Derivations and extensions of lie color algebra

Acta Mathematica Scientia 2008,28B(4):933–948 http://actams.wipm.ac.cn

DERIVATIONS AND EXTENSIONS OF LIE COLOR ALGEBRA∗




Zhang Qingcheng (

)

)

Zhang Yongzheng (

School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China E-mail: [email protected]; [email protected]

Abstract In this article, the authors obtain some results concerning derivations of finitely generated Lie color algebras and discuss the relation between skew derivation space SkDer(L) and central extension H 2 (L, F ) on some Lie color algebras. Meanwhile, they generalize the notion of double extension to quadratic Lie color algebras, a sufficient condition for a quadratic Lie color algebra to be a double extension and further properties are given. Key words Derivation, central extension, double extension, quadratic Lie color algebra 2000 MR Subject Classification

1

17B40, 17B50, 17B56, 17B70

Introduction

Lie color algebras are the natural generalization of Lie algebras and Lie superalgebras. In recent years, they have become an interesting subject of mathematics and physics. The PBW basis and Ado theorem of Lie color algebras have been obtained. Moreover, the cohomology groups of Lie color algebras were introduced and investigated (see [1]–[3]). In [4] and [5], the authors discussed the delta methods and primeness criteria for universal enveloping algebras of Lie color algebras, respectively. Lie color algebras of Witt type and Weyl type were constructed and studied in [6, 7]. The representations of Lie color algebras were explicitly described in [8]. As is well known, derivations and extensions of Lie algebras, Lie superalgebras, and Lie color algebras are very important subjects. In [9, 10], the author gave a classical description of central extensions H 2 (L, F ) of Lie algebra L over prime characteristic field by means of derivations and skew derivations. Note that a classification of quadratic Lie algebras was given by A.Medina and P.Revoy in [11] with this notion of double extension. In [12], the author generalized the notion of double extension to quadratic Lie superalgebras. This article is devoted to the study of derivations, central extensions, and double extension of Lie color algebras. Our present works are motivated by the results and methods in the Lie algebra case (see [9]–[12]), and would be based on some articles on Lie algebras and Lie superalgebras (see [13]–[18]). ∗ Received

December 28, 2005; revised December 26, 2006. Supported by National Natural Science Foundation of China (10271076)

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This article is organized as follows: Section 2 gives some results of derivations of Lie color algebras. In Section 3 the relation between skew derivation space SkDer(L) and central extension H 2 (L, F ) on some Lie color algebras is established. Finally, Section 4 generalizes the notion of double extension to quadratic Lie color algebras and gives the sufficient condition for a quadratic Lie color algebra to be a double extension.

2

Derivations of Lie Color Algebra

Throughout this section, F denotes a field of any characteristic. Let F ∗ = F \{0} be the group of units of F . All gradations are with respect to an additively written Abelian group. We will denote the degree of a nonzero homogeneous element x by σ(x). Definition 2.1 Let G be an abelian group. A map ε : G × G −→ F ∗ is called a skewsymmetric 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 ), (3) ε(g , h)ε(h , g) = 1. If x and y are homogeneous, and ε is skew-symmetric bicharacter, then, we shorten the notation by writing ε(x, y) instead of ε(σ(x) , σ(y)).  Definition 2.2 A Lie color algebra L is a G-graded vector space L = Lg with a g∈G

graded bilinear map [ , ]: L × L −→ L satisfying [Lξ , Lη ] ⊂ Lξ+η for every ξ, η ∈ G, and [x, y] = −ε(x, y)[y, x], ε(z, x)[x, [y, z]] + ε(x, y)[y, [z, x]] + ε(y, z)[z, [x, y]] = 0, for every x ∈ Lx , y ∈ Ly , and z ∈ Lz . Example 1 If G := {0} and ε(0, 0) := 1, then every Lie algebra is a Lie color algebra. Example 2 If G is an arbitrary commutative group and ε(x, y) := 1 for every x, y ∈ G, then every G-graded Lie algebra is a Lie color algebra. ¯ := (−1)αβ for every α, β ∈ Z, then every Example 3 If G := Z2 = {¯ 0, ¯ 1} and ε(¯ α, β) Lie superalgebra is a Lie color algebra. Example 4 If G := Z+ denotes the set of non-negative integers and ε(α, β) := (−1)αβ for every α, β ∈ G, then every G-graded Lie superalgebra is a Lie color algebra (see [1]–[6]). A color subalgebra is a G-graded subalgebra B of L such that [B, B] ⊂ B. A color ideal of L is a G-graded subalgebra I of L such that [L, I] ⊂ I. Note that by Definition 2.2 a color ideal I also satisfies [I, L] ⊂ I. As is well known, the K-linear map [, ] : A × A −→ A defined by [a, b] := ab − ε(a, b)ba for all a, b ∈ A, where A is a G-graded associative F -algebra, which gives rise to a Lie color algebra structure on A called the associated Lie color algebra of A and will be denoted by L(A).  Let L be a Lie color algebra. A L-module M is a G-graded vector space M = Mg such g∈G

that Lg .Mh ⊆ Mg+h , for every g, h ∈ G and [x, y].m = x.(y.m) − ε(x, y)y.(x.m),

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for every m ∈ M, x, y ∈ L. Note that a homomorphism ρ : L → End(M ) of Lie color algebras defines a L-module structure on M and vice vera. Let V and W be L-modules, and f ∈ Hom(V, W )g , g ∈ G such that f (x.m) = ε(g, x)x.f (m), for every x ∈ L, m ∈ M , then f is called a color homomorphism of L-modules. If ε(g, x) = 1, then f is called a homomorphism of L-modules. Let L be a Lie color algebra over an algebraically closed field F . We shall always assume ˜ := {f ∈ L∗ | f (Lg ) = 0, that the group G is generated by {g ∈ G; Lg = 0}. Note that L and L for all but finitely many g ∈ G} are color L-modules where the gradation of the latter is being ˜ g := {f ∈ L∗ | f (Lh ) = 0, ∀h ∈ G\{−g}}. given by (L) Definition 2.3 Let V be a L-module. A linear mapping ϕ : L → V is called a derivation if ϕ([x, y]) = ε(ϕ, x)x.ϕ(y) − ε(ϕ + x, y)y.ϕ(x), for every x, y ∈ L. The derivations of the form x → x.v(v ∈ V ) are called inner. We say that a derivation ϕ has degree g(deg(ϕ) = g) if ϕ = 0 and ϕ(Lh ) ⊂ Vg+h , for all h ∈ G. We let DerF (L, V ) and InnF (L, V ) denote the spaces of derivations and inner derivations, respectively, and write DerF (L, V )g := {ϕ ∈ DerF (L, V ); deg(ϕ) = g} ∪ {0}. Note that H 1 (L, V ) := DerF (L, V )/InnF (L, V ) is the first cohomology group of L with coefficients in V . In this section, L is assumed to be finitely generated (as a Lie color algebra). It is easily seen that this forces G to be finitely generated.  Proposition 2.1 Let V be a L-module. Then DerF (L, V ) = DerF (L, V )g . g∈G

Proof For each element g ∈ G, let pg : L → Lg and πg : V → Vg denote the canonical projections. According to our general assumption there is a finite subset S ⊂ L generating L. Let ϕ : L → V be a derivation. Then there are finite sets Q, R ⊂ G such that  S⊂ Lg , (∗) g∈Q

and ϕ(S) ⊂ For g ∈ G, we put ϕg :=

 h∈G



Vg .

(∗∗)

g∈R

πg+h ϕph . Because for xh ∈ Lh and xk ∈ Lk , we have

ϕg ([xh , xk ]) = πg+h+k ϕ([xh , xk ]) = πg+h+k (ε(ϕ, h)xh ϕ(xk ) − ε(ϕ + h, k)xk ϕ(xh )) = ε(ϕ, h)xh πg+k (ϕ(xk )) − ε(ϕ + h, k)xk πg+h (ϕ(xh )) = ε(ϕ, h)xh ϕg (xk ) − ε(ϕ + h, k)xk ϕg (xh ), it shows that ϕg is contained in DerF (L, V )g . Let T := {g − h | g ∈ R, h ∈ Q}. Then T is finite and we obtain, observing (*) and (**), for y ∈ S   ϕ(y) = πg ϕ(y) = πg ϕph (y) g∈R

g∈R h∈Q

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    ( π(g−h)+h ϕph (y)) = ( πq+h ϕph (y)) h∈Q g∈R

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h∈Q q∈T

πq+h ϕph (y) =

q∈T h∈Q

This shows that the derivations ϕ and  ϕ= ϕq . This proves the assertion.

 q∈T



πq+h ϕph (y) =

q∈T h∈G



ϕq (y).

q∈T

ϕq coincide on S. As S generates L, we obtain

q∈T

Proposition 2.2 Let V be a L-module such that (a) H 1 (L0 , Vg ) = 0, ∀g ∈ G \ {0}, (b) HomL0 (Lg , Vh ) = 0, g = h. Then DerF (L, V ) = DerF (L, V )0 + InnF (L, V ). Proof Let ϕ : L → V be a color derivation. According to Proposition 2.1, we can  decompose ϕ into its homogeneous components: ϕ = ϕg , ϕg ∈ DerF (L, V )g . Suppose that g∈G

g = 0. Then ϕg |L0 is a color derivation from L0 into the L0 -module Vg . By virtue of (a) ϕg |L0 is inner, i.e., there is vg ∈ Vg such that ϕg (x) = x.vg , for all x ∈ L0 . Consider ψg : L → V, ψg (x) = ϕg (x) − ε(g, x)x.vg , ∀x ∈ L. Then ψg is a color derivation of degree g, which vanishes on L0 . Hence, for every x0 ∈ L0 , x ∈ Lh , ψg ([x0 , x]) = ε(g, 0)x0 ψg (x) − ε(g + 0, h)x.ψg (x0 ) = x0 .ψg (x). Thus, ψg is a color homomorphism of L0 -modules and condition (b) entails the vanishing of ψg on Lh for every h ∈ G. Consequently, ϕg ∈ InnF (L, V ), proving the desired result. Remark 1 Let L be a Lie color algebra and V a L-module. Suppose that τ : L → V is a color homomorphism of L-modules of degree r, i.e., τ (Lg ) ⊂ Vg+r , for all g ∈ G. If σ : G → F is Z-linear, then the linear mapping ϕσ,τ : L → V given by ϕσ,τ (xg ) = σ(g)τ (xg ), for every xg ∈ Lg , g ∈ G is a derivation of degree r. Proof For every xg ∈ Lg , xh ∈ Lh , we have ϕσ,τ ([xg , xh ]) = σ(g + h)τ ([xg , xh ]) = (σ(g) + σ(h))τ ([xg , xh ]) = σ(g)τ ([xg , xh ]) + σ(h)τ ([xg , xh ]) = −ε(g, h)ε(r, h)σ(g)xh τ (xg ) + σ(h)ε(r, g)xg τ (xh ) = ε(r, g)xg ϕσ,τ (xh ) − ε(g + r, h)xh ϕσ,τ (xg ). This is the desired result. Let J
L be an ideal and consider the L-submodule V J := {v ∈ V | x.v = 0, ∀x ∈ J} of V.  Remark 2 Let A ⊂ L be a color subalgebra, J
L a color ideal. If L = A J, then every derivation ϕ : A → V L can be extended to a derivation ϕˆ : L → V by setting ϕ(J) ˆ = 0. Proof Let ϕˆ : L → V such that ⎧ ⎨ ϕ(x), x ∈ A, ϕ(x) ˆ = ⎩ 0, x ∈ J. Then we have ϕ([a, ˆ x]) = 0, a ∈ A, x ∈ J. But ε(ϕ, ˆ a)a.ϕ(x) ˆ − ε(ϕˆ + a, x)x.ϕ(a) ˆ = −ε(ϕˆ + a, x)x.ϕ(a) ˆ = −ε(ϕˆ + a, x)x.ϕ(a).

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Since ϕ(a) ∈ V J , then x.ϕ(a) = 0, i.e., ϕ([a, ˆ x]) = ε(ϕ, ˆ a)a.ϕ(x) ˆ − ε(ϕˆ + a, x)x.ϕ(a). ˆ Hence ϕˆ is a color derivation and ϕ(J) ˆ = 0.

3

Central Extensions and Skew Derivations

In this section, we will discuss the relation between skew derivation space SkDer(L) and central extension H 2 (L, F ) on some Lie color algebra L. Throughout this section, all Lie color algebras and modules are assumed to be finitedimensional over a field F of characteristic p = 2. Definition 3.1 Let g1 , g2 be Lie color algebras. A Lie color algebra g is an extension μ λ of g2 by g1 , if there is an exact sequence of Lie color algebras 0 → g1 → g → g2 → 0, where λ and μ are Lie color algebras homomorphisms. The kernel η of μ is said to be the kernel of the extension. μ λ Definition 3.2 An extension 0 → g1 → g → g2 → 0, of g2 by g1 with kernel η is said to be trivial if there exists a graded ideal of g supplementary to η in g. It is called central if η is a subset of C(g), where C(g) is the center of g. Theorem 3.1 Let L, L , A, B be Lie color algebras over field F. (1) If L is an extension of B by A, and there is an isomorphism between L and L, then  L is also an extension of B by A. (2) Let L be an extension of B by A and so is L . If a homomorphism f : L −→ L such that λ = f λ, μ = μ f, then f : L −→ L is an isomorphism of Lie color algebras. Lemma 3.2 The extension L of two solvable Lie color algebras is solvable. Let L be a Lie color algebra over field F . Then F has a trivial structure of a G-graded L-module: the G-gradation of F is defined by F = F0 , Fg = 0, where g ∈ G, g = 0, and the representation of L in F is equal to zero. Let L∗ = HomF (L, F ). We know that L∗ is a G-graded space. Let (x.f )(y) := −ε(x, f )f ([x, y]), where x ∈ L, y ∈ L, f ∈ L∗ . Then L∗ is a G-graded L-module. Let V and W be G-graded L-modules and λ : V × W −→ F a bilinear form. λ is called non-degenerate if V ⊥ = W ⊥ = 0. λ is called L-invariant if λ(x.v, w) = −ε(x, v)λ(v, x.w), x ∈ L, v ∈ V, w ∈ W . λ is called an associative form of L if V = W = L and λ([x, y], z) = λ(x, [y, z]), ∀x, y, z ∈ L. Every invariant bilinear form of L is associative by the definition of a Lie color algebra. A linear mapping f : V −→ W is said to be skew with respect to λ if λ(v1 , f (v2 )) = −ε(v1 , f + v2 )ε(f, v2 )λ(v2 , f (v1 )), v1 , v2 ∈ V. If ϕ : L −→ L is skew and ϕ ∈ DerF (L), we call ϕ a skew derivation of L. Let SkDer(L) denote the subspace of DerF (L) which consist of all skew derivations. Definition 3.3 Let V and W be G-graded L-modules. Suppose f ∈ HomF (V, W )g , where g ∈ G , is called a color-homomorphism if f (x.v) = ε(f, x)x.f (v), ∀v ∈ V, x ∈ L. f ∈ HomF (V, W )g is said to be a color-isomorphism if f is a color-homomorphism and f is a bijection. If not mentioned explicitly, all homogeneous elements mean to be homogeneous with respect to G-gradation.

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Suppose that λ : L × V −→ F is a homogeneous and non-degenerate bilinear form. For every f ∈ HomF (L, V ), there exists fλ ∈ HomF (L, V ) such that λ(x, f (y)) = ε(x, f + y)ε(f, y)λ(y, fλ (x)), x, y ∈ L. Proposition 3.3 Let L be simple Lie color algebra, and assume that λ : L × V −→ F is a non-degenerate and homogeneous invariant bilinear form. Then the following statements hold: (1) Let D : L −→ V be a derivation. Then D + Dλ is a color-homomorphism of Lmodules. (2) If V is not color-isomorphic to L as an L-module, then every homogeneous derivation D : L −→ V is skew. Proof (1) According to the invariance of λ, we have λ(x, Dλ ([y, z]) = ε(D, y + z)ε(x, y + z)ε(x, D)ε(y + z, D + x)ε(D, x)λ(x, Dλ ([y, z])) = ε(D, y + z)ε(x, y + z)ε(x, D)λ([y, z], D(x)) = −ε(D, y + z)ε(x, y + z)ε(x, D)ε(y, z)λ(z, y.D(x)) = ε(D + x, z)ε(x, D)ε(y, z)λ(z, −ε(D, x)x.D(y) + D([x, y]) = ε(D + x + y, z)ε(z, x)λ([x, z], D(y)) +ε(D + x + y, z)ε(x, D)λ(z, D([x, y])) = ε(D + x + y, z)ε(z, x)λ([x, z], D(y)) + ε(D, y)λ([x, y], Dλ (z)) = ε(D + y, z)λ(x, z.D(y)) + ε(D, y)λ(x, y.Dλ (z)). Consequently, Dλ ([y, z]) = ε(D + y, z)z.D(y) + ε(D, y)y.Dλ (z). Hence, (D + Dλ )(y.z) = D([y, z]) + Dλ ([y, z]) = ε(D, y)y.D(z) − ε(D + y, z)z.D(y) + ε(D + y, z)z.D(y) + ε(D, y)y.Dλ (z) = ε(D, y)y.(D + Dλ )(z). This implies that D + Dλ is a color-homomorphism of L-modules. (2) As L is simple and dimF L = dimF V, D + Dλ is either 0 or a color-isomorphism. The latter possibility contradicts the assumption pertaining to V , and hence, D = −Dλ and D is skew. Let L be simple. The vector space L∗ obtains the structure of an L-module by means of (x.f )(y) = −ε(x, f )f ([x, y]). The canonical bilinear form λ : L × L∗ −→ F , such that λ(x, f ) = ε(x, f )f (x) is non-degenerate and invariant. Lemma 3.4 Let L be simple. If L does not possess any non-trivial associative form, then every homogeneous derivation from L into L∗ is skew with respect to the canonical form. Proof This is a direct result of Proposition 3.3(2). Please look up the definition and some information with respect to cohomology groups of Lie superalgebras in Ref. [2] and [15].

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Lemma 3.5 The mapping ψ : C 2 (L, F ) −→ C 1 (L, L∗ ), f → f¯ such that f¯(x)(y) = f (x, y) induces a homomorphism H 2 (L, F ) −→ H 1 (L, L∗ ). Proof By virtue of the definition of ψ , we have degψ = 0. To verify Lemma 3.5, we only need to verify that ψ(Z 2 (L, F )) ⊆ Z 1 (L, L∗ ) and ψ(B 2 (L, F )) = B 1 (L, L∗ ). If f ∈ Z 2 (L, F ), by virtue of df (x, y)(z) = 0, we have df¯(x, y)(z) = −f ([x, y], z) + f (x, [y, z]) − ε(x, y)f (y, [x, z]) = 0. Then ψ(f ) ∈ Z 1 (L, L∗ ). As C 1 (L, F ) = L∗ = C 0 (L, L∗ ), ψ(dg)(x)(y) = dg(x, y) = −g([x, y]) and dg(x)(y) = (ε(g, x)x.g)(y) = −g([x, y]) for ∀g ∈ C 1 (L, F ), i.e., ψ(B 2 (L, F )) = B 1 (L, L∗ ). Proposition 3.6 Let L be a Lie color algebra and let λ : L × L∗ −→ F denote the canonical form. Then the following statements hold: (1) Imψ =< {f + B 1 (L, L∗ ) | f ∈ Z 1 (L, L∗ )∩SkDer(L, L∗ )} >; (2) If L is simple and does not possess any nondegenerate associative form, then H 2 (L, F ) ∼ = 1 ∗ H (L, L ). Proof (1) Assume that f : L → L∗ is a skew 1-cocycle. Obviously, λ(x, f (y)) = −ε(x, f + y)ε(f, y)λ(y, f (x)), and ε(x, f + y)f (y)(x) = −ε(x, f + y)ε(f, y)ε(y, f + x)f (x)(y). So we have f (x)(y) = −ε(x, y)f (y)(x) and g(x, y) := f (x)(y), which defines a 2-cochain. Since f is a 1-cocycle, we obtain f ([x, y]) = ε(f, x)x.f (y) − ε(f + x, y)y.f (x), i.e., f is a derivation. Since d(g)(x, y, z) = −g([x, y], z) + ε(y, z)g([x, z], y) − ε(x, y + z)g([y, z], x) = −f ([x, y])(z) + ε(y, z)f ([x, z])(y) − ε(x, y + z)f ([y, z])(x) = −f ([x, y])(z) − ε(x, y)f (y)([x, z]) + f (x)([y, z]) = −f ([x, y])(z) + ε(f, x)(x.f (y))(z) − ε(f + x, y)(y.f (x))(z) = 0, g is a 2-cocycle with ψ(g)(x) = f (x), ∀x ∈ L. Consequently, ψ(¯ g ) = f¯. Conversely, let f¯ be an element of Imψ. Then there is g ∈ C 2 (L, F ) such that f¯ = ψ(¯ g) = ∗ g¯1 . Consequently, f (x) = g1 (x) + d(h)(x) = g1 (x) + ε(h, x)x.h for some h ∈ L . This shows that f (x)(y) = g(x, y) − h([x, y]). Because g is alternating, we obtain f (x)(y) = −ε(x, y)f (y)(x), i.e., f is skew. It is clear that f is a derivation. (2) As every element of Z 1 (L, L∗ ) is a derivation, the conclusion follows from Lemma 3.5. Lemma 3.7 Suppose that L has a nondegenerate homogeneous associative form. Then there exists a color-isomorphic ϕ : L −→ L∗ . Proof Let λ be a nondegenerate associative form on L. We define ϕ : L → L∗ by means of ϕ(x)(y) := λ(x, y). The mapping ϕ is linear. As kerϕ =rad(λ) = 0, and ϕ is injective, hence ϕ is bijective. Let x, y, z be elements of L. Then we obtain ϕ([z, x])(y) = λ([z, x], y) = −ε(z, x)λ(x, [z, y]) = −ε(z, x)ϕ(x)([z, y]) = ε(ϕ, z)(z.ϕ(x))(y), i.e., ϕ is a color-isomorphism. Theorem 3.8 Suppose that L is simple and λ a nondegenerate associative form of L. Then SkDerF (L)/adL is color isomorphic to H 2 (L, F ).

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Proof Define a linear mapping σ : SkDerF (L)/adL → H 2 (L, F ) D + adL → f + B 2 (L, F ), where f (x, y) = λ(D(x), y). As λ is color symmetric and D is skew, f ∈ C 2 (L, f ). With the associativity of λ and the definition of skew derivation, we have df = 0. Hence f ∈ Z 2 (L, F ). If D = adx, x ∈ L, then f = d(−D(x)), D(x) ∈ C 1 (L, F ). It follows that the definition of σ is valid. Assume that f ∈ B 2 (L, F ), then there is g ∈ C 1 (L, F ) = L∗ such that f = dg. Applying Lemma 3.7, there is x ∈ L such that ϕ(x) = g. Let y, z be elements of L. Then we obtain λ(D(y), z) = f (y, z) = dg(y, z) = −g([y, z]) = −ϕ(x)([y, z]) = −λ(x, [y, z]) = −λ([x, y], z). Since λ is a nondegenerate associative form, we obtain D(y) = −[x, y] = ad(−x)(y), i.e., D ∈ adL. Consequently, σ is injective. Applying Lemma 3.7 and Proposition 3.6, we can prove σ is surjective. Let σ(D + adL) = f + B 2 (L, F ) and σ(x.D + adL) = g + B 2 (L, F ), where x ∈ L. Then g(y, z) = λ((x.D)(y), z) = λ(x.D(y) − ε(x, D)D([x, y]), z), and ε(x, λ)(x.f )(y, z) = −ε(x, f + λ)f ([x, y], z) − ε(x, λ + f + y)f (y, [x, z]) = −ε(x, D)λ(D([x, y]), z) − ε(x, D + y)λ(D(y), [x, z]) = −ε(x, D)λ(D([x, y]), z) + λ(x.D(y), z). Consequently, σ(x.D + adL) = ε(x, λ)(x.f ) + B 2 (L, F ) = ε(σ, x)x.σ(D + adL), i.e., σ is a color-isomorphism.

4

The Double Extension of Lie Color Algebras

In this subsection, we consider finite dimensional Lie color algebras over a field F of characteristic 0. Definition 4.1 Let L be a Lie color algebra. A bilinear form B on L is said to be color symmetric if B(x, y) = ε(x, y)B(y, x), ∀x, y ∈ L. B is said to be color invariant if B([x, y], z) = B(x, [y, z]), ∀x, y, z ∈ L. Definition 4.2 Let L be a Lie color algebra with a bilinear form B. (L, B) is called quadratic if B is color symmetric, non-degenerate, and color invariant. In this case, B is called an invariant scalar product on L. Definition 4.3 Let (L, B) be a quadratic Lie color algebra. A color ideal I of L is called non-degenerate (resp. degenerate) if the restriction of B to I is a non-degenerate (resp. degenerate) bilinear form. Definition 4.4 We say that a quadratic Lie color algebra (L, B) is B-irreducible if L does not contain any nontrivial and non-degenerate color ideal.

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Definition 4.5 Let L be a Lie color algebra and B a bilinear form on L, and let D be a homogenous color derivation of L. D is called color skew-symmetric if B(D(x), y) = −ε(D, x)B(x, D(y)), for all x, y ∈ L. Definition 4.6 A bilinear form w of L × L in F is said to be a 2-cocycle on L if ε(z, x)w(x, [y, z]) + ε(x, y)w(y, [z, x]) + ε(y, z)w(z, [x, y]) = 0, for all x, y, z ∈ L. The G-gradation of L × L induces a G-gradation on Z 2 (L, F ), which is the set of all 2-cocycle on L. Lemma 4.1 Let (L, B) be a quadratic Lie color algebra. 1 If D is a homogeneous color skew-symmetric derivation of L, then w defined by w(x, y) := B(D(x), y),

∀x, y ∈ L

is a homogeneous color skew-symmetric 2-cocycle on L. 2 Conversely, if w is a homogeneous color skew-symmetric 2-cocycle on L, there exists a unique homogeneous color skew-symmetric derivation D of L such that w(x, y) = B(D(x), y), ∀x, y ∈ L. Proof 1 Let D be a homogeneous color skew-symmetric derivation of L. Let x, y and z be three homogeneous elements of L, respectively. Then w(z, [x, y]) = B(D(z), [x, y]) = −ε(D, z)B(z, D([x, y])) = −ε(D, z)B(z, [D(x), y]) − ε(D, z)ε(D, x)B(z, [x, D(y)]) = ε(D, z)ε(D + x, y)B([z, y], D(x)) − ε(D, z)ε(D, x)B([z, x], D(y)) = ε(z, x)B(D(x), [z, y]) − ε(z, y)ε(x, y)B(D(y), [z, x]) = ε(z, x)w(x, [z, y]) − ε(z, y)ε(x, y)w(y, [z, x]). Thus, ε(z, x)w(x, [y, z]) + ε(x, y)w(y, [z, x]) + ε(y, z)w(z, [x, y]) = 0. 2 Conversely, we consider the map ϕ : L −→ L∗ defined by ϕ(x)(y) = B(x, y), ∀x, y ∈ L. Let w be a homogeneous color skew-symmetric 2-cocycle on L such that wx (. , y) = w(x, y), ∀y ∈ L. By virtue of Lemma 3.7, there exists a unique z in L such that wx = ϕ(z). Then w(x, y) = B(z, y),

∀y ∈ L.

Put D(x) = z. Then D is a well-defined map by the uniqueness of z. Clearly D is linear and homogeneous. Because w is color skew-symmetric, D is also color skew-symmetric. Now, we have to show that D is a color derivation. Let x, y and z be three homogeneous elements in L. We write that w is a 2-cocycle: w(x, [y, z]) = w([x, y], z) + ε(x, y)w(y, [x, z]),

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which is equivalent to B(D(x), [y, z]) = B(D([x, y]), z) + ε(x, y)B(D(y), [x, z]). Because B is non-degenerate, this equality implies that D([x, y]) = [D(x), y] − ε(x, y)[D(y), x] = [D(x), y] + ε(D, x)[x, D(y)]. Proposition 4.2 Let (L1 , B1 ) be a quadratic Lie color algebra. Suppose that L2 is a Lie color algebra and ψ : L2 −→SkDer(L1 ) ⊂ Der(L1 ) is a homomorphism of Lie color algebra. Let ϕ : L1 × L1 −→ L∗2 be the map defined by ϕ(x, y)(z) := ε(x + y, z)B1 (ψ(z)(x), y),

∀x, y ∈ L1 , z ∈ L2 .

Then 1 the vector space L1 ⊕ L∗2 with the product [x + f, y + g] = [x, y]1 + ϕ(x, y),

x, y ∈ L1 , f, g ∈ L∗2

is a Lie color algebra. It is a central extension of L1 by L∗2 . 2 The homomorphism ψ can be extended to a homomorphism of Lie color algebra ψ˜ : L2 → Der(L1 ⊕ L∗2 ) defined by ˜ ψ(z)(x + f ) = ψ(z)(x) + π(z)(f ),

x ∈ L1 , z ∈ L2 , f ∈ L∗2 ,

where π is the coadjoint representation of L2 . Proof 1 We show that L1 ⊕ L∗2 with the product is a Lie color algebra. We choose the gradation (L1 ⊕ L∗2 )α = (L1 )α ⊕ (L∗2 )α , α ∈ G. Let x + f be in (L1 ⊕ L∗2 )x and y + g be in (L1 ⊕ L∗2 )y . An easy computation shows that ϕ(x, y) = −ε(x, y)ϕ(y, x), and then [x + f, y + g] = −ε(x, y)[y + g, x + f ]. Let z + h be in (L1 ⊕ L∗2 )z , then ε(z, x)[x + f, [y + g, z + h]] + ε(x, y)[y + g, [z + h, x + f ]] + ε(y, z)[z + h, [x + f, y + g]] = ε(z, x)[x, [y, z]1 ]1 + ε(z, x)ϕ(x, [y, z]1 ) + ε(x, y)[y, [z, x]1 ]1 +ε(x, y)ϕ(y, [z, x]1 ) + ε(y, z)[z, [x, y]1 ]1 + ε(y, z)ϕ(z, [x, y]1 ) = ε(z, x)ϕ(x, [y, z]1 ) + ε(x, y)ϕ(y, [z, x]1 ) + ε(y, z)ϕ(z, [x, y]1 ). Suppose that T is a homogeneous element in L2 . We consider map WT : L1 × L1 −→ F defined by WT (x, y) = B1 (ψ(T )(x), y) = ε(T, x + y)ϕ(x, y)(T ). By Lemma 4.1 , WT is a 2-cocycle, which follows the Jacobi identity. So the vector space L1 ⊕ L∗2 with the product is a Lie color algebra.

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Moreover, L1 ⊕ L∗2 is a central extension of L1 by L∗2 , because one has λ

μ

0 −→ L∗2 −→ L1 ⊕ L∗2 −→ L1 −→ 0, where μ(x + f ) = x, for x ∈ L1 and f ∈ L∗2 . Then L∗2 = Kerμ, and λ is an injective homomorphism. 2 Let z be an element of L2 . Then ψ(z) is a color skew-symmetric derivation of L1 . We extend ψ(z) for the derivation of L1 ⊕ L∗2 by  + f ) := ψ(z)(x) + π(z)(f ), ψ(z)(x  is a color derivation. If x + f where π is the coadjoint representation of L2 . We first proof ψ(z) ∗ and y + g are two homogeneous elements in L1 ⊕ L2 , then   ψ(z)([x + f, y + g]) = ψ(z)([x, y]1 + ϕ(x, y)) = ψ(z)[x, y]1 + π(z)ϕ(x, y) = [ψ(z)(x), y]1 + ε(z, x)[x, ψ(z)(y)]1 + π(z)ϕ(x, y). On the other hand  + f ), y + g] = [ψ(z)(x) + π(z)(f ), y + g] = [ψ(z)(x), y]1 + ϕ(ψ(z)(x), y) [ψ(z)(x  + g)] = [x + f, ψ(z)(y) + π(z)(g)] = [x, ψ(z)(y)]1 + ϕ(x, ψ(z)(y)). [x + f, ψ(z)(y Thus,   + f ), y + g] − ε(z, x)[x + f, ψ(z)(y  + g)] ψ(z)([x + f, y + g]) − [ψ(z)(x = ε(z, x)[x, ψ(z)(y)]1 + π(z)ϕ(x, y) − ϕ(ψ(z)(x), y) −ε(z, x)[x, ψ(z)(y)]1 − ε(z, x)ϕ(x, ψ(z)(y)) = π(z)ϕ(x, y) − ϕ(ψ(z)(x), y) − ε(z, x)ϕ(x, ψ(z)(y)). By an easy computation one can show that π(z)ϕ(x, y) = ϕ(ψ(z)(x), y) + ε(z, x)ϕ(x, ψ(z)(y)).  is a color derivation of L1 ⊕ L∗ . We define ψ by So ψ(z) 2  = ψ(z). ψ : L2 −→ Der(L1 ⊕ L∗2 ), ψ(z) Then ψ is a homomorphism of Lie color algebra . Proposition 4.3 The hypothesis are the same as in the Proposition 4.2. The vector space L = L2 ⊕ L1 ⊕ L∗2 , with the product [x2 + x1 + f, y2 + y1 + g] : = [x2 , y2 ]2 + [x1 , y1 ]1 + ψ(x2 )(y1 ) − ε(x1 , y1 )ψ(y2 )(x1 ) +π(x2 )(g) − ε(x, y)π(y2 )(f ) + ϕ(x1 , y1 ), where x2 + x1 + f (resp. y2 + y1 + g ) is homogeneous of degree x (resp. y) in L, is a Lie color algebra.

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Proof By Proposition 4.2, ψ is a homomorphism of Lie color algebras. Then L with this product is a Lie color algebra, which is a semi-direct product of L2 and L1 ⊕ L∗2 with respect to ψ (see [3]). Proposition 4.4 Let L be the Lie color algebra of Proposition 4.3, and let γ be a color symmetric invariant bilinear form on L2 . Then the bilinear form B defined on L by B(x2 + x1 + f, y2 + y1 + g) := B1 (x1 , y1 ) + γ(x2 , y2 ) + f (y2 ) + ε(x, y)g(x2 ), where x2 + x1 + f (resp. y2 + y1 + g) is homogeneous of degree x (resp. y) in L, is an invariant scalar product on L. Proof It is obvious that B is color symmetric. To show that B is non-degenerate, take homogeneous element x2 + x1 + f in L⊥ , then 0 = B(x2 + x1 + f, y1 ) = B1 (x1 , y1 ), for all y1 in L1 , so x1 = 0, then 0 = B(x2 + f, g) = ε(x, y)g(x2 ), for all g in L∗2 , so x2 = 0, then 0 = B(f, y2 ) = f (y2 ), for all y2 in L2 , so f = 0. It remains to show that B is color invariant, and let x = x2 + x1 + f, y = y2 + y1 + g and z = z2 + z1 + h be three homogeneous elements in L. Let x, y, z denote their respective degrees. Then B([x, y], z) = B1 ([x1 , y1 ], z1 ) + B1 (ψ(z2 )(y1 ), z1 ) − ε(x, y)B1 (ψ(y2 )(x1 ), z1 ) +γ([x2 , y2 ], z2 ) + π(x2 )(g)(z2 ) − ε(x, y)π(y2 )(f )(z2 ) +ϕ(x1 , y1 )(z2 ) + ε(x + y, z)h([x2 , y2 ]). On the other hand B(y, [x, z]) = B1 (y1 , [x1 , z1 ]) + B1 (y1 , ψ(x2 )(z1 )) − ε(x, z)B1 (y1 , ψ(z2 )(x1 )) γ(y2 , [x2 , z2 ]) + B1 (ψ(y2 )(x1 ), z1 ) + g([x2 , y2 ]) +ε(x, y)ε(x + y, z)h([x2 , y2 ]) − ε(x, y)f ([y2 , z2 ]). Now, because B1 is color invariant, we deduce that B is color invariant. Hence, (L, B) is a quadratic Lie color algebra. Proposition 4.5 Let L be a Lie color algebra, and DerL a derivation algebra. Then the vector space DerL ⊕ L with the product [D1 + x1 , D2 + x2 ] := [D1 , D2 ] + [x1 , x2 ] + D1 (x2 ) − ε(x1 , D2 )D2 (x1 ) is a Lie color algebra. It is an extension of DerL by L and is called the holomorph of L. Proof We choose the gradation: (DerL ⊕ L)α = (DerL)α ⊕ Lα , α ∈ G. Let D1 + x1 be in (DerL ⊕ L)x1 and D2 + x2 be in (DerL ⊕ L)x2 . An easy computation shows that [D1 + x1 , D2 + x2 ] = −ε(x1 , x2 )[D2 + x2 , D1 + x1 ].

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Let D3 + x3 be in (DerL ⊕ L)x3 . Then ε(x3 , x1 )[D1 + x1 , [D2 + x2 , D3 + x3 ]] + ε(x1 , x2 )[D2 + x2 , [D3 + x3 , D1 + x1 ]] +ε(x2 , x3 )[D3 + x3 , [D1 + x1 , D2 + x2 ]] = ε(x3 , x1 )([D1 , [D2 , D3 ]] + [x1 , [x2 , x3 ]] + [x1 , D2 (x3 )] − ε(x2 , x3 )[x1 , D3 (x2 )] +D1 ([x2 , x3 ]) + D1 D2 (x3 ) − ε(x2 , x3 )D1 D3 (x2 ) − ε(x1 , x2 + x3 )[D2 , D3 ](x1 )) +ε(x1 , x2 )([D2 , [D3 , D1 ]] + [x2 , [x3 , x1 ]] + [x2 , D3 (x1 )] − ε(x3 , x1 )[x2 , D1 (x3 ) +D2 ([x3 , x1 ]) + D2 D3 (x1 ) − ε(x3 , x1 )D2 D1 (x3 ) − ε(x2 , x3 + x1 )[D3 , D1 ](x2 )) +ε(x2 , x3 )([D3 , [D1 , D2 ]] + [x3 , [x1 , x2 ]] + [x3 , D1 (x2 )] − ε(x1 , x2 )[x3 , D2 (x1 )] +D3 ([x1 , x2 ] + D3 D1 (x2 ) − ε(x1 , x2 )D3 D2 (x1 ) − ε(x3 , x1 + x2 )[D1 , D2 ](x3 )) = 0. Thus, DerL ⊕ L is a Lie color algebra. Obviously, there exists an isomorphism between L and color ideal {0 + x | x ∈ L} of DerL ⊕ L. There also exists another isomorphism between DerL and color subalgebra {D + 0 | D ∈ DerL} of DerL ⊕ L. Then DerL ⊕ L is an extension of DerL by L. In this subsection we suppose that (L, B) is a B-irreducible quadratic Lie color algebra, that J is a maximal color ideal of L such that L = J ⊕ V, where V is a Lie sub-color algebra of L. Lemma 4.6 If (L, B) is a quadratic Lie color algebra and I is a color ideal of L. Then ¯ defined by the Lie color algebra (I + I ⊥ )/I ∩ I ⊥ , with the bilinear form B ¯ x, y¯) = B(x, y), B(¯

x, y ∈ I + I ⊥

is quadratic. ¯ is well defined. We show that B ¯ is non-degenerate. Let x Proof B ¯ be in (I + I ⊥ )/I ∩ I ⊥ such that ¯ x, y¯) = 0, ∀¯ B(¯ y ∈ (I + I ⊥ )/I ∩ I ⊥ . Let x = j + j  be a representative element of x¯, where j ∈ I, j  ∈ I ⊥ . Then ¯ x, α B(¯ ¯ ) = 0,

α ∈ I.

¯ x, α But B(¯ ¯ ) = B(x, α) = B(j + j  , α) = B(j, α) = 0. So j ∈ I ∩ I ⊥ . We then deduce that ⊥ x ∈ I ∩ I and x¯ = 0. Proposition 4.7 If A = J ⊥ ⊕ V , then B restricted to A is non-degenerate and L = J ⊥ ⊕ A⊥ ⊕ V. Proof If J is maximal, then J ⊥ ⊂ J is a minimal color ideal of L. Since L is B-irreducible, we deduce that the restrictions of B to J and J ⊥ are non-degenerate. Now, we have to show that B restricted to A is non-degenerate. Let a = j + v ∈ A ∩ A⊥ , where j ∈ J ⊥ , v ∈ V. For all b = j  + v  in A, we have B(j + v, j  + v  ) = B(j, v  ) + B(v, j  ) + B(v, v  ).

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Take b = j  in J ⊥ , then B(j + v, j  ) = B(v, j  ) = 0. We deduce that v = 0 as element of J ∩ V . On the other hand B(a, j) ∈ B(J, j) = 0, since j ∈ J ⊥ . So B(j, J ⊕ V ) = 0. But B is non-degenerate, then j = 0. We deduce that L = A ⊕ A⊥ . Now, J ⊥ ⊂ J and A⊥ ⊂ J, then J ⊥ ⊕ A⊥ ⊂ J. But dim(J ⊥ ⊕ A⊥ ) = dimL − dimV = dimJ. So L = J ⊥ ⊕ A⊥ ⊕ V, and the restrictions of B to A⊥ and J ⊥ × V are non-degenerate. The restrictions of B to ¯ is a J ⊥ × A⊥ and A⊥ × V are identically zero. Denote J/J ⊥ = W and by Lemma 4.6, (W, B) quadratic color algebra. Proposition 4.8 On A⊥ there exists a structure of quadratic Lie color algebra, which ¯ is an isomorphic to (W, B). Proof If v is homogenous in V and x is homogeneous in A⊥ ⊆ J, then, [v, x] = j + a ∈ J = J ⊥ ⊕ A⊥ , where j ∈ J ⊥ and a ∈ A⊥ . Let v  in V , then B(j, v  ) = B([v, x] − a, v  ) = B([v, x], v  ) = −ε(v, x)B(x, [v, v  ]) = 0. So j ∈ V ⊥ and then j ∈ L⊥ = {0}. Thus, [V, A⊥ ] ⊂ A⊥ . Let x, y in A⊥ . Then [x, y] = α(x, y) + β(x, y), where α(x, y) ∈ J ⊥ and β(x, y) ∈ A⊥ . Now, it is easy to see that β is a structure of Lie color algebra on A⊥ which is denoted by [, ]A⊥ The restriction B  of B to A⊥ is A⊥ color invariant. Thus B  is a scalar product on A⊥ . Considering the linear map θ of A⊥ to W defined by θ(x) = x ¯,

∀x ∈ A⊥ ,

it is easy to verify that θ is an isomorphism. Moreover, θ is an isomorphism of Lie color algebras. If x and y are in A⊥ , then x, y¯]W = [θ(x), θ(y)]W . θ([x, y]A⊥ ) = β(x, y) = α(x, y) + β(x, y) = [x, y] = [¯ Proposition 4.9 The quadratic Lie color algebra L = J ⊥ ⊕ A⊥ ⊕ V is isomorphic to the ¯ by V . double extension of (W, B) Proof We consider the map υ of J ⊥ to V ∗ defined by υ : J ⊥ −→ V ∗ , x −→ υ(x) = B(x, .). Since B is non-degenerate, υ is one to one. It is the same as the map δ : V −→ (J ⊥ )∗ , x −→ δ(x) = B(x, .). So dimJ ⊥ = dimV ∗ and υ is an isomorphism of G-graded vector spaces. Now on J ⊥ and V ∗ there are structures of V -modules defined by

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1 v.j = [v, j], v ∈ V, j ∈ J ⊥ , 2 (v.f )(v  ) = −ε(v, f )f ([v, v  ]), v, v  ∈ V, f ∈ V ∗ . Let ψ be the linear map from V to DerW defined by ψ(v)(¯ x) = [v, x], v ∈ V, x ¯ ∈ W. (∗ ∗ ∗) Then ψ is a homomorphism of Lie color algebras. Now, if ϕ is the map from W × W to V ∗ defined by ¯ ϕ(¯ x, y¯)(v) = ε(x + y, v)B(ψ(v)(¯ x), y¯), x ¯, y¯ ∈ W, v ∈ V, ¯ by then, by Proposition 4.8, the Lie color algebra (V ⊕ W ⊕ V ∗ ) is a double extension of (W, B) V by means of the homomorphism ψ, where T is an invariant scalar product on V ⊕ W ⊕ V ∗ . Let τ be the map from L to V ⊕ W ⊕ V ∗ defined by τ (j + a + v) = v + θ(a) + υ(j), where j ∈ J ⊥ , a ∈ A⊥ and v ∈ V . Since θ and υ are bijective, τ is a bijection from L to V ⊕ W ⊕ V ∗ . It remains to show that τ is an isomorphism of Lie color algebras. Let x = j + a + v and x = j  + a + v  be homogeneous elements in V ⊕ W ⊕ V ∗ with degrees x and x respectively. Since J ⊥ is a minimal color ideal, [J, J ⊥ ] = {0}. On the other hand [J ⊥ , A⊥ ] = {0}. So, [x, x ] = [j, v  ] + [v, j  ] + α(a, a ) + β(a, a ) + [a, v  ] + [v, a ] + [v, v  ]. Now, by the definition of the map υ and the structure of V -module V ∗ . We have υ([j, v  ]) = B([j, v  ], .) = −ε(x, x )v  .υ(j). If v ∈ V , then υ(α(a, a ))(v) = B(α(a, a ), v) = B(a, [a , v]) ¯ = ε(x + x , x)B([v, a], a ) = ε(x + x , x)B(ψ(v)(¯ a), a ¯ ) = ϕ(¯ a, a ¯ )(v). We then deduce that τ ([x, x ]) = v.υ(j) − ε(x, x )v  .υ(j) + ϕ(¯ a, a¯ ) +θ([a, a ]A⊥ ) + θ([a, v  ]) + θ([v, a ]) + [v, v  ] = v.υ(j  ) − ε(x, x )v  .υ(j) + ϕ(¯ a, a¯ ) +[¯ a, a¯ ] + [a, v  ] + [v, a ] + [v, v  ] W

= [v, v  ] + [¯ a, a¯ ]W + ψ(v)(a¯ ) − ε(x, x )ψ(v  )(¯ a)     ¯ +v.υ(j ) − ε(x, x )v .υ(j) + ϕ(¯ a, a ) = [τ (x), τ (x )]. Next, B(x, x ) = B(j, v  ) + B(a, a ) + B(v, j  ) + B(v, v  ) ¯ a, a¯ ) + B(v, v  ) + υ(j)(v  ) + ε(x, x )υ(j  )(v) = B(¯ = T  (τ (x), τ (x )),

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where T  is the scalar product defined on V ⊕ W ⊕ V ∗ by ¯ T  (v + w + f, V  + w + f  ) = B(w, w ) + B(v, v  ) + f (v  ) + ε(v, v  )f  (v). ¯ by V by means of ψ defined by (***). Thus, we have shown that L is double extension of (W, B) References 1 Scheunert M. Generalized Lie algebras. J Math Phys, 1979, 20: 712–720 2 Scheunert M, Zhang R. Cohomology of Lie superalgebras and their generalizations. J Math Phys, 1998, 39: 5024–5061 3 Scheunert M. Theory of Lie superalgebras. Berlin, Heidelbeg, New York: Springer, 1979. 270 4 Wilson M. Delta methods in enveloping algebras of Lie colour algebras. J Algebra, 1995, 175: 661–696 5 Price K. Primeness criteria for universal enveloping algebras of Lie color algebras. J Algebra, 2001, 235: 589–607 6 Passman D. Simple Lie color algebras of Witt type. J Algebra, 1998, 208: 698–721 7 Su Y, Zhao K, Zhu L. Simple Lie color algebras of Weyl type. Israel J Math, 2003, 137: 109–123 8 Feldvoss J. Representations of Lie colour algebras. Adv Math, 2001, 157: 95–137 9 Farnsteiner R. Central extensions and invariant forms of greaded Lie algebras. Algebras, Groups and Geometries, 1986, 3: 431–455 10 Farnsteiner R. Derivations and central extensions of finitely generated graded Lie algebras. J Algebra, 1988, 118: 33–45 11 Medina A, Revoy P. Algebres de Lie et produit scalaire invariant. Ann Scient Ec Norm Sup 4e serie, 1985, 18: 553–561 12 Benamor H, Benayadi S. Double extension of quadratic Lie superalgebras. Commun in Algebra, 1999, 27: 67–88 13 Gould M, Zhang R, Bracken A. Lie bi-superalgebras and graded classical Yang-Baxter equation. Reviews in Mathematical Physics, 1999, 3: 223–240 14 Leites D. Cohomology of Lie superalgebras. Funct Anal Appl, 1975, 9: 75–79 15 Tripathy K., Patra M. Cohomology theory and deformations of Z2 -gradalgebras. J Math Phys, 1990, 31: 2822–2831 16 Humphreys J. Introduction to Lie Algebras and Representation Theory. New York: Springer-Verlag, 1972. 1–293 17 Jacobson N. Lie Algebras. New York: Dover Publ, 1979 18 Zhang Q C, Zhang Y Z. Derivation algebras of the modular Lie supperalgebras W and S of Cartan type. Acta Mathematica Scientia, 2000, 20B(1): 137–144