider the first summand in (3.2).Note that by (3.1) and the fact that L 0 = H we have (3.4) Consider now the third summand in (3.2).We have Suppose that 0 = β + γ.By Lemma 2.6-(2), one gets (β + γ)φ −1 ∈ Π.Therefore, we get {β, γ} a connection from Third, we must show that ) * , thanks to the given definition.Now taking into account Theorem 3.6.The following assertions hold: (1) For any α ∈ Π, the involutive Hom-Lie color subalgebra , taking into account Proposition 3.4 and Proposition 3.5, we have As we also have by Proposition (2) The simplicity of L implies I [α] = L. From here, it is clear that [α] = Π and Theorem 3.7.For a vector space complement where any Proof.Each I [α] is well defined and by Theorem 3.6-( 1), an involutive ideal of L.
It is clear that Finally, Proposition 3.5 gives us [ Let us denote by Z(L) the center of L, that is, Corollary 3.9.If L is a perfect split involutive regular Hom-Lie color algebra, then L is the direct sum of the involutive ideals given in Theorem 3.6-( 1), . Now, by Z(L) = 0 and Proposition 3.5, the direct character of the sum is clear.

The simple components
In this section, we focus on the simplicity of split involutive regular Hom-Lie color algebras L by centering our attention in those of maximal length.Proof.It is analogous to the proof of Lemma 4.1 in [7].
Taking into account the above lemma, observe that the grading of I together with Lemma 2.5-(2) allow us to assert that Let us introduce the concepts of root-multiplicativity and maximal length in the framework of split involutive regular Hom-Lie color algebras, in a similar way to the ones for split regular Hom-Lie color algebras in [7].Definition 4.3.We say that a split involutive regular Hom-Lie color algebra L is root-multiplicative if given α ∈ Π λ and β ∈ Π µ , with λ, µ ∈ Λ, such that α + β ∈ Π, then [L λ α , L µ β ] = 0. Definition 4.4.We say that a split involutive regular Hom-Lie color algebra L is of maximal length if for any α ∈ Π λ with λ ∈ Λ, we have dim L kλ kα = 1 for k ∈ {±1}.Observe that for a split involutive regular Hom-Lie color algebra L of maximal length, (4.1) allows us to assert that given any nonzero graded ideal I of L we can write where Π λ I := {α ∈ Π : From here, since α 0 = 0, there exist β ∈ Π and µ ∈ Λ such that α Now, let us take any β ∈ Π such that β / ∈ {±α 0 φ z : z ∈ Z}.Since α 0 and β are connected, we have a connection {α 1 , α 2 , . . ., α k }, k ≥ 2, from α 0 to β satisfying the following conditions: for some n ∈ N, and for some m ∈ N and ε ∈ {±1}.
Theorem 4.6.Let L be a perfect split involutive regular Hom-Lie color algebra of maximal length and root-multiplicative.Then L is simple if and only if it has all its nonzero roots connected.
Proof.The first implication is Theorem 3.6-(2).To prove the converse, consider I a nonzero ideal of L = H ⊕ ( α∈Π L α ).By (4.7), we have H ⊂ I. Given any α ∈ Π, by the fact that α = 0 and the maximal length of L we have and so α∈Π L α ⊂ I. From here and H ⊂ I, we conclude that I = L. Therefore, L is simple.where any I [α] is a minimal involutive ideal of L, and each one being a simple split involutive regular Hom-Lie color algebra having all its nonzero roots connected.
Proof.By Corollary 3.9, we can write L = [α]∈Π/∼ I [α] as direct sum of the family of ideals

Example
In this section, an example is provided to clarify the results in Sections 3 and 4. The process of the example is described in four steps as follows.
Step 1 (Lie color algebra pso(2m + 1, 2n)).Let Λ = Z 2 × Z 2 and the skewsymmetric bi-character on Λ is defined as As a linear space, the Λ-graded Lie superalgebra L is a direct sum of four graded components: 1) .
Then, the root vectors and the corresponding root spaces are given by: Rev. Un.Mat.Argentina, Vol.60, No. 1 (2019)

Theorem 4 . 7 .
Let L be a perfect split involutive regular Hom-Lie color algebra of maximal length and root-multiplicative.ThenL = [α]∈Π/∼ I [α] , where each I [α] is a split involutive regular Hom-Lie color algebra having as root system Π I [α] = [α].In order to apply Theorem 4.6 to each I [α] , we have to observe that the root-multiplicativity of L and Lemma 3.5 show that Π I [α] has all of its elements Π I [α] -connected, that is, connected through connections contained in Π I [α] .We also get that any of the I [α] is root-multiplicative as consequence of the rootmultiplicativity of L. Clearly, I [α] is of maximal length, and finally Z I [α] (I [α] ) = 0, as consequence of Lemma 3.5, Theorem 4.6, and Z(L) = 0. We can therefore apply Theorem 4.6 to any I [α] so as to conclude that I [α] is simple.It is clear that the decomposition L = [α]∈Π/∼ I [α] satisfies the assertions of the theorem.