Generalized Lie Triple Derivations of Lie Color Algebras and Their Subalgebras

: Consider a Lie color algebra, denoted by L . Our aim in this paper is to study the Lie triple derivations TDer ( L ) and generalized Lie triple derivations GTDer ( L ) of Lie color algebras. We discuss the centroids, quasi centroids and central triple derivations of Lie color algebras, where we show the relationship of triple centroids, triple quasi centroids and central triple derivation with Lie triple derivations and generalized Lie triple derivations of Lie color algebras L . A classiﬁcation of Lie triple derivations algebra of all perfect Lie color algebras is given, where we prove that for a perfect and centerless Lie color algebra, TDer ( L ) = Der ( L ) and TDer ( Der ( L )) = Inn ( Der ( L )) .


Introduction
The generalization of Lie algebra is introduced by Ree [1], which is now known as Lie color algebra. Lie color algebra plays an important role in theoretical physics, as expalained in [2,3]. Montgomery [4] proved that Simple Lie color algebra can be obtained from associative graded algebra, while the Ado theorem and the PBW theorem of Lie color algebra were proven by Scheunert [5]. In the last two decades, Lie color algebra has developed as an interesting topic in mathematics and physics [6][7][8][9][10].
The concept of derivations contributes significantly in the different mathematical fields such as in associative (non-associative) rings and operator algebras. In algebra, derivation is usually a linear map that satisfies the Leibniz rule. Researchers have worked on the concept of derivations, generalized derivations, centroids and quasi centroids with different perspectives in [11][12][13][14]. In fact there are various forms of derivations in algebra (Lie algebra) such as double derivations, triple derivations, and n-derivations. In the present article, we focus on the Lie triple derivations, which are first introduced by Müller [15]. Later on, various authors investigated the triple derivations in different algebraic settings. Wang and Xiao [16] studied the Lie triple derivations of incidence algebras, which is a type of operator algebra. Triple derivations of another operator algebra called nest algebra was discussed by Zhang [17]. Xiao and Wei [18] have researched the Lie triple derivations of triangular algebra. Furthermore, Lie triple derivations of some von Neumann algebra are studied by Qi [19], where it is proven that a Lie triple derivation of von Neumann algebra is the sum of a derivation algebra and a special additive map that sends the commutator to zero. Zhou [20] studied the triple derivations of perfect Lie algebra, where it is proven that Lie triple derivations of the perfect Lie algebra are in fact a derivation algebra. Moreover, every Lie triple derivation of the derivation algebra is an inner derivation. Later, this work was extended to Lie superalgebras in [21].
Our purpose in this paper is to discuss the Lie triple derivations TDer(L) (generalized Lie triple derivations GTDer(L)) of a Lie color algebra L. We discuss centroids and quasi centroids of Lie color algebras and evaluate some important results. In addition, our main

•
Lie triple derivations algebra coincide with Lie derivations algebra. • Lie triple derivations of derivations algebra coincide with inner derivations algebra.
This paper is organized as follows; in Section 2, we recall some important definitions and notions related to Lie color algebras L. Along with presenting some interesting propositions, we show that Lie triple derivations TDer(L) (generalized triple derivations GTDer(L)) of a Lie color algebra L form a subalgebra of the general linear Lie color algebra Kl(L). In Section 3, we define triple centroid TC(L) and triple quasi centroid TQC(L) of Lie color algebra. We show that, for centerless Lie color algebra, centroid and quasi centroid belong to commutative Lie color algebra, as explained in the literature [20,22,23]. Furthermore, we obtain the relation of centroid and quasi centroid with Lie triple derivations (generalized Lie triple derivations) of Lie color algebra L. In Section 4, we prove our main results in Theorems 2 and 3, where we prove that for the perfect and centerless Lie color algebras L, we have TDer(L) = Der(L) and TDer(Der(L)) = Inn(Der(L)). We prove our results by giving some interesting lemmas.

On the Lie Triple Derivations of Lie Color Algebras
Consider a Lie color algebra L over a field F with a characteristic denoted by Char(F ), satisfying Char(F ) = 2. The operation of L is denoted by [., .]. Let F * = F \{0} be the group of units of F and hg(L) be the set of all homogeneous elements in L. Suppose that x is a homogeneous element and its degree is represented by σ(x). We use G to denote a fixed abelian group, and θ, µ, λ are some notions for the elements of G. A Lie color algebra L is called perfect if its derived subalgebra [L, L] is equal to itself L. The center of L is denoted by Z(L). To introduce the concept of a Lie color algebra, we recall the bicharacter of an abelian group. Definition 1. Let F be a field and G be an abelian group. A map : G × G → F * is called a skew-symmetric bicharacter on G if the following identities hold, for all f , g, h ∈ G: 1. ( (g, h) (h, g) = 1.
With the notation of bicharacter, we can use it to define Lie color algebras as follows. For simplicity, we use (s, t) a shorthand notation for (σ(s), σ(t)) in the next definition. [ [s, t] = − (s, t)[t, s], and 3. ( Example 1. 1. If G = Z 2 (the additive group of integers modulo 2) and if one defines as (i, j) := (−1) ij for all i, j ∈ Z 2 , then Lie color algebras are just Lie superalgebras.

2.
If (i, j) := 1 for all i, j ∈ G, then a Lie color algebra is a G-graded Lie algebra [24].

3.
Suppose that A = ⊕ g∈G A g is an associative G-graded algebra and is a skew-symmetric bicharacter on G. Let [s, t] = st − (s, t)ts (the -commutator of s, t) for all s ∈ A σ , t ∈ A θ . Then (A, [., .]) turns out to be a Lie color algebra.
For any two vector spaces V and W, we use Hom(V, W) to denote the space of all linear mappings from V to W in the sequel. It is easy to check that Kl(L) = ⊕ θ∈G Kl θ (L) is a Lie color algebra over F with the bracket for all D θ , D µ ∈ hg(Kl(L)).

Definition 4.
A homogeneous derivation of degree θ of a Lie color algebra L = ⊕ θ∈G L θ is an element D ∈ Kl θ (L) such that for all s ∈ hg(L), t ∈ L.
Let Der θ (L) be the set of homogeneous derivations in Kl θ (L). Then Der(L) := ⊕ θ∈G Der θ (L) is a Lie color subalgebra of Kl(L) and is called the derivation algebra of L.
As a generalization of derivations of a Lie color algebra, we introduce the concept of Lie triple derivations of a Lie color algebra as follows.
Definition 5. Let L = ⊕ θ∈G L θ be a Lie color algebra and D ∈ Kl θ (G). Then D is called Lie for all s, t ∈ hg(L), u ∈ L.
It is obvious that every derivation in Lie color algebra is indeed a Lie triple derivation, but the converse is not always true in general. The set of all Lie triple derivations of degree θ of L is denoted by TDer θ (L). Let TDer(L) = ⊕ θ∈G TDer θ (L). Next, we will prove that TDer(L) is a Lie color subalgebra of Kl(L). Proposition 1. Suppose that L := ⊕ θ∈G L θ is a Lie color algebra. Then TDer(L) is a Lie color subalgebra of Kl(L).
A further generalization of Lie triple derivation is a generalized Lie triple derivation defined as follows. Definition 6. Let L = ⊕ θ∈G L θ be a Lie color algebra and D ∈ Kl θ (L). Then D is called a generalized Lie triple derivation of L if there is E ∈ TDer θ (L) related to D such that, for all s, t ∈ hg(L), u ∈ L.
It is obvious that a Lie triple derivation is a generalized Lie triple derivation with D = E, so TDer(L) ⊆ GTDer(L), but its converse is not always true.
We denote all generalized Lie triple derivations on a Lie color algebra by GTDer(L) = ⊕ θ∈G GTDer θ (L). Just like the set of all Lie triple derivations, the set of all generalized Lie triple derivations also forms a Lie color subalgebra of the Lie color algebra of linear maps Kl(L).

Lemma 1 ([25]
). For any s, t, u ∈ hg(L), D ∈ GTDer(L) and E ∈ TDer(L) related to D, we can obtain that Proof. Consider that D ∈ GTDer θ (L) and E is the Lie triple derivation related to D.
Applying D ∈ GTDer(L) on Jacobi identity: we get that The set of all central Lie triple derivations of Lie color algbera L is denoted by ZTDer(L) = ⊕ θ∈G ZTDer θ (L).

Remark 1.
It could be remarked that a central Lie triple derivation is a generalized Lie triple derivation with E = 0.
As stated in [26], central Lie derivations algebra is an ideal of Lie derivation algebra, and we want to check whether central Lie triple derivations ZTDer(L) are ideal of TDer(L) algebra or not.

Proof. By Equation (2), it is not hard to show that ZTDer(L) is subalgebra of TDer(L)
and GTDer(L). Next, we show that ZTDer(L) is an ideal. For any D 1 ∈ ZTDer θ1 (L) and for any s, t ∈ hg(L), u ∈ L. Similarly, we have Thus, we have [D 1 , D 2 ] ∈ ZTDer θ1+θ2 (L). This completes the proof.

Centroids and Quasi Centroids
In this section, we define the Lie triple centroid and Lie triple quasi centroid of a Lie color algebra L. We determine their relationship with TDer(L) and GTDer(L). We also discuss the relation between the center Z(L) and these maps. In order to achieve our goal, we first give some definitions, and then we move forward to our main results [23,27,28].
for any s, t ∈ hg(L), u ∈ L.
It should be noted that the second condition in Equation (20) follows from the use of Jacobi identity.
for any s, t ∈ hg(L), u ∈ L.
Proposition 5. Let L be a Lie color algebra, for any generalized Lie triple derivation D and Lie triple derivation E related to D; we have D − E ∈ TC(L).
Proof. Proof is obvious from Equation (12). Proposition 6. Let L be a Lie color algebra, for any generalized Lie triple derivation D ∈ GTDer(L) and Lie triple derivation E ∈ TDer(L) related to D; we have (D − E) ∈ TQC(L).

Proof. From Lemma 1, it is clear that
Therefore, by Definition 9, we can say (D − E) ∈ TQC(L). This completes the proof. Proof. We only need to prove the first case; the other case can be proven in a similar way. For any s, t ∈ hg(L) and u ∈ L, D 1 ∈ TC θ 1 (L), D 2 ∈ TC θ 2 (L), we find that Proof. Let D 1 ∈ TC θ 1 (L) and D 2 ∈ TQC θ 2 (L). Then for s, t ∈ hg(L) and u ∈ L, we have From Equation ( [TDer(L), TC(L)] ⊆ TC(L).
Proposition 9. Let L be a Lie color algebra. Then we have the following results: 1.
Proof. As with previous results, we only show the proof of first result; the second case can be obtained similarly. Suppose that D 1 ∈ GTDer θ 1 (L), D 2 ∈ TQC θ 2 (L), s, t ∈ hg(L), u ∈ L. Then On the other hand, we have By using Definition 9, we find that Thus, we have [D 1 , D 2 ] ∈ TQC θ1+θ2 (L). This completes the proof.

Classification of Triple Derivations of Perfect Lie Color Algebras
In the final section, we classify (generalized) Lie triple derivations of all perfect Lie color algebras. Let us recall some useful definitions. For the similar results related to Lie algebras and Lie superalgebras, readers are referred to [20,21].

Definition 10.
For any s ∈ L, we can define adjoint derivation ad : L → L such that ad(s)(t) = [s, t], for all t ∈ L. The set of all such derivations of L is denoted by Inn(L).
For convenience, we assume L to be a finite dimensional perfect Lie color algebra throughout this section. It is easy to see that Der(L) and Inn(L) are subalgebras of the Lie color algebras of TDer(L). Furthermore, we have the following lemmas. Proof. Let D ∈ TDer θ (L) and s ∈ L. Since L is a perfect Lie color algebra, there exists s = ∑ i∈I [s i 1 , s i 2 ] for some finite index set I such that s i 1 , s i 2 ∈ hg(L). For any arbitrary t ∈ L, we have By the arbitrariness of t, Inn(L) is an ideal of Lie color algebra TDer(L).
In fact, there is another connection between Der(L) and TDer(L). First let us give some lemmas. Proof. By the proof of Lemma 2, if L is perfect and has zero center, then we can define a linear endomorphism δ D on L as follows. For any s = ∑ i∈I [s i 1 , s i 2 ] ∈ L, we have This definition of δ D (s) does not depend on the choice of expression of s. To prove it, we take where s can also be expressed in the other form s = ∑ j∈J [t j 1 , t j 2 ]. Since D ∈ TDer(L), for any u ∈ L, we have that Hence, [(γ − ω), u] = 0, for any u ∈ L, i.e., γ − ω ∈ Z(L). As the center is zero, we have that γ = ω. Hence, δ D is well defined. Furthermore, it follows from the proof of Lemma (2) immediately that [D, ad(s)] = adδ D (s). Finally, to prove that δ D is a derivation of L. for any D ∈ TDer(L), s, t ∈ L, we have [D, ad[s, t]] = adδ D ([s, t]). By using Jacobi identity for Lie color algebras, we have for any s, t ∈ hg(L), u ∈ L. Since D ∈ TDer(L), we can obtain that The second main result of this article is given as follows.
The proof of the theorem follows from the following lemmas.
Proof. Since L is a perfect Lie color algebra, for any element s = ∑ i∈I [s i 1 , s i 2 ] ∈ L, we have From Lemma 2, it is not hard to see that D(Inn(L)) ⊆ Inn(L).
Lemma 6. Let L be a perfect Lie color algebra with the zero center. If D ∈ TDer(Der(L)), then for any s ∈ L, there exists d ∈ Der(L), such that D(ad(s)) = ad(d(s)).
Now it is not difficult to prove Theorem 3 by using above lemmas.