Generalized Derivations of Lie Color Algebras | SpringerLink

Results. Math. 63 (2013), 923–936 c 2012 Springer Basel AG 1422-6383/13/030923-14 published online March 10, 2012 DOI 10.1007/s00025-012-0241-2

Results in Mathematics

Generalized Derivations of Lie Color Algebras Liangyun Chen, Yao Ma and Lin Ni Abstract. In this paper, we give some basic properties of the generalized derivation algebra GDer(L) of a Lie color algebra L. In particular, we prove that GDer(L) = QDer(L) + QC(L), the sum of the quasiderivation algebra and the quasicentroid. We also prove that QDer(L) can be embedded as derivations in a larger Lie color algebra. Mathematics Subject Classification. 17B75. Keywords. Lie Color Algebras, Generalized Derivations, Quasiderivations, Centroids, Quasicentroids.

0. Introduction As a natural generalization of Lie algebras and Lie superalgebras [14], Lie color algebras play an important role in theoretical physics [18,23–25]. Ree [22] introduced generalized Lie algebras, which are called Lie color algebras today. Scheunert [23] proved the PBW theorem and the Ado theorem of Lie color algebras. Montgomery [20] obtained the simple Lie color algebras from associative graded algebras and proved the Herstein’s theorem. In recent years, Lie color algebras have become an interesting subject of mathematics and physics [1,3,5,21–25,27]. As is well known, derivation and generalized derivation algebras are very important subjects both in the research of Lie algebras and Lie superalgebras. In the study of Levi factors in derivation algebras of nilpotent Lie algebras, the generalized derivations, quasiderivations, centroids, and quasicentroids play key roles [2]. Melville dealt particularly with the centroids of nilpotent Lie Supported by NNSF of China (No. 11171055), NSF of Jilin province (No.201115006), Scientific Research Foundation for Returned Scholars Ministry of Education of China and the Fundamental Research Funds for the Central Universities (No. IISSXT146).

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algebras [19]. The most important and systematic research on the generalized derivation algebras of a Lie algebra and their Lie subalgebras was due to Leger and Luks [17]. In article [2], some nice properties of the generalized derivation algebras and their subalgebras, for example, of the quasiderivation algebras and of the centroids have been obtained. In particular, they investigated the structure of the generalized derivation algebras and characterized the Lie algebras satisfying certain conditions. Meanwhile, they also pointed that there exist some connections between quasiderivations and cohomology of Lie algebras. For the generalized derivation algebras of more general non-associative algebras, the readers will be referred to [6–13,15,16,19,26]. The purpose of this paper is to generalize some beautiful results in [2,28] to the generalized derivation superalgebras. In this paper, we mainly study the derivation algebra Der(L), the center derivation algebra ZDer(L), the quasiderivation algebra QDer(L), and the generalized derivation algebra GDer(L) of a Lie color algebra L. We proceed as follows. Firstly we recall some basic definitions and propositions which will be used in what follows. Then we give some basic properties of the generalized derivation algebras and their Lie subalgebras, show that the quasicentroid of a Lie color algebra is also a Lie color algebra if only and if it is an associative color algebra. Finally we prove that the quasiderivations of L can be embedded as derivations in a larger Lie color algebra L and obtain a direct sum decomposition of Der(L) when the annihilator of L is equal to zero.

1. Preliminaries Throughout this paper L is assumed to be a Lie color algebra over a field F of characteristic = 2 with the bracket [, ] : (a, b) → [a, b]. We denote the set of all homogeneous elements in a Lie color algebra L by hg(L). The notation σ(x) means that x is a homogeneous element and σ(x) represents the degree of x. Let G be an additive group and the notations θ, μ, λ, γ denote the elements of G. Our notations and terminologies are standard as may be found in [1,21–25,27,28]. Definition 1.1. [3] Let F be a field, Let G be an abelian group. A map ε : G × G → F ∗ is called a skew-symmetric bicharacter on G if the following identities hold, for all f, g, h ∈ G (1) ε(f, g + h) = ε(f, g)ε(f, h), (2) ε(g + h, f ) = ε(g, f )ε(h, f ), (2) ε(g, h)ε(h, g) = 1. Definition 1.2. [3] A Lie color algebra L is a G-graded vector space L = L with a graded bilinear map [, ] : L × L → L satisfying g∈G g (1) [Lθ , Lμ ] ⊆ Lθ+μ ,

∀θ, μ ∈ G,

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(2) [x, y] = −ε(x, y)[y, x], (3) ε(z, x)[x, [y, z]] + ε(x, y)[y, [z, x]] + ε(y, z)[z, [x, y]] = 0, where x ∈ Lσ(x) , y ∈ Lσ(y) , z ∈ Lσ(z) , σ(x), σ(y), σ(z) ∈ G. Definition 1.3. [3] Let Plθ (L) = {D ∈ Hom(L, L) : D(Lμ ) ⊆ Lθ+μ for all μ ∈ G}. ThenPl(L) = ⊕θ∈G Plθ (L) is a Lie color algebra over F with the bracket [Dθ , Dμ ] = Dθ Dμ − ε(θ, μ)Dμ Dθ , for all Dθ , Dμ ∈ hg(pl(L)). A homogeneous derivation of degree θ of L is an element D ∈ Plθ (L) such that [D(x), y] + ε(θ, x)[x, D(y)] = D([x, y]), for all x ∈ hg(L), y ∈ L. We denote the set of homogeneous derivations of degree θ of L by Derθ (L), then Der(L) := θ∈G Derθ (L) is a Lie color subalgebra of Pl(L) and is called the derivation algebra of L. Definition 1.4. An element D ∈ Plθ (L) is said to be a homogeneous generalized derivation of degree θ, if there exist elements D , D ∈ Plθ (L) such that [D(x), y] + ε(θ, x)[x, D (y)] = D ([x, y]),

(1.1)

for all x ∈ hg(L), y ∈ L. Definition 1.5. We call D ∈ Plθ (L) a homogeneous quasiderivation of degree θ, if there exists an element D ∈ Plθ (L) such that [D(x), y] + ε(θ, x)[x, D(y)] = D ([x, y]),

(1.2)

for all x ∈ hg(L), y ∈ L. Let GDerθ (L) and QDerθ (L) be the sets of homogeneous generalized derivations and of degree θ, respectively. of homogeneous quasiderivations GDer(L) := θ∈G GDerθ (L) and QDer(L) := θ∈G QDerθ (L) denote the sets of generalized derivations and of quasiderivations, respectively. It is easy to verify that both GDer(L) and QDer(L) are Lie color subalgebras of Pl(L) (see Proposition 2.1). Definition 1.6. If C(L) := θ∈G Cθ (L), with Cθ (L) consisting of D ∈ Plθ (L) satisfying [D(x), y] = ε(θ, x)[x, D(y)] = D([x, y]), for all x ∈ hg(L), y ∈ L, then C(L) is called the centroid of L. Definition 1.7. If QC(L) := θ∈G QCθ (L) and QCθ (L) consisting of D ∈ Plθ (L) such that [D(x), y] = ε(θ, x)[x, D(y)] for all x ∈ hg(L), y ∈ L, then QC(L) is called the quasicentroid of L. Define ZDer(L) := θ∈G Derθ (L), where Derθ (L) consists of D ∈ Plθ (L) satisfying that [D(x), y] = D([x, y]) = 0 for all x ∈ hg(L), y ∈ L. Then it is easy to see that ZDer(L) is a Lie color ideal of Der(L).

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It is easy to verify that ZDer(L) ⊆ Der(L) ⊆ QDer(L) ⊆ GDer(L) ⊆ Pl(L).

2. Generalized Derivation Algebras and Their Lie Color Subalgebras First, we give some basic properties of center derivation algebra, quasiderivation algebra and the generalized derivation algebra of a Lie color algebra. Proposition 2.1. Let L be a Lie color algebra. Then the following statements hold: (1) GDer(L), QGer(L) and C(L) are Lie color subalgebras of Pl(L). (2) ZDer(L) is a Lie color ideal of Der(L). Proof. (1)

Assume that Dθ , Dμ ∈ hg(GDer(L)), x ∈ hg(L) and y ∈ L. Then

[Dθ Dμ (x), y] = Dθ ([Dμ (x), y]) − ε(θ, μ + x)[Dμ (x), Dθ (y)] = Dθ (Dμ ([x, y]) − ε(μ, x)[x, Dμ (y)] −ε(θ, μ + x)([Dμ ([x, Dθ (y)]) − ε(μ, x)[x, Dμ Dθ (y)]) = Dθ Dμ ([x, y]) − ε(μ, x)Dθ ([x, Dμ (y)]) −ε(θ, μ + x)Dμ ([x, Dθ (y)]) + ε(θ, μ)ε(θ + μ, x)[x, Dμ Dθ (y)], and [Dμ Dθ (x), y] = Dμ Dθ ([x, y]) − ε(θ, x)Dμ ([x, Dθ (y)]) −ε(μ, θ + x)Dθ ([x, Dμ (y)]) + ε(μ, θ)ε(μ + θ, x)[x, Dθ Dμ (y)]. Thus for all x ∈ hg(L) and y ∈ L, we have [[Dθ , Dμ ](x), y] + ε(θ + μ, x)[x, [Dθ , Dμ ](y)] = [Dθ , Dμ ]([x, y]). Since both [Dθ , Dμ ] and [Dθ , Dμ ] are in Plθ+μ (L), [Dθ , Dμ ] ∈ GDerθ+μ (L) for all θ, μ ∈ G. Thus GDer(L) is a Lie color subalgebra of Pl(L). Similarly QGer(L) is a Lie color subalgebra of Pl(L). Assume that Dθ , Dμ ∈ hg(C(L)), x ∈ hg(L), and y ∈ L. Then [Dθ Dμ (x), y] = Dθ ([Dμ (x), y]) = ε(μ, x)Dθ ([x, Dμ (y)]) = ε(θ + μ, x)[x, Dθ Dμ (y)], and [Dμ Dθ (x), y] = ε(μ + θ, x)[x, Dμ Dθ (y)]. Thus [[Dθ , Dμ ](x), y] = [Dθ Dμ (x), y] − ε(θ, μ)[Dμ Dθ (x), y] = ε(θ + μ, x)[x, [Dθ , Dμ ](y)].

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Note that [[Dθ , Dμ ](x), y] = [Dθ Dμ (x), y] − ε(θ, μ)[Dμ Dθ (x), y] = Dθ Dμ ([x, y]) − ε(θ, μ)Dμ Dθ ([x, y]) = [Dθ , Dμ ]([x, y]), then [Dθ , Dμ ] ∈ Cθ+μ (L), which implies that C(L) is a Lie color subalgebra of Pl(L). (2) Assume that Dθ ∈ hg(ZDer(L)), Dμ ∈ hg(Der(L)), x ∈ hg(L), and y ∈ L. Then we have [[Dθ , Dμ ](x), y] = [Dθ Dμ (x), y] − ε(θ, μ)[Dμ Dθ (x), y] = −ε(θ, μ)Dμ ([Dθ (x), y]) + ε(μ, x)[Dθ (x), Dμ (y)] = 0, and [Dθ , Dμ ]([x, y]) = Dθ Dμ ([x, y]) − ε(θ, μ)Dμ Dθ ([x, y]) = Dθ ([Dμ (x), y]) + ε(μ, x)Dθ ([x, Dμ (y)]) = 0. Hence [Dθ , Dμ ] ∈ hg(ZDer(L)), and so (2) holds.

Lemma 2.2. Let L be a Lie color algebra. Then (1) [Der(L), C(L)] ⊆ C(L). (2) [QDer(L), QC(L)] ⊆ QC(L). (3) [QC(L), QC(L)] ⊆ QDer(L). (4) C(L) ⊆ QDer(L). Proof. (1) Let Dθ ∈ Derθ (L), Dμ ∈ Cμ (L). Then for all x ∈ hg(L) and y ∈ L, we have [Dθ Dμ (x), y] = Dθ ([Dμ (x), y]) − ε(θ, μ + x)[Dμ (x), Dθ (y)] = Dθ Dμ ([x, y]) − ε(θ, μ + x)ε(μ, x)[x, Dμ Dθ (y)], and [Dμ Dθ (x), y] = Dμ ([Dθ (x), y]) = Dμ Dθ ([x, y]) − ε(θ, x)Dμ ([x, Dθ (y)]) = Dμ Dθ ([x, y]) − ε(θ + μ, x)[x, Dμ Dθ (y)]. Hence, [[Dθ , Dμ ](x), y] = [Dθ Dμ (x), y] − ε(θ, μ)[Dμ Dθ (x), y] = Dθ Dμ ([x, y]) − ε(θ, μ)Dμ Dθ ([x, y]) = [Dθ , Dμ ]([x, y]).

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On the other hand, [Dθ Dμ (x), y] = Dθ ([Dμ (x), y]) − ε(θ, μ + x)[Dμ (x), Dθ (y)] = ε(μ, x)Dθ ([x, Dμ (y)]) − ε(θ, μ + x)ε(μ, x)[x, Dμ Dθ (y)] = ε(μ, x)[Dθ (x), Dμ (y)] + ε(θ + μ, x)[x, Dθ Dμ (y)] −ε(θ, μ)ε(θ + μ, x)[x, Dμ Dθ (y)], [Dμ Dθ (x), y] = ε(μ, θ + x)[Dθ (x), Dμ (y)]. Then [[Dθ , Dμ ](x), y] = [Dθ Dμ (x), y] − ε(θ, μ)[Dμ Dθ (x), y] = ε(θ + μ, x)[x, Dθ Dμ (y)] − ε(θ, μ)ε(θ + μ, x)[x, Dμ Dθ (y)] = ε(θ + μ, x)[x, [Dθ , Dμ ](y)]. Thus, [Dθ , Dμ ] ∈ Cθ+μ (L), and we get [Der(L), C(L)] ⊆ C(L). (2)

Similar to the proof of (1).

(3)

Assume that Dθ , Dμ ∈ hg(QC(L)), then for all x ∈ hg(L) and y ∈ L we have [[Dθ , Dμ ](x), y] + ε(θ + μ, x)[x, [Dθ , Dμ ](y)] = 0. Hence [Dθ , Dμ ] ∈ QDerθ+μ (L), as desired.

(4)

Assume that Dθ ∈ Cθ (L), for all x ∈ hg(L) and y ∈ L, we have [Dθ (x), y] + ε(θ, x)[x, Dθ (y)] = 2Dθ ([x, y]). Thus Dθ ∈ QDerθ (L).

Theorem 2.3. Let L be a Lie color algebra. Then GDer(L) = QDer(L)+QC(L). Proof. Let Dθ ∈ GDerθ (L). Then for all x, y ∈ hg(L), there exist Dθ , Dθ ∈ Plθ (L) such that [Dθ (x), y] + ε(θ, x)[x, Dθ (y)] = Dθ ([x, y]). Since ε(θ + x, y)[y, Dθ (x)] + ε(x, y)[Dθ (y), x] = ε(x, y)Dθ ([y, x]), [Dθ (y), x] + D +D D −D ε(θ, y)[y, Dθ (x)] = Dθ ([y, x]). Hence Dθ ∈ GDerθ (L), and θ 2 θ , θ 2 θ ∈ Plθ (L). For all x, y ∈ hg(L), we have Dθ + Dθ Dθ + Dθ (x), y + ε(θ, x) x, (y) = Dθ ([x, y]), 2 2 and

Dθ − Dθ Dθ − Dθ (x), y − ε(θ, x) x, (y) = 0, 2 2

which imply that

Dθ +Dθ 2

∈ QDerθ (L) and

Dθ −Dθ 2

∈ QCθ (L).

Corollary 2.4. Let L be a Lie color algebra. Then QC(L) + [QC(L), QC(L)] is a Lie color ideal of GDer(L).

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Proof. By Theorem 2.3 and Lemma 2.2(2), (3), we have QC(L) + [QC(L), QC(L)] ⊆ GDer(L), and [QC(L) + [QC(L), QC(L)], GDer(L)] = [QC(L) + [QC(L), QC(L)], QDer(L) + QC(L)] ⊆ QC(L) + [QC(L), QC(L)] + [[QC(L), QC(L)], QDer(L)]. It is easy to verify [[QC(L), QC(L)], QDer(L)] ⊆ [QC(L), QC(L)] by Jacobi identity. Thus [QC(L)+[QC(L), QC(L)], GDer(L)] ⊆ QC(L)+[QC(L), QC(L)]. Theorem 2.5. Let L be a Lie color algebra and Z(L) the center of L. Then [C(L), QC(L)] ⊆ Hom(L, Z(L)). Moreover, if Z(L) = {0}, then [C(L), QC(L)] = {0}. Proof. Assume that Dθ ∈ Cθ (L), Dμ ∈ QCμ (L), and x, y ∈ hg(L). Then [[Dθ , Dμ ](x), y] = [Dθ Dμ (x), y] − ε(θ, μ)[Dμ Dθ (x), y] = Dθ ([Dμ (x), y]) − ε(μ, x)[Dθ (x), Dμ (y)] = Dθ ([Dμ (x), y]) − ε(μ, x)Dθ ([x, Dμ (y)]) = Dθ ([Dμ (x), y] − ε(μ, x)[x, Dμ (y)]) = 0. Hence [Dθ , Dμ ](x) ∈ Z(L), and [Dθ , Dμ ] ∈ Hom(L, Z(L)) as desired. Furthermore, if Z(L) = {0}, it is clear that [C(L), QC(L)] = {0}. Theorem 2.6. Let L be a Lie color algebra, if Z(L) = {0}, then C(L) = QDer(L) ∩ QC(L). Proof. Suppose that Dθ ∈ QDerθ (L) ∩ QCθ (L), then there exists Dθ ∈ Plθ (L) such that [Dθ (x), y] + ε(θ, x)[x, Dθ (y)] = Dθ ([x, y]) and [Dθ (x), y] = ε(θ, x)[x, Dθ (y)]. For convenience, let Dθ = 2Eθ , then we have Eθ ([x, y]) = [Dθ (x), y] = ε(θ, x)[x, Dθ (y)],

(2.1)

for all x ∈ hg(L), y ∈ L. Note that for all x, y, z ∈ hg(L), we have [Eθ ([x, y]), z] = [[Dθ (x), y], z] = [Dθ (x), [y, z]] + ε(y, z)[[Dθ (x), z], y] = ε(θ, x)[x, Dθ ([y, z])] + ε(y, z)[Eθ ([x, z]), y] = Eθ ([x, [y, z]]) − ε(x + y, z)[Eθ ([z, x]), y]. So ε(z, x)[Eθ ([x, y]), z] + ε(y, z)[Eθ ([z, x]), y] = ε(z, x)Eθ ([x, [y, z]]).

(2.2)

Adding the three equations obtained from (2.2) by cyclically permuting x, y, z, we get 2(ε(z, x)[Eθ ([x, y]), z] + ε(y, z)[Eθ ([z, x]), y] + ε(x, y)[Eθ ([y, z]), x]) = 0. Thus, ε(z, x)[Eθ ([x, y]), z] + ε(y, z)[Eθ ([z, x]), y] = −ε(x, y)[Eθ ([y, z]), x] = ε(θ + z, x)[x, Eθ ([y, z])].

(2.3)

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By (2.1), (2.2) and (2.3), we have [x, Eθ ([y, z])] = [x, Dθ ([y, z])]. Hence [x, Dθ ([y, z])] = [x, [Dθ (y), z]] = ε(θ, y)[x, [y, Dθ (z)]]. If Z(L) = {0}, then Dθ ([y, z]) = [Dθ (y), z] = ε(θ, y)[y, Dθ (z)], for all y, z ∈ hg(L), i.e., Dθ ∈ Cθ (L). Thus QDer(L) ∩ QC(L) ⊆ C(L). The inverse of the inclusion is obvious, and the statement holds immediately. Proposition 2.7. Let L be a Lie color algebra, if Z(L) = {0}, then QC(L) is a Lie color algebra if and only if [QC(L), QC(L)] = 0. Proof. (⇒) Assume that Dθ , Dμ ∈ hg(QC(L)), x ∈ hg(L) and y ∈ L. Since QC(L) is a Lie color algebra, [Dθ , Dμ ] ∈ hg(QC(L)), then [[Dθ , Dμ ](x), y] = ε(θ + μ, x)[x, [Dμ , Dθ ](y)]. From the proof of Lemma 2.2 (3), we have [[Dθ , Dμ ](x), y] = −ε(θ + μ, x)[x, [Dμ , Dθ ](y)]. Hence [[Dθ , Dμ ](x), y] = 0, i.e., [Dθ , Dμ ] = 0. (⇒) It is clear. Definition 2.8. [4] Let L be a G-graded color algebra, if the multiplication satisfies the following identities: x · y = ε(x, y)y · x, ε(z, x + w)(((x · y) · w) · z − (x · y) · (w · z)) +ε(x, y + w)(((y · z) · w) · x − (y · z) · (w · x)) +ε(y, z + w)(((z · x) · w) · y − (z · x) · (w · y)) = 0, for all x, y, z, w ∈ hg(L). Then we call L a Jordan color algebra. Proposition 2.9. Let L be a Lie color algebra, with the operation Dλ • Dθ = Dλ Dθ + ε(λ, θ)Dθ Dλ , for all elements Dλ , Dθ ∈ hg(Pl(L)), Pl(L) is a Jordan color algebra. Proof. Assume that Dλ , Dθ , Dμ , Dγ ∈ hg(Pl(L)), we have Dλ • Dθ = Dλ Dθ + ε(λ, θ)Dθ Dλ = ε(λ, θ)(Dθ Dλ + ε(θ, λ)Dλ Dθ ) = ε(λ, θ)Dθ • Dλ . Since ((Dλ • Dθ ) • Dμ ) • Dγ = ((Dλ Dθ + ε(λ, θ)Dθ Dλ ) • Dμ ) • Dγ = ((Dλ Dθ + ε(λ, θ)Dθ Dλ )Dμ + ε(λ + θ, μ)Dμ (Dλ Dθ + ε(λ, θ)Dθ Dλ )) • Dγ = Dλ Dθ Dμ Dγ + ε(λ, θ)Dθ Dλ Dμ Dγ + ε(λ + θ, μ)Dμ Dλ Dθ Dγ +ε(λ+θ, μ)ε(λ, θ)Dμ Dθ Dλ Dγ +ε(λ+θ + μ, γ)Dγ Dλ Dθ Dμ +ε(λ+θ+μ, γ)ε(λ, θ)Dγ Dθ Dλ Dμ +ε(λ+θ+μ, γ)ε(λ+θ, μ)Dγ Dμ Dλ Dθ +ε(λ + θ + μ, γ)ε(λ + θ, μ)ε(λ, θ)Dγ Dμ Dθ Dλ ,

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and (Dλ • Dθ ) • (Dμ • Dγ ) = (Dλ Dθ + ε(λ, θ)Dθ Dλ ) • (Dμ Dγ + ε(μ, γ)Dγ Dμ ) = (Dλ Dθ + ε(λ, θ)Dθ Dλ )(Dμ Dγ + ε(μ, γ)Dγ Dμ ) +ε(λ + θ, μ + γ)(Dμ Dγ + ε(μ, γ)Dγ Dμ )(Dλ Dθ + ε(λ, θ)Dθ Dλ ) = Dλ Dθ Dμ Dγ + ε(μ, γ)Dλ Dθ Dγ Dμ + ε(λ, θ)Dθ Dλ Dμ Dγ +ε(λ, θ)ε(μ, γ)Dθ Dλ Dγ Dμ + ε(λ + θ, μ + γ)Dμ Dγ Dλ Dθ +ε(λ + θ, μ + γ)ε(λ, θ)Dμ Dγ Dθ Dλ + ε(λ + θ, μ + γ)ε(μ, γ)Dγ Dμ Dλ Dθ +ε(λ + θ, μ + γ)ε(μ, γ)ε(λ, θ)Dγ Dμ Dθ Dλ . We have ε(γ, λ + μ)(((Dλ • Dθ ) • Dμ ) • Dγ − (Dλ • Dθ ) • (Dμ • Dγ )) = ε(λ+θ+γ, μ)ε(γ, λ)Dμ Dλ Dθ Dγ +ε(λ + θ+γ, μ)ε(λ, θ)ε(γ, λ)Dμ Dθ Dλ Dγ +ε(θ, γ)Dγ Dλ Dθ Dμ + ε(λ, θ)ε(θ, γ)Dγ Dθ Dλ Dμ − ε(γ, λ)Dλ Dθ Dγ Dμ −ε(λ, θ)ε(γ, λ)Dθ Dλ Dγ Dμ − ε(λ + θ + γ, μ)ε(θ, γ)Dμ Dγ Dλ Dθ −ε(λ + θ + γ, μ)ε(λ, θ)ε(θ, γ)Dμ Dγ Dθ Dλ . Therefore, we get ε(γ, λ + μ)(((Dλ • Dθ ) • Dμ ) • Dγ − (Dλ • Dθ ) • (Dμ • Dγ )) +ε(λ, θ + μ)(((Dθ • Dγ ) • Dμ ) • Dλ − (Dθ • Dγ ) • (Dμ • Dλ )) +ε(θ, γ + μ)(((Dγ • Dλ ) • Dμ ) • Dθ − (Dγ • Dλ ) • (Dμ • Dθ )) = 0, and so the statement holds.

Corollary 2.10. Let L be a Lie color algebra, with respect to the operation Dλ •Dθ = Dλ Dθ +ε(λ, θ)Dθ Dλ , for all elements Dλ , Dθ ∈ hg(QC(L)), QC(L) is a Jordan color algebra. Proof. We need only to show that Dλ • Dθ ∈ QC(L) for all Dλ , Dθ ∈ hg(QC(L)). For all x, y ∈ hg(L), we have [Dλ • Dθ (x), y] = [Dλ Dθ (x), y] + ε(λ, θ)[Dθ Dλ (x), y] = ε(λ, θ + x)[Dθ (x), Dλ (y)] + ε(θ, x)[Dλ (x), Dθ (y)] = ε(λ, θ)ε(λ + θ, x)[x, Dθ Dλ (y)] + ε(λ + θ, x)[x, Dλ Dθ (y)] = ε(λ + θ, x)[x, Dλ • Dθ (y)]. Hence Dλ • Dθ ∈ QC(L).

Theorem 2.11. Let L be a Lie color algebra. Then QC(L) is a Lie color algebra with [Dλ , Dθ ] = Dλ Dθ − ε(λ, θ)Dθ Dλ if and only if QC(L) is also an associative color algebra with respect to composition.

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Proof. (⇐) For all Dλ , Dθ ∈ hg(QC(L)), we have Dλ Dθ ∈ QC(L) and Dθ Dλ ∈ QC(L), so [Dλ , Dθ ] = Dλ Dθ − ε(λ, θ)Dθ Dλ ∈ QC(L). Hence, QC(L) is a Lie color algebra. (⇒) Note that Dλ Dθ = Dλ • Dθ + [Dλ2,Dθ ] and by Corollary 2.10, we have Dλ • Dθ ∈ QC(L), [Dλ , Dθ ] ∈ QC(L). It follows that Dλ Dθ ∈ QC(L) as desired.

3. The Quasiderivations of Lie Color Algebras In this section, we will prove that the quasiderivations of L can be embedded as derivations in a larger Lie color algebra L and obtain a direct sum decomposition of Der(L) when the annihilator of L is equal to zero. Proposition 3.1. Let L = g∈G Lg be a Lie color algebra over F and t an inde˘ g := Lg [tF [t]/(t3 )] = {Σ(xg ⊗t+yg ⊗t2 ) : xg , yg ∈ Lg }. terminant. We define L ˘ ˘ is a Lie color algebra with the operation [xλ ⊗ti , xθ ⊗ ˘ Let L = g∈G Lg , then L tj ] = [xλ , xθ ] ⊗ ti+j , for all xλ , xθ ∈ hg(L), i, j ∈ {1, 2}. Proof. For all xλ , xθ , xμ ∈ hg(L), and i, j, k ∈ {1, 2}, we have [xλ ⊗ ti , xθ ⊗ tj ] = [xλ , xθ ] ⊗ ti+j = −ε(λ, θ)[xθ , xλ ] ⊗ ti+j = −ε(λ, θ)[xθ ⊗ tj , xλ ⊗ ti ], and [xλ ⊗ ti , [xθ ⊗ tj , xμ ⊗ tk ]] = [xλ , [xθ , xμ ]] ⊗ ti+j+k = ([[xλ , xθ ], xμ ] + ε(λ, θ)[xθ , [xλ , xμ ]]) ⊗ ti+j+k = [[xλ , xθ ], xμ ] ⊗ ti+j+k + ε(λ, θ)[xθ , [xλ , xμ ]] ⊗ ti+j+k = [[xλ ⊗ ti , xθ ⊗ tj ], xμ ⊗ tk ] + ε(λ, θ)[xθ ⊗ tj , [xλ ⊗ ti , xμ ⊗ tk ]]. ˘ is a Lie color algebra. Hence L

For notational convenience, we write xt(xt2 ) in place of x ⊗ t(x ⊗ t2 ). If U is a G− graded subspace of L such that L = U ⊕ [L, L], then ˘ = Lt + Lt2 = Lt + [L, L]t2 + U t2 , L or more precisely, ˘= L

g∈G

˘g = L

(Lg t + [L, L]g t2 + Ug t2 ).

g∈G

˘ satisfying Now we define a mapping ϕ : QDer(L) → Pl(L) ϕ(D)(at + bt2 + ut2 ) = D(a)t + D (b)t2 ,

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where D ∈ QDerθ (L), and D is in equation (1.2), a ∈ hg(L), b ∈ hg([L, L]), u ∈ hg(U ) and d(a) = d(b) = d(u). Proposition 3.2. (1) d(ϕ) = 0. (2) ϕ is injective and ϕ(D) does not depend on the choice of D . ˘ (3) ϕ(QDer(L)) ⊆ Der(L). Proof. (1) It is clear. (2) If ϕ(Dλ ) = ϕ(Dθ ), then for all a ∈ hg(L), b ∈ hg([L, L]) and u ∈ hg(U ), we have ϕ(Dλ )(at + bt2 + ut2 ) = ϕ(Dθ )(at + bt2 + ut2 ), that is, Dλ (a)t + Dλ (b)t2 = Dθ (a)t + Dθ (b)t2 , so Dλ (a) = Dθ (a). Hence Dλ = Dθ , and ϕ is injective. Suppose that there exists D such that ϕ(D)(at + bt2 + ut2 ) = D(a)t + D (b)t2 , and [D(x), y] + ε(D, x)[x, D(y)] = D ([x, y]), then we have D ([x, y]) = D ([x, y]), thus D (b) = D (b). Hence ϕ(D)(at + bt2 + ut2 ) = D(a)t + D (b)t2 = D(a)t + D (b)t2 , which implies ϕ(D) is determined by D. (3) We have [xλ ti , xθ tj ] = [xλ , xθ ]ti+j = 0, for all i + j ≥ 3. Thus, to show ˘ we need only to check the validness of the following ϕ(D) ∈ Der(L), equation ϕ(D)([xt, yt]) = [ϕ(D)(xt), yt] + ε(D, x)[xt, ϕ(D)(yt)]. For all x, y ∈ hg(L), we have ϕ(D)([xt, yt]) = ϕ(D)([x, y]t2 ) = D ([x, y])t2 = ([D(x), y] + ε(D, x)[x, D(y)])t2 = [D(x)t, yt] + ε(D, x)[xt, D(y)t] = [ϕ(D)(xt), yt] + ε(D, x)[xt, ϕ(D)(yt)]. ˘ Therefore, for all D ∈ QDer(L), we have ϕ(D) ∈ Der(L).

˘ ϕ as Proposition 3.3. Let L be a Lie color algebra with Ann(L) = {0} and L, ˘ = ϕ(QDer(L)) ⊕ ZDer(L). ˘ be defined above. Then Der(L) ˘ = Lt2 . For all g ∈ Der(L), ˘ we Proof. Since Ann(L) = {0}, we have Ann(L) 2 2 ˘ ˘ ˘ ˘ have g(Ann(L)) ⊆ Ann(L), hence g(U t ) ⊆ g(Ann(L)) ⊆ Ann(L) = Lt . Now we define a mapping f : Lt + [L, L]t2 + U t2 → Lt2 such that ⎧ ⎨ g(x) ∩ Lt2 , x ∈ Lt; x ∈ U t2 ; f (x) = g(x), ⎩ 0, x ∈ [L, L]t2 . ˘ L] ˘ ⊆ ˘ L]) ˘ = f ([L, L]t2 ) = 0, [f (L), It is clear that f is linear. Note that f ([L, 2 2 ˘ [Lt , Lt + Lt ] = 0, hence f ∈ ZDer(L). Since (g − f )(Lt) = g(Lt) ∩ Lt ⊆ ˘ L]) ˘ ⊆ [L, ˘ L] ˘ = [L, L]t2 , there Lt, (g − f )(U t2 ) = 0 and (g − f )([L, L]t2 ) = g([L, exist D, D ∈ Pl(L) such that for all a ∈ L, b ∈ [L, L], (g − f )(at) = D(a)t, (g − ˘ and by the definition of Der(L), ˘ we f )(bt2 ) = D (b)t2 . Since g − f ∈ Der(L)

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have [(g − f )(a1 t), a2 t] + ε(g − f, a1 )[a1 t, (g − f )(a2 t)] = (g − f )([a1 t, a2 t]) for all a1 , a2 ∈ L. Hence [D(a1 ), a2 ] + ε(D, a1 )[a1 , D(a2 )] = D ([a1 , a2 ]). Thus ˘ = D ∈ QDer(L). Therefore, g − f = ϕ(D) ∈ ϕ(QDer(L)), and we have Der(L) ˘ ϕ(QDer(L)) + ZDer(L). ˘ there exists an element D ∈ QDer(L) For all f ∈ ϕ(QDer(L)) ∩ ZDer(L), such that f = ϕ(D). Then f (at + bt2 + ut2 ) = ϕ(D)(at + bt2 + ut2 ) = D(a)t + ˘ we D (b)t2 for all a ∈ L, b ∈ [L, L]. On the other hand, since f ∈ ZDer(L), 2 2 2 ˘ have f (at + bt + ut ) ∈ Ann(L) = Lt . That is to say, D(a) = 0 for all a ∈ L and so D = 0. Hence f = 0. ˘ = ϕ(QDer(L)) ⊕ ZDer(L) ˘ as desired. Therefore Der(L)

4. Conclusions and Discussion In this paper, we mainly study the derivation algebra Der(L), the center derivation algebra ZDer(L), the quasiderivation algebra QDer(L), and the generalized derivation algebra GDer(L) of a Lie color algebra L, which are used to interpret the structures of Lie color algebra. It is just the beginning of the study on generalized derivation of Lie color algebras, so it might be interesting to consider many nice properties on δ−derivations of Lie algebras in [2,6–13,15,16,19,26] for Lie color algebras. In particular, it is quite possible to obtain some more explicit properties of center derivation algebra, quasiderivation algebra and the generalized derivation algebra of a Lie color algebra, respectively. Moreover, we have proved that the quasiderivation algebra can be embedded as derivations in a larger Lie color algebra. Similarly, it is still worth finding certain more examples of Lie color algebras.

Acknowledgements The authors would like to thank the referee for carefully reading and checking the paper both for its English as well as its mathematics.

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Received: October 12, 2011. Accepted: February 23, 2012.