Constructions and Cohomology of color Hom-Lie algebras
aa r X i v : . [ m a t h . R A ] J u l CONSTRUCTIONS AND COHOMOLOGY OF COLOR HOM-LIE ALGEBRAS
K. ABDAOUI, F. AMMAR AND A. MAKHLOUF
Abstract.
The main purpose of this paper is to define representations and a cohomology of color Hom-Lie algebras and to study some key constructions and properties. We describe Hartwig-Larsson-SilvestrovTheorem in the case of Γ-graded algebras, study one-parameter formal deformations, discuss α k -generalizedderivation and provide examples. Introduction
Color Hom-Lie algebras are the natural generalizations of Hom-Lie algebras and Hom-Lie superalgebras. Inrecent years, they have become an interesting subject of mathematics and physics. A cohomology of colorLie algebras were introduced and investigated, see [26, 28], and the representations of color Lie algebraswere explicitly described in [4]. As is well known, derivations and extensions of Hom-Lie algebras, Hom-Liesuperalgebras and color Hom-Lie algebras are very important subjects. Color Hom-Lie algebras were studiedin [34]. In the particular case of Hom-Lie superalgebras, cohomology theory of was provided in [3].This paper focusses on Γ-graded Hom-algebras with Γ is an abelian group. Mainly, we prove a Γ-gradedversion of a Hartwig-Larsson-Silvestrov Theorem and we study representations and cohomology of colorHom-Lie algebras.The paper is organized as follows. In section 1, we recall definitions and some key constructions of colorHom-Lie algebras and we provide a list of twists of color Hom-Lie algebras. Section 2 is dedicated to describeand prove the Γ-graded version Hartwig-Larsson-Silvestrov Theorem, which was proved for Hom-Lie algebrasin [15, Theorem 5] and for Hom-Lie superalgebras in [5, Theorem 4 . sl c , [ ., . ] α , ε, α ). In section 4, we study formal deformationsof color Hom-Lie algebras. In last section, we study the homogeneous α k -generalized derivations and the α k -centroid of color Hom-Lie algebras and we give some properties generalizing the homogeneous generalizedderivations discussed in [8]. Moreover in Proposition 5 . α -derivation of color Hom-Liealgebras gives rise to a color Hom-Jordan algebras.1. Definitions, Constructions and Examples
In the following we summarize definitions of color Hom-Lie and color Hom-associative algebraic structures(see [34]) generalizing the well known color Lie and color associative algebras.Throughout the article we let K be an algebraically closed field of characteristic 0 and K ∗ = K \{ } be thegroup of units of K . • Let Γ be an abelian group. A vector space V is said to be Γ-graded, if there is a family ( V γ ) γ ∈ Γ of vectorsubspace of V such that V = M γ ∈ Γ V γ . An element x ∈ V is said to be homogeneous of the degree γ ∈ Γ if x ∈ V γ , γ ∈ Γ, and in this case, γ iscalled the color of x . As usual, denote by x the color of an element x ∈ V . Thus each homogeneous element x ∈ V determines a unique group of element x ∈ Γ by x ∈ V x . Fortunately, We can almost drop the symbol” − ”, since confusion rarely occurs. In the sequel, we will denote by H ( V ) the set of all the homogeneouselements of V .Let V = L γ ∈ Γ V γ and V ′ = L γ ∈ Γ V ′ γ be two Γ-graded vector spaces. A linear mapping f : V −→ V ′ is saidto be homogeneous of the degree υ ∈ Γ if f ( V γ ) ⊆ V ′ γ + υ , ∀ γ ∈ Γ . if in addition, f is homogeneous of degree zero, i.e. f ( V γ ) ⊆ V ′ γ holds for any γ ∈ Γ, then f is said to beeven. • An algebra A is said to be Γ-graded if its underlying vector space is Γ-graded, i.e. A = L γ ∈ Γ A γ , andif, furthermore A γ A γ ′ ⊆ A γ + γ ′ , for all γ, γ ′ ∈ Γ. It is easy to see that if A has a unit element e , it follows e ∈ A . A subalgebra of A is said to be graded if its graded as a subspace of A .Let A ′ be another Γ-graded algebra. A homomorphism f : A −→ A ′ of Γ-graded algebras is by definition ahomomorphism of the algebra A into the algebra A ′ , which is, in addition an even mapping. Definition 1.1.
Let K be a field and Γ be an abelian group. A map ε : Γ × Γ → K ∗ is called a skew-symmetricbi-character on Γ if the following identities hold , for all a, b, c in Γ(1) ε ( a, b ) ε ( b, a ) = 1 , (2) ε ( a, b + c ) = ε ( a, b ) ε ( a, c ) , (3) ε ( a + b, c ) = ε ( a, c ) ε ( b, c ) . The definition above implies, in particular, the following relations ε ( a,
0) = ε (0 , a ) = 1 , ε ( a, a ) = ± , for all a ∈ Γ . If x and x ′ are two homogeneous elements of degree γ and γ ′ respectively and ε is a skew-symmetric bi-character, then we shorten the notation by writing ε ( x, x ′ ) instead of ε ( γ, γ ′ ). Definition 1.2. [34]
A color Hom-Lie algebra is a quadruple ( A , [ ., . ] , ε, α ) consisting of a Γ -graded vectorspace A , a bi-character ε , an even bilinear mapping [ ., . ] : A × A → A ( i.e. [ A a , A b ] ⊆ A a + b for all a, b ∈ Γ) and an even homomorphism α : A → A such that for homogeneous elements we have [ x, y ] = − ε ( x, y )[ y, x ] ( ε -skew symmetric ) . (cid:9) x,y,z ε ( z, x )[ α ( x ) , [ y, z ]] = 0 ( ε -Hom-Jacobi condition ) . where (cid:9) x,y,z denotes summation over the cyclic permutation on x, y, z . In particular, if α is a morphism of color Lie algebras (i.e. α ◦ [ ., . ] = [ ., . ] ◦ α ⊗ ), then we call ( A , [ ., . ] , ε, α )a multiplicative color Hom-Lie algebra.Observe that when α = Id , the ε -Hom-Jacobi condition (1 .
2) reduces to the usual ε -Jacobi condition (cid:9) x,y,z ε ( z, x )[ x, [ y, z ]] = 0for all x, y, z ∈ H ( A ) . Example 1.1.
A color Lie algebra ( A , [ ., . ] , ε ) is a color Hom-Lie algebra with α = Id , since the ε -Hom-Jacobi condition reduces to the ε -Jacobi condition when α = Id . Definition 1.3.
Let ( A , [ ., . ] , ε, α ) be a color Hom-Lie algebra. It is called (1) multiplicative color Hom-Lie algebra if α [ x, y ] = [ α ( x ) , α ( y )] . (2) regular color Hom-Lie algebra if α is an automorphism. (3) involutive color Hom-Lie algebra if α is an involution, that is α = Id . Let ( A , [ ., . ] , ε, α ) be a regular color Hom-Lie algebra. It was observed in [9] that the composition methodusing α − leads to a color Lie algebra. Proposition 1.1.
Let ( A , [ ., . ] , ε, α ) be a regular color Hom-Lie algebra. Then ( A , [ ., . ] α − = α − ◦ [ ., . ] , ε ) isa color Lie algebra.Proof. Note that [ ., . ] α − is ε -skew-symmetric because [ ., . ] is ε -skew-symmetric and α − is linear.For x, y, z ∈ H ( A ), we have: (cid:9) x,y,z ε ( z, x )[ x, [ y, z ] α − ] α − = (cid:9) x,y,z ε ( z, x ) α − [ x, [ y, z ] α − ]= α − ( (cid:9) x,y,z ε ( z, x )[ α ( x ) , [ y, z ]])= 0 . (cid:3) Remark 1.1.
In particular the proposition is valid when α is an involution. The following theorem generalizethe result of [34] . In the following starting from a color Hom-Lie algebra and an even color Lie algebraendomorphism, we construct a new color Hom-Lie algebra. ONSTRUCTIONS AND COHOMOLOGY OF COLOR HOM-LIE ALGEBRAS 3
We recall in the following the definition of Hom-associative color algebra which provide a different wayfor constructing color Hom-Lie algebra by extending the fundamental construction of color Lie algebras fromassociative color algebra via commutator bracket multiplication. This structure was introduced by [34] .
Definition 1.4. [34]
A color Hom-associative algebra is a triple ( A , µ, α ) consisting of a Γ -graded linearspace A , an even bilinear map µ : A×A → A ( i.e µ ( A a , A b ) ⊂ A a + b ) and an even homomorphism α : A → A such that (1.1) µ ( α ( x ) , µ ( y, z )) = µ ( µ ( x, y ) , α ( z )) . In the case where µ ( x, y ) = ε ( x, y ) µ ( y, x ) , we call the Hom-associative color algebra ( A , µ, α ) a commutativeHom-associative color algebra. Remark 1.2.
We recover classical associative color algebra when α = Id A and the condition (1 . is theassociative condition in this case. Proposition 1.2. [34]
Let ( A , µ, α ) be a Hom-associative color algebra defined on the linear space A bythe multiplication µ and an even homomorphism α . Then the quadruple ( A , [ ., . ] , ε, α ) , where the bracket isdefined for x, y ∈ H ( A ) by [ x, y ] = µ ( x, y ) − ε ( x, y ) µ ( y, x ) is a color Hom-Lie algebra. Theorem 1.1.
Let ( A , [ ., . ] , ε, α ) be a color Hom-Lie algebra and β : A −→ A be an even color Lie algebraendomorphism. Then ( A , [ ., . ] β , ε, β ◦ α ) , where [ x, y ] β = β ◦ [ x, y ] , is a color Hom-Lie algebra.Moreover, suppose that ( A ′ , [ ., . ] ′ , ε ) is a color Lie algebra and α ′ : A ′ −→ A ′ is color Lie algebra endomor-phism. If f : A −→ A ′ is a Li color algebra morphism that satisfies f ◦ β = α ′ ◦ f then f : ( A , [ ., . ] β , ε, β ◦ α ) −→ ( A ′ , [ ., . ] ′ , ε, α ′ ) is a morphism of color Hom-Lie algebras.Proof. Obviously [ ., . ] β is ε -skew-symmetric and we show that ( A , [ ., . ] β , ε, β ◦ α ) satisfies the ε -Hom-Jacobicondition 1 .
2. Indeed (cid:9) x,y,z ε ( z, x )[ β ◦ α ( x ) , [ y, z ] β ] β = (cid:9) x,y,z ε ( z, x )[ β ◦ α ( x ) , β ([ y, z ])] β = β ( (cid:9) x,y,z ε ( z, x )[ α ( x ) , [ y, z ]])= 0 . The second assertion follows from f ([ x, y ] β ) = f ([ β ( x ) , β ( y )]) = [ f ◦ β ( x ) , f ◦ β ( y )] ′ = [ α ′ ◦ f ( x ) , α ′ ◦ f ( y )] ′ = [ f ( x ) , f ( y )] ′ α ′ . (cid:3) Example 1.2.
Let ( A , [ ., . ] , ε ) be a color Lie algebra and α be a color Lie algebra morphism, then ( A , [ ., . ] α = α ◦ [ ., . ] , ε, α ) is a multiplicative color Hom-Lie algebra. Definition 1.5.
Let ( A , [ ., . ] , ε, α ) be a multiplicative color Hom-Lie algebra and n ≥ . Define the nthderived Hom-algebra of A by A ( n ) = ( A , [ ., . ] ( n ) = α n − ◦ [ ., . ] , ε, α n ) . Note that A (0) = A , A (1) = ( A , [ ., . ] (1) = α ◦ [ ., . ] , ε, α ) , and A ( n +1) = ( A n ) . Corollary 1.1.
Let ( A , [ ., . ] , ε, α ) be a color Hom-Lie algebra. Then the nth derived Hom-algebra of AA ( n ) = ( A , [ ., . ] ( n ) = α n − ◦ [ ., . ] , ε, α n ) is also a color Hom-Lie algebra for each n ≥ . K. ABDAOUI, F. AMMAR AND A. MAKHLOUF
Examples of twists of color Hom-Lie algebras.
In this section we provide examples of color Hom-Lie algebras. We use the classification of color Lie algebra provided in [23] and the twisting principle.In Examples 1 . .
4, Γ is the group Z , and ε : Γ × Γ −→ C is defined by the matrix[ ε ( i, j )] = − − − − − − . Elements of this matrix specify in the natural way values of ε on the set { (1 , , , (1 , , , (0 , , } ×{ (1 , , , (1 , , , (0 , , } ⊂ Z × Z (The elements (1 , , , (1 , , , (0 , ,
1) are ordered and numberedby the numbers 1 , , ε on other elements from Z × Z do not affect themultiplication [ ., . ]. Example 1.3.
The graded analogue of sl (2 , C ) is defined as complex algebra with three generators e , e and e satisfying the commutation relations e e + e e = e , e e + e e = e , e e + e e = e . Let sl (2 , C ) be Z -graded linear space sl (2 , C ) = sl (2 , C ) (1 , , ⊕ sl (2 , C ) (1 , , ⊕ sl (2 , C ) (0 , , with basis e ∈ sl (2 , C ) (1 , , , e sl (2 , C ) (1 , , , e ∈ sl (2 , C ) (0 , , . The homogeneous subspaces of sl (2 , C ) graded bythe elements of Z different from (1 , , , (1 , , and (1 , , are zero and so are omitted. The bilinearmultiplication [ ., . ] : sl (2 , C ) × sl (2 , C ) −→ sl (2 , C ) defined, with respect to the basis { e , e , e } , by theformulas [ e , e ] = e e − e e = 0 , [ e , e ] = e e + e e = e , [ e , e ] = e e − e e = 0 , [ e , e ] = e e + e e = e , [ e , e ] = e e − e e = 0 , [ e , e ] = e e + e e = e , makes sl (2 , C ) into a three-dimensional color-Lie algebra.By using [34, Theorem 3.14] , we provide this Lie color algebra with a Hom structure, for that we considera linear even map α : sl (2 , C ) −→ sl (2 , C ) checking α [ x, y ] = [ α ( x ) , α ( y )] , ∀ x, y ∈ H ( sl (2 , C )) in such way ( sl (2 , C ) , [ ., . ] α = α ◦ [ ., . ] , α ) is a color Hom-Lie algebra. The morphism of sl c are given with respect to thebasics { e , e , e } by α ( e ) = a e + a e + a e ,α ( e ) = b e + b e + b e ,α ( e ) = c e + c e + c e , where a i , b i , c i ∈ C .Then, we obtain the following lists of twisted color Hom-Lie algebra ( sl c , [ ., . ] α = α ◦ [ ., . ] , ε, α ) in Table :Table 1 ONSTRUCTIONS AND COHOMOLOGY OF COLOR HOM-LIE ALGEBRAS 5 [ e , e ] α = − e [ e , e ] α = − e [ e , e ] α = − e α = − − [ e , e ] α = e α = − − [ e , e ] α = e [ e , e ] α = − e [ e , e ] α = − e [ e , e ] α = e [ e , e ] α = e α = − − [ e , e ] α = − e α = [ e , e ] α = − e [ e , e ] α = − e [ e , e ] α = e [ e , e ] α = − e [ e , e ] α = e α = − − [ e , e ] α = − e α = − −
10 1 0 [ e , e ] α = e [ e , e ] α = e [ e , e ] α = − e [ e , e ] α = e [ e , e ] α = e α = − − [ e , e ] α = − e α = [ e , e ] α = − e [ e , e ] α = − e [ e , e ] α = e [ e , e ] α = − e [ e , e ] α = e α = − − [ e , e ] α = − e α = − − [ e , e ] α = e [ e , e ] α = e [ e , e ] α = e [ e , e ] α = − e [ e , e ] α = − e α = [ e , e ] α = e α = − − [ e , e ] α = − e [ e , e ] α = − e [ e , e ] α = e [ e , e ] α = e [ e , e ] α = − e α = [ e , e ] α = e α = − − [ e , e ] α = − e [ e , e ] α = e [ e , e ] α = − e [ e , e ] α = − e [ e , e ] α = − e α = − − [ e , e ] α = e α = −
11 0 00 − [ e , e ] α = e [ e , e ] α = − e [ e , e ] α = e [ e , e ] α = e [ e , e ] α = − e α = [ e , e ] α = e α = − − [ e , e ] α = − e [ e , e ] α = e [ e , e ] α = − e [ e , e ] α = − e [ e , e ] α = − e α = − − [ e , e ] α = e α = − −
11 0 0 [ e , e ] α = e [ e , e ] α = − e [ e , e ] α = e [ e , e ] α = e [ e , e ] α = − e α = [ e , e ] α = e α = − − [ e , e ] α = − e [ e , e ] α = e [ e , e ] α = − e [ e , e ] α = − e [ e , e ] α = − e α = −
10 1 0 − [ e , e ] α = e α = − − [ e , e ] α = e [ e , e ] α = − e Remark 1.3.
The morphisms algebras of sl viewed as a color Lie algebra are all automorphisms. Example 1.4.
The graded analogue of the Lie algebra of the group of plane of motions is defined as a complexalgebra with three generators e , e and e satisfying the commutation relations e e + e e = e , e e + e e = e , e e + e e = 0 . The linear space A spanned by e , e , e can be made into a Γ -graded ε -Lie algebra. Thegrading group Γ and the commutation factor ε are the same as in Example . . But the multiplication [ ., . ] is different and is defined by [ e , e ] = e e − e e = 0 , [ e , e ] = e e + e e = e , [ e , e ] = e e − e e = 0 , [ e , e ] = e e + e e = e , [ e , e ] = e e − e e = 0 , [ e , e ] = e e + e e = 0 . By using [34, Theorem 3.14] , we provide the following color Lie algebras with a Hom structure. We consideran even linear map α : A −→ A checking α [ x, y ] = [ α ( x ) , α ( y )] for all x, y ∈ H ( A ) in such way ( A , [ ., . ] α = α ◦ [ ., . ] , α ) is a color Hom-Lie algebra. Then, in Table , we obtain the following color Hom-Lie algebras :Table 2 ONSTRUCTIONS AND COHOMOLOGY OF COLOR HOM-LIE ALGEBRAS 7 [ e , e ] α = − b e − b e [ e , e ] α = b e + b e [ e , e ] α = b e + b e α = − b − b b − b [ e , e ] α = b e + b e α = b b b b [ e , e ] α = 0 [ e , e ] α = 0[ e , e ] α = 0 [ e , e ] α = 0[ e , e ] α = 0 α = a a [ e , e ] α = 0 α = c [ e , e ] α = a e + a e [ e , e ] α = c e where a i , b i , c i ∈ C , i = 1 , . Remark 1.4.
It is easy to check that the even bilinear multiplications in Examples . and . satisfy thethree following axioms :Γ − grading axiom : [ A γ , A η ] ⊆ A γ + η , ∀ γ, η ∈ Γ .ε -skew-symmetry condition : [ x, y ] = − ε ( x, y )[ y, x ] .ε -Jacoby identity : (cid:9) x,y,z ε ( z, x )[ x, [ y, z ]] = 0 for nonzero homogeneous x, y and z and where (cid:9) x,y,z denotes summation over the cyclic permutation on x, y, z .
2. Γ -graded version Hartwig-Larsson-Silvestrov Theorem
In this section, we describe and prove the Hartwig-Larsson-Silvestrov Theorem [15, Theorem 5] for nongradedalgebras, in the case of Γ-graded algebra. A Z -graded version of this theorem was given in [5]. We aim toconsider it deeply for color algebras case.Let A = L γ ∈ Γ A γ be an associative color algebra. We assume that A is color commutative, that is forhomogeneous elements x, y in A , the identity xy = ε ( x, y ) yx holds. Let σ : A −→ A be an even coloralgebra endomorphism of A . Then A is color bimodule over itself, the left (resp. right) action is defined by x · l y = σ ( x ) y (resp. y · r x = yx ). For simplicity, we denote the module multiplication by a dot and the colormultiplication by juxtaposition. In the sequel, the elements of A are supposed to be homogeneous. Definition 2.1.
Let d ∈ Γ . A color σ -derivation ∆ d on A is an endomorphism satisfying ( CD ) : ∆ d ( A γ ) ⊆ A γ + d , ( CD ) : ∆ d ( xy ) = ∆ d ( x ) y + ε ( d, x ) σ ( x )∆ d ( y ) , ∀ x, y ∈ H ( G ) . In particular for d = 0 , we have ∆( xy ) = ∆( x ) y + σ ( x )∆( y ) , then ∆ is called even color σ -derivation. Theset of all color σ -derivation is denoted by Der εσ ( A ) = L P γ ∈ Γ Der γε ( A ) .The structure of A -color module of Der σ ( A ) is as usual. Let ∆ ∈ Der σ ( A ) , the annihilator Ann (∆) is theset of all x ∈ H ( G ) such that x · ∆ = 0 . We set A · ∆ = { x · ∆ : x ∈ H ( G ) } to be a color A -module of Der σ ( A ) .Let σ : A −→ A be a fixed endomorphism, ∆ an even color σ -derivation ∆ ∈ Der εσ ( A ) and δ be an elementin A . Then Theorem 2.1. If (2.2) σ ( Ann (∆)) ⊆ Ann (∆) holds, the map [ ., . ] σ : A · ∆ × A · ∆ −→ A · ∆ defined by setting :(2.3) [ x · ∆ , y · ∆] σ = ( σ ( x ) · ∆) ◦ ( y · ∆) − ε ( x, y )( σ ( y ) · ∆) ◦ ( x · ∆) where ◦ denotes the composition of functions, is a well-defined color algebra bracket on the Γ -graded space A · ∆ and satisfies the following identities for x, y ∈ H ( G )[ x · ∆ , y · ∆] σ = ( σ ( x )∆( y ) − ε ( x, y ) σ ( y )∆( x )) · ∆ , ∀ x, y ∈ H ( G ) . (2.4) [ x · ∆ , y · ∆] σ = − ε ( x, y )[ y · ∆ , x · ∆] σ , ∀ x, y ∈ A . (2.5) K. ABDAOUI, F. AMMAR AND A. MAKHLOUF
In addition, if (2.6) ∆( σ ( x )) = δσ (∆( x )) , ∀ x ∈ H ( G ) holds, then (2.7) (cid:9) x,y,z ε ( z, x ) (cid:16) [ σ ( x ) · ∆ , [ y · ∆ , z · ∆] σ ] σ + δ [ x · ∆ , [ y · ∆ , z · ∆] σ ] σ (cid:17) = 0 , ∀ x, y, z ∈ H ( A ) for all x, y and z in H ( G ) .Proof. We must first show that [ ., . ] σ is a well-defined function. That is,if x · ∆ = x · ∆, then(2.8) [ x · ∆ , y · ∆] σ = [ x · ∆ , y · ∆] σ and(2.9) [ y · ∆ , x · ∆] σ = [ y · ∆ , x · ∆] σ for x , x , y ∈ H ( A ). Now x · ∆ = x · ∆ is equivalent to ( x − x ) ∈ Ann (∆). Therefore, using theassumption (2 . σ ( x − x ) ∈ Ann (∆). Then since | x | = | x | and σ ( x − x ) ∈ Ann (∆), weobtain: [ x · ∆ , y · ∆] σ − [ x · ∆ , y · ∆] σ = ( σ ( x ) · ∆) ◦ ( y · ∆) − ε ( x , y )( σ ( y ) · ∆) ◦ ( x · ∆)= − ( σ ( x ) · ∆) ◦ ( y · ∆) + ε ( x , y )( σ ( y ) · ∆) ◦ ( x · ∆)= ( σ ( x − x ) · ∆) ◦ ( y · ∆) − ε ( x , y )( σ ( y ) · ∆) ◦ (( x − ε ( y, x ) ε ( x , y ) x ) · ∆) . Which shows (2 . .
9) is analogous.Next we prove (2 . A · ∆ is closed under [ ., . ] σ .Let x, y, z ∈ H ( G ) be arbitrary. Then, since ∆ is an even color σ -derivation on A we have:[ x · ∆ , y · ∆] σ ( z ) = ( σ ( x ) · ∆)( y · ∆)( z ) − ε ( x, y )( σ ( y ) · ∆)( x · ∆)( z )= σ ( x ) · ∆( y ∆( z )) − ε ( x, y ) σ ( y )∆( x ∆( z ))= σ ( x ) (cid:16) ∆( y )∆( z ) + σ ( y )∆ ( z ) (cid:17) − ε ( x, y ) σ ( y ) (cid:16) ∆( x )∆( z ) + σ ( x )∆ ( z ) (cid:17) = (cid:16) σ ( x )∆( y ) − ε ( x, y ) σ ( y )(∆( x ) (cid:17) ∆( z ) + (cid:16) σ ( x ) σ ( y ) − ε ( x, y ) σ ( y ) σ ( x ) (cid:17) ∆ ( z ) . Since A is color commutative, the last term is zero. Thus (2 .
2) is true. The ε -skew-symmetry condition (2 . .
3. Using the linearity of σ and ∆ and the definition of [ ., . ] σ on the formula(2 . ., . ] σ is an even bilinear map.It remains to prove (2 . .
2) and that ∆ is an even color σ -derivation on A , we get ε ( z, x )[ σ ( x ) · ∆ , [ y · ∆ , z · ∆] σ ] σ = ε ( z, x )[ σ ( x ) · ∆ , ( σ ( y ) · ∆( z ) − ε ( y, z ) σ ( z ) · ∆( y )) · ∆] σ = ε ( z, x ) (cid:16) σ ( x )∆( σ ( y )∆( z ) − ε ( y, z ) σ ( x ) σ ( z )∆( y )+ σ ( ε ( y, z ) ε ( x, z + y ) σ ( z )∆( y ) − ε ( x, z + y ) σ ( y )∆( z ))∆( σ ( x )) (cid:17) · ∆= ε ( z, x ) (cid:16) σ ( x )∆( σ ( y )∆( z ) + σ ( x ) σ ( y )∆ ( z ) − ε ( y, z )∆( σ ( z ))∆( y ) − ε ( y, z ) σ ( z )∆ ( y )) − ( ε ( x, y ) σ ( y ) σ (∆( z )) − ε ( y, z ) ε ( x, z + y ) σ ( z ) σ (∆( y )))∆( σ ( x ) (cid:17) · ∆ , (2.10)where σ = σ ◦ σ and ∆ = ∆ ◦ ∆. Applying cyclic summation to the second and fourth term in (2 .
10) andsince A is color commutative, we get (cid:9) x,y,z ε ( z, x ) (cid:16) ( σ ( x )∆( σ ( y )∆ ( z ) − ε ( y, z ) σ ( x ) σ ( z )∆ ( y )) · ∆ (cid:17) = (cid:9) x,y,z ε ( z, x ) (cid:16) ( σ ( x )∆( σ ( y )∆ ( z ) − ε ( x, y ) σ ( y ) σ ( x )∆ ( z )) · ∆ (cid:17) = 0 . ONSTRUCTIONS AND COHOMOLOGY OF COLOR HOM-LIE ALGEBRAS 9
Similarly, if we apply cyclic summation to the fifth and sixth term in (2 .
10) and use the relation (2 .
6) weobtain: (cid:9) x,y,z ε ( z, x ) (cid:16) − ε ( x, y ) σ ( y ) σ (∆( z ))∆( σ ( x )) + ε ( y, z ) ε ( x, y + z ) σ ( z )∆ ( y )∆( σ ( x )) (cid:17) = (cid:9) x,y,z ε ( z, x ) (cid:16) − ε ( x, y ) σ ( y ) σ (∆( z )) δσ (∆( x )) + ε ( y, z ) ε ( x, y + z ) σ ( z )∆ ( y ) δσ (∆( x )) (cid:17) = δ (cid:16) (cid:9) x,y,z ε ( z, x )(( − ε ( x, y ) σ ( y ) σ (∆( z )) σ (∆( x )) + ε ( x, y ) σ ( y ) σ (∆( x )) σ (∆( z ))) · ∆) (cid:17) = 0 , where we again use the color commutativity of A . Consequently, the only terms in the right hand side of(2 .
10) which do not vanish when take cyclic summation are (cid:9) x,y,z ε ( z, x )[ σ ( x ) · ∆ , [ y · ∆ , z · ∆] σ ] σ = (cid:9) x,y,z ε ( z, x ) (cid:16) ( σ ( x )∆( σ ( y ))∆( z ) − ε ( y, z ) σ ( x )∆( σ ( z )∆( y )) · ∆ (cid:17) . (2.11)We now consider the other term in (2 . .
6) we have[ y · ∆ , z · ∆] σ = ( σ ( y )∆( z ) − ε ( y, z ) σ ( z )∆( y )) · ∆ . Using first this and then (2 . δ [ x · ∆ , [ y · ∆ , z · ∆] σ ] σ = δ [ x · ∆ , ( σ ( y )∆( z ) − ε ( y, z ) σ ( z )∆( y )) · ∆] σ = δ (cid:16) σ ( x )∆( σ ( y )∆( z ) − ε ( y, z ) σ ( z )∆( y )) − ε ( x, y + z ) σ ( σ ( y )∆( z ) − ε ( y, z ) σ ( z )∆( y ))) · ∆( x ) (cid:17) = δ (cid:16) ( ε ( y, z ) σ ( x )∆ ( z ) σ ( y ) + ε ( y, z ) σ ( x ) σ (∆( z ))∆( σ ( y )) − σ ( x )∆ ( y ) σ ( z ) − σ ( x ) σ (∆( y ))∆( σ ( z )) − ε ( x, y ) ε ( y, z ) − σ (∆( z )) σ ( y )∆( x ) + σ (∆( y )) σ ( z )∆( x )) · ∆ (cid:17) = ε ( y, z ) δσ ( x )∆ ( z ) σ ( y ) + ε ( y, z ) δσ ( x ) σ (∆( z ))∆( σ ( y )) − δσ ( x )∆ ( y ) σ ( z ) − δσ ( x ) σ (∆( y ))∆( σ ( z )) − ε ( x, y ) ε ( y, z ) − δσ (∆( z )) σ ( y )∆( x ) + δσ (∆( y )) σ ( z )∆( x )) · ∆ . Using (2 . (cid:16) ε ( y, z ) δσ ( x )∆ ( z ) σ ( y ) + ε ( y, z ) σ ( x )∆( σ ( z ))∆( σ ( y )) − δσ ( x )∆ ( y ) σ ( z ) − δσ ( x ) σ (∆( y ))∆( σ ( z )) − ε ( x, y ) ε ( y, z )∆( σ ( z )) σ ( y )∆( x ) + ∆( σ ( y )) σ ( z )∆( x ) (cid:17) · ∆= (cid:16) ε ( y, z ) δσ ( x )∆ ( z ) σ ( y ) − δσ ( x )∆ ( y ) σ ( z ) − ε ( x, y ) ε ( y, z )∆( σ ( z )) σ ( y )∆( x ) + ∆( σ ( y )) σ ( z )∆( x ) (cid:17) · ∆ . The first two terms of this last expression vanish after a cyclic summation, so we get (cid:9) x,y,z ε ( z, x ) δ [ x · ∆ , [ y · ∆ , z · ∆] σ ] σ = (cid:9) x,y,z ε ( z, x ) (cid:16) ( − ε ( x, y ) ε ( y, z )∆( σ ( z )) σ ( y )∆( x ) + ∆( σ ( y )) σ ( z )∆( x )) · ∆ (cid:17) . (2.12)Finally, combining this with (2 .
11) we deduce (cid:9) x,y,z ε ( z, x )([ σ ( x ) · ∆ , [ y · ∆ , z · ∆] σ ] σ + δ [ x · ∆ , [ y · ∆ , z · ∆] σ ] σ )= (cid:9) x,y,z ε ( z, x )[ σ ( x ) · ∆ , [ y · ∆ , z · ∆] σ ] σ + (cid:9) x,y,z ε ( z, x ) δ [ x · ∆ , [ y · ∆ , z · ∆] σ ] σ = (cid:9) x,y,z ε ( z, x ) (cid:16) ( σ ( x )∆( σ ( y ))∆( z ) − ε ( y, z ) σ ( x )∆( σ ( z )∆( y )) · ∆ (cid:17) + (cid:9) x,y,z ε ( z, x ) (cid:16) ( − ε ( x, y ) ε ( y, z )∆( σ ( y )) σ ( x )∆( z ) + ∆( σ ( z )) σ ( x )∆( y )) · ∆ (cid:17) = 0 , as was to be shown. The proof is complete. (cid:3) Cohomology and Representations of color Hom-Lie algebras
In this section we define a family of cohomology complexes of color Hom-Lie algebras, discuss the represen-tations in connection with cohomology and provide an example of computation.3.1.
Cohomology of color Hom-Lie algebras.
We extend to color Lie algebras, the concept of A -moduleintroduced in [3, 10, 30], and then define a family of cohomology complexes for colr Hom-Lie algebras.Let ( A , [ ., . ] , ε, α ) be a color Hom-Lie algebra, ( M, β ) be a pair of Γ-graded vector space M and an evenhomomorphism of vectors spaces β : M −→ M , and[ ., . ] M : A × M −→ M ( g, m ) [ g, m ] M be an even bilinear map satisfying [ A γ , M γ ] ⊆ M γ + γ for all γ , γ ∈ Γ. Definition 3.1.
The triple ( M, [ ., . ] M , β ) is said to be A -module if the even bilinear map [ ., . ] M satisfies (3.13) β ([ x, m ] M ) = [ α ( x ) , β ( m )] M and (3.14) [[ x, y ] , β ( m )] M = [ α ( x ) , [ y, m ]] M − ε ( x, y )[ α ( y ) , [ x, m ]] M . The cohomology of color Lie algebras was introduced in [28]. In the following, we define cohomologycomplexes of color Hom-Lie algebras.The set C n ( A , M ) of n -cochains on space A with values in M , is the set of n -linear maps f : A n −→ M satisfying f ( x , ..., x i , x i +1 , ..., x n ) = − ε ( x i , x i +1 ) f ( x , ..., x i +1 , x i , ..., x n ) , ∀ ≤ i ≤ n − . For n = 0, we have C ( A , M ) = M .The map f is called even (resp. of degree γ ) when f ( x , ..., x i , ..., x n ) ∈ M for all elements ( x , ..., x n ) ∈ A ⊗ n (resp. f ( x , ..., x i , ..., x n ) ∈ M γ for all elements ( x , ..., x n ) ∈ A ⊗ n of degree γ ).A n -cochain on A with values in M is defined to be a n -cochain f ∈ C n ( A , M ) such that it is compatiblewith α and β in the sense that f ◦ α = β ◦ f .Denote by C nα,β ( A , M ) the set of n -cochains:(3.15) C nα,β ( A , M ) = { f : A −→ M : f ◦ α = β ◦ f } . We extend this definition to the case of integers n < C nα,β ( A , M ) = { } if n < − C ( A , M ) = M. A homogeneous element f ∈ C nα,β ( A , M ) is called n -cochain.Next, for a given integer r , we define the coboundary operator δ nr . Definition 3.2.
We call, for n ≥ and for a any integer m , a n -coboundary operator of the color Hom-Liealgebra ( A , [ ., . ] , ε, α ) the linear map δ nr : C nα,β ( A , M ) −→ C n +1 α,β ( A , M ) defined by δ nr ( f )( x , ...., x n ) =(3.16) X ≤ s With the above notations, for any f ∈ C nα,β ( A , M ) , we have δ nr ( f ) ◦ α = β ◦ δ nr ( f ) , ∀ n ≥ . Thus, we obtain a well defined map δ nr : C nα,β ( A , M ) −→ C n +1 α,β ( A , M ) . Proof. Let f ∈ C nα,β ( A , M ) and ( x , ...., x n ) ∈ H ( A ⊗ n +1 ), we have δ nr ( f ) ◦ α ( x , ...., x n )= δ nr ( f )( α ( x ) , ...., α ( x n ))= X ≤ s Let ( A , [ ., . ] , ε, α ) be a color Hom-Lie algebra and ( M, β ) be an A -module. Then the pair ( L n ≥ C nα,β , δ nr ) is a cohomology complex. That is the maps δ nr satisfy δ nr ◦ δ n − r = 0 , ∀ n ≥ , ∀ r ≥ Proof. For any f ∈ C n − ( A , M ), we have δ nr ◦ δ n − r ( f )( x , ...., x n ) = X s 19) we have δ n − r ( f )( α ( x ) , ..., α ( x s − ) , [ x s , x t ] , α ( x s +1 ) , ..., b x t , ..., α ( x n ))= X s ′ 20) implies that[ α r + n − ( x s ) , δ n − r ( f )( x , ..., b x s , .., x n )] M = [ α r + n − ( x s ) , X s ′ 22) + (3 . 25) + (3 . . Using (3 . 14) and (3 . . 31) = [ α r + n − ([ x s , x t ]) , f ( α ( x ) , ...α ( x s − ) , \ α ([ x s , x t ]) , α ( x s +1 ) ...., b x t , ..., α ( x n ))] M = [ α n + r − ( x s ) , [ α r + n − ( x t ) , f ( x , ..., d x s,t , .., x n )] M ] M − [ α r + n − ( x t ) , [ α r + n − ( x s ) , f ( x , ..., d x s,t , .., x n )] M ] M . Thus X s 31) + n X s =0 ( − s ε ( γ + x + ... + x s − , x s )(3 . n X s =0 ( − s ε ( γ + x + ... + x s − , x s )(3 . 37) = 0 . By a simple calculation, we get X s 30) + n X s =0 ( − s ε ( γ + x + ... + x s − , x s )(3 . 35) = 0 , X s 33) + n X s =0 ( − s ε ( γ + x + ... + x s − , x s )(3 . 34) = 0 , and X s 23) + (3 . X s 21) + (3 . X s 24) + (3 . . Therefore δ nr ◦ δ n − r = 0. Which completes the proof. (cid:3) Let Z nr ( A , M ) (resp. B nr ( A , M )) denote the kernel of δ nr (resp. the image of δ n − r ). The spaces Z nr ( A , M )and B nr ( A , M ) are graded submodules of C nα,β ( A , M ) and according to Proposition 3.1, we have(3.38) B nr ( A , M ) ⊆ Z nr ( A , M ) . The elements of Z nr ( A , M ) are called n -cocycles, and the elements of B nr ( A , M ) are called the n -coboundaries.Thus, we define a so-called cohomology groups H nr ( A , M ) = Z nr ( A , M ) B nr ( A , M ) . We denote by H nr ( A , M ) = L d ∈ Γ ( H nr ( A , M )) d the space of all r -cohomology group of degree d of the colorHom-Lie algebra A with values in M .Two elements of Z nr ( A , M ) are said to be cohomologeous if their residue class modulo B nr ( A , M ) coincide,that is if their difference lies in B nr ( A , M ).3.2. Adjoint representations of color Hom-Lie algebras. In this section, we generalize to color Hom-Lie algebras some results from [3] and [30]. Let ( A , [ ., . ] , ε, α ) be a multiplicative color Hom-Lie algebra. Weconsider A represents on itself via bracket with respect to the morphism α . Definition 3.3. Let ( A , [ ., . ] , ε, α ) be a color Hom-Lie algebra. A representation of A is a triplet ( M, ρ, β ) ,where M is a Γ -graded vector space, β ∈ End ( M ) and ρ : A −→ End ( M ) is an even linear map satisfying (3.39) ρ ([ x, y ]) ◦ β = ρ ( α ( x )) ◦ ρ ( y ) − ε ( x, y ) ρ ( α ( y )) ◦ ρ ( x ) , ∀ x, y ∈ H ( A ) . Now, we discuss the adjoint representations of a color Hom-Lie algebra. Proposition 3.1. Let ( A , [ ., . ] , ε, α ) be a color Hom-Lie algebra and ad : A −→ End ( A ) be an operatordefined for x ∈ H ( A ) by ad ( x )( y ) = [ x, y ] . Then ( A , ad, α ) is a representation of A .Proof. Since A is color Hom-Lie algebra, the ε -Hom-Jacobi condition on x, y, z ∈ H ( A ) is ε ( z, x )[ α ( x ) , [ y, z ]] + ε ( y, z )[ α ( z ) , [ x, y ]] + ε ( x, y )[ α ( y ) , [ z, x ]] = 0and may be written ad ([ x, y ])( α ( z )) = ad ( α ( x ))( ad ( y )( z )) − ε ( x, y ) ad ( α ( y ))( ad ( x )( z )) . Then the operator ad satisfies ad ([ x, y ]) ◦ α = ad ( α ( x )) ◦ ad ( y ) − ε ( x, y ) ad ( α ( y )) ◦ ad ( x ) . Therefore, it determines a representation of the color Hom-Lie algebra A . (cid:3) We call the representation defined in the previous Proposition the adjoint representation of the colorHom-Lie algebra A . Definition 3.4. The α s -adjoint representation of the color Hom-Lie algebra ( A , [ ., . ] , ε, α ) , which we denoteby ad s , is defined by ad s ( a )( x ) = [ α s ( a ) , x ] , ∀ a, x ∈ H ( A ) . Lemma 3.2. With the above notations, we have ( A , ad s ( . )( . ) , α ) is a representation of the color Hom-Liealgebra ( A , [ ., . ] , ε, α ) . ad s ( α ( x )) ◦ α = α ◦ ad s ( x ) .ad s ([ x, y ]) ◦ α = ad s ( α ( x )) ◦ ad s ( y ) − ε ( x, y ) ad s ( α ( y )) ◦ ad s ( x ) . ONSTRUCTIONS AND COHOMOLOGY OF COLOR HOM-LIE ALGEBRAS 15 Proof. First, the result follows from ad s ( α ( x ))( α ( y )) = [ α s +1 ( x ) , α ( y )]= α ([ α s ( x ) , y ])= α ◦ ad s ( x )( y )and ad s ([ x, y ])( α ( z )) = [ α s ([ x, y ]) , α ( z )] = [[ α s ( x ) , α s ( y )] , α ( z )]= ε ( x + y, z ) ε ( z, y ) ε ( z, x )[ α s +1 ( x ) , [ α s ( y ) , z ]+ ε ( x + y, z ) ε ( z, y ) ε ( x, y )[ α s +1 ( y ) , [ z, α s ( x )]= [ α s +1 ( x ) , [ α s ( y ) , z ] + ε ( x + y, z )[ α s +1 ( y ) , [ z, α s ( x )]= [ α s +1 ( x ) , [ α s ( y ) , z ] − ε ( x, y )[ α s +1 ( y ) , [ α s ( x ) , z ]= [ α s +1 ( x ) , ad s ( y )( z )] − ε ( x, y )[ α s +1 ( y ) , ad s ( x )( z )]= ad s ( α ( x )) ◦ ad s ( y )( z ) − ε ( x, y ) ad s ( α ( y )) ◦ ad s ( x )( z ) . Then ad s ([ x, y ]) ◦ α = ad s ( α ( x )) ◦ ad s ( y ) − ε ( x, y ) ad s ( α ( y )) ◦ ad s ( x ). (cid:3) The set of n -cochains on A with coefficients in A which we denote by C nα ( A , A ), is given by C nα ( A , A ) = { f ∈ C n ( A , A ) : f ◦ α ⊗ n = α ◦ f } . In particular, the set of 0-cochains is given by C α ( A , A ) = { x ∈ H ( A ) : α ( x ) = x } . Proposition 3.2. Associated to the α s -adjoint representation ad s , of the color Hom-Lie algebra ( A , [ ., . ] , ε, α ) , D ∈ C α,ad s ( A , A ) is -cocycle if and only if D is an α s +1 -derivation of the color Hom-Lie algebra ( A , [ ., . ] , ε, α ) of degree γ . (i.e. D ∈ ( Der α s +1 ( A )) γ ) .Proof. The conclusion follows directly from the definition of the coboundary δ . D is closed if and only if δ ( D )( x, y ) = − D ([ x, y ]) + ε ( γ, x )[ α s +1 ( x ) , D ( y )] − ε ( γ + x, y )[ α s +1 ( y ) , D ( x )] = 0 . So D ([ x, y ]) = [ D ( x ) , α s +1 ( y )] + ε ( γ, x )[ α s +1 ( x ) , D ( y )]which implies that D is an α s +1 -derivation of ( A , [ ., . ] , ε, α ) of degree γ . (cid:3) The α − -adjoint representation ad − . Proposition 3.3. Associated to the α − -adjoint representation ad − , we have H ( A , A ) = C α ( A , A ) = { x ∈ H ( A ) : α ( x ) = x } .H ( A , A ) = Der α ( A ) . Proof. For any 0-cochain x ∈ C α ( A , A ), we have δ ( x )( y ) = ε ( x, y )[ α − ( y ) , x ] = 0 , ∀ y ∈ H ( A ) . Therefore any 0-cochain is closed. Thus, we have H ( A , A ) = C α ( A , A ) = { x ∈ H ( A ) : α ( x ) = x } . Since there is not exact 1-cochain, by Proposition 3.2, we have H ( A , A ) = Der α ( A ) . Let w ∈ C α ( A , A ) be an even ε -skew-symmetric bilinear operator commuting with α . Considering a t-parametrized family of bilinear operations[ x, y ] t = [ x, y ] + tw ( x, y ) . Since w commute with α , α is a morphism with respect to the bracket [ ., . ] t for every t . If all bracket[ ., . ] t endow that w generates a deformation of the color Hom-Lie algebra ( A , [ ., . ] , ε, α ). By computing the ε -Hom-Jacobi condition of [ ., . ] t , this is equivalent to (cid:9) x,y,z ε ( z, x ) (cid:16) w ( α ( x ) , [ y, z ]) + [ α ( x ) , w ( y, z )] (cid:17) = 0 , (3.40) (cid:9) x,y,z ε ( z, x ) w (cid:16) α ( x ) , w ( y, z ) (cid:17) = 0 . (3.41)Obviously, (3 . 40) means that w is an even 2-cocycle with respect to the α − -adjoint representation ad − .Furthermore, (3 . 41) means that w must itself defines a color Hom-Lie algebra structure on A . (cid:3) The α -adjoint representation ad . Proposition 3.4. Associated to the α -adjoint representation ad , we have H ( A , A ) = { x ∈ H ( A ) : α ( x ) = x, [ x, y ] = 0 } .H ( A , A ) = Der α ( A ) Inn α ( A ) . Proof. For any 0-cochain, we have d ( x )( y ) = [ α ( x ) , y ] = [ x, y ].Therefore, the set of 0-cocycle Z ( A , A ) is given by Z ( A , A ) = { x ∈ C α ( A , A ) : [ x, y ] = 0 , ∀ y ∈ H ( A ) } . As, B ( A , A ) = { } , we deduce that H ( A , A ) = { x ∈ H ( A ) : α ( x ) = x, [ x, y ] = 0 } For any f ∈ C α ( A , A ), we have δ ( f )( x, y ) = − f ([ x, y ]) + ε ( f, x )[ α ( x ) , f ( y )] − ε ( f + x, y )[ α ( y ) , f ( x )]= − f ([ x, y ]) + ε ( f, x )[ α ( x ) , f ( y )] + [ f ( x ) , α ( y )] . Therefore the set of 1-cocycles is exactly the set of α -derivations Der α ( A ) . Furthermore, it is obvious that any exact 1-coboundary is of the form of [ x, . ] for some x ∈ C α ( A , A ).Therefore, we have B ( A , A ) = Inn α ( A ). Which implies that H ( A , A ) = Der α ( A ) Inn α ( A ) . (cid:3) The coadjoint representation f ad . In this subsection, we explore the dual representations and coadjointrepresentations of color Hom-Lie algebras. Let ( A , [ ., . ] , ε, α ) be a color Hom-Lie algebra and ( M, ρ, β ) be arepresentation of A . Let M ∗ be the dual vector space of M . We define a linear map e ρ : A −→ End ( M ∗ ) by e ρ ( x ) = − t ρ ( x ). Let f ∈ M ∗ , x, y ∈ H ( A ) and m ∈ M . We compute the right handside of the identity (3 . e ρ ( α ( x )) ◦ e ρ ( y ) − ε ( x, y ) e ρ ( α ( y )) ◦ e ρ ( x ))( f )( m )= ( e ρ ( α ( x )) ◦ e ρ ( y )( f ) − ε ( x, y ) e ρ ( α ( y )) ◦ e ρ ( x )( f ))( m )= − e ρ ( y )( f )( ρ ( α ( x )( m )) + ε ( x, y ) e ρ ( x )( f )( ρ ( α ( y )( m ))= f ( ρ ( y ) ◦ ρ ( α ( x ))( m )) − ε ( x, y ) f ( ρ ( x ) ◦ ρ ( α ( y ))( m ))= f ( ρ ( y ) ◦ ρ ( α ( x ))( m ) − ε ( x, y ) ρ ( x ) ◦ ρ ( α ( y ))( m )) . On the other hand, we set that the twisted map for e ρ is e β = t β , the left hand side of (3 . 39) writes e ρ ([ x, y ]) ◦ e β ( f )( m ) = e ρ ([ x, y ])( f ◦ β )( m )= − f ◦ β ( ρ ([ x, y ])( m )) . Therefore, we have the following Proposition: Proposition 3.5. Let ( A , [ ., . ] , ε, α ) be a color Hom-Lie algebra and ( M, ρ, β ) be a representation of A .Let M ∗ be the dual vector space of M . The triple ( M ∗ , e ρ, e β ) , where e ρ : A −→ End ( M ∗ ) is given by e ρ ( x ) = − t ρ ( x ) , defines a representation of color Hom-Lie algebra ( A , [ ., . ] , ε, α ) if and only if (3.42) β ◦ ρ ([ x, y ]) = ρ ( x ) ◦ ρ ( α ( y )) − ε ( x, y ) ρ ( y ) ◦ ρ ( α ( x )) . We obtain the following characterization in the case of adjoint representation. Corollary 3.1. Let ( A , [ ., . ] , ε, α ) be a color Hom-Lie algebra and ( A , ad, α ) be the adjoint representation of A , where ad : A −→ End ( A ) . We set f ad : A −→ End ( A ∗ ) and f ad ( x )( f ) = − f ◦ ad ( x ) . Then ( A ∗ , f ad, e α ) isa representation of A if and only if α ◦ ad ([ x, y ]) = ad ( x ) ◦ ad ( α ( y )) − ε ( x, y ) ad ( y ) ◦ ad ( α ( x )) , ∀ x, y ∈ H ( A ) . ONSTRUCTIONS AND COHOMOLOGY OF COLOR HOM-LIE ALGEBRAS 17 Example. Let ( sl c , C ) be a color Lie algebra such that sl c = ⊕ γ ∈ Γ X γ and Γ = Z × Z , with X (0 , = 0 , X (1 , = e , X (0 , = e , X (1 , = e . The algebra ( sl c , C ) has a homogeneous basis { e , e , e } with degree given by | e | = γ = (1 , , | e | = γ = (0 , , | e | = γ = (1 , . The bracket [ ., . ] in sl c is given by[ e , e ] = e , [ e , e ] = e , [ e , e ] = e . Then ( sl c , [ ., . ]) is a color Lie algebra. According to Theorem 1 . 1, the triple ( sl c , [ ., . ] α = α ◦ [ ., . ] , α ) is a colorHom-Lie algebra such that the bracket [ ., . ] α and the even linear map α are defined by[ e , e ] α = e , α ( e ) = − e [ e , e ] α = − e , α ( e ) = − e [ e , e ] α = − e , α ( e ) = e . Let ψ ∈ C ad ( sl c , sl c ). The 2-coboundary is defined by equation (3 . . Now, suppose that ψ is a 2-cocycleof sl c . Then ψ satisfies ψ ( α ( x ) , [ y, z ] α ) = ψ ([ x, y ] α , α ( z )) − ε ( γ, x )[ α ( x ) , ψ ( y, z )] α + ε ( γ + x, y )[ α ( y ) , ψ ( x, z )] α − ε ( γ + x + y, z )[ α ( z ) , ψ ( x, y )] α − ε ( y, z ) ψ ([ x, z ] α , α ( y ))(3.43)By plugging the following triples( e , e , e ) , ( e , e , e ) , ( e , e , e ) , ( e , e , e ) , ( e , e , e ) , ( e , e , e ) , · · · ( e , e , e ) , ( e , e , e ) . respectively in (3 . Case 1: If γ = γ = (1 , • Z γ ( sl c , sl c ) = { ψ : ψ ( e i , e i ) = 0 , ψ ( e i , e j ) = ψ ( e j , e i ) , ∀ i = j, i = 1 , , } = { ψ : ψ ( e , e ) = a e + a e , ψ ( e , e ) = a e + a e + a e , ψ ( e , e ) = a e } , • B γ ( sl c , sl c ) = { δf : δf ( e i , e i ) = 0 , δf ( e , e ) = a e , δf ( e , e ) = a e , δf ( e , e ) = 0 , ∀ i = 1 , , } . Then H γ ( sl c , sl c ) = { ψ : ψ ( e , e ) = a e , ψ ( e , e ) = a e + a e } . Case 2: If γ = γ = (0 , • Z γ ( sl c , sl c ) = { ψ : ψ ( e i , e i ) = 0 , ψ ( e i , e j ) = ψ ( e j , e i ) , ∀ i = j, i = 1 , , } = { ψ : ψ ( e i , e i ) = 0 , ψ ( e , e ) = a e , ψ ( e , e ) = 0 , ψ ( e , e ) = a e } , • B γ ( sl c , sl c ) = { δf : δf ( e i , e i ) = 0 , δf ( e , e ) = a e , δf ( e , e ) = 0 , δf ( e , e ) = a e , ∀ i = 1 , , } Then H γ ( sl c , sl c ) = { } . Case 3: If γ = γ = (1 , • Z γ ( sl c , sl c ) = { ψ : ψ ( e i , e i ) = 0 , ψ ( e i , e j ) = ψ ( e j , e i ) , ∀ i = j, i = 1 , , } = { ψ : ψ ( e i , e i ) = 0 , ψ ( e , e ) = 0 , ψ ( e , e ) = a e + a e , ψ ( e , e ) = a e − a e } , • B γ ( sl c , sl c ) = { δf : δf ( e i , e i ) = 0 , δf ( e , e ) = 0 , δf ( e , e ) = a e + a e , δf ( e , e ) = a e − a e , ∀ i = 1 , , } Then H γ ( sl c , sl c ) = { } .So H ( sl c , sl c ) = H γ ( sl c , sl c ) ⊕ H γ ( sl c , sl c ) ⊕ H γ ( sl c , sl c ) = { ψ : ψ ( e , e ) = a e , ψ ( e , e ) = a e + a e } . Formal deformations of color Hom-Lie algebras Formal deformations of color Hom-Lie algebras.Definition 4.1. Let ( A , [ ., . ] , ε, α ) be a color Hom-Lie algebra. A one parameter formal deformation of A is given by K [[ t ]] -bilinear map [ ., . ] t : A [[ t ]] × A [[ t ]] −→ A [[ t ]] of the form [ ., . ] t = P i ≥ t i [ ., . ] i where [ ., . ] i is aneven K -bilinear map [ ., . ] i : A [[ t ]] × A [[ t ]] −→ A [[ t ]] (extended to be even K [[ t ]] -bilinear) and satisfying for all x, y, z ∈ H ( A ) the following conditions (4.44) [ x, y ] t = − ε ( x, y )[ y, x ] t , (4.45) (cid:9) x,y,z ε ( z, x )[ α ( x ) , [ y, z ] t ] t = 0 . The deformation is said to be of order k if [ ., . ] t = k P i ≥ t i [ ., . ] i . Remark 4.1. The ε -skew symmetry of [ ., . ] t is equivalent to the ε -skew symmetry of [ ., . ] i for i ∈ Z ≥ . Condition (4 . 45) is called deformation equation of the color Hom-Lie algebra and it is equivalent to (cid:9) x,y,z X i,j,k ≥ ε ( z, x ) t i + j [ α ( x ) , [ y, z ] i ] j = 0i.e (cid:9) x,y,z X i,s ≥ ε ( z, x ) t s [ α ( x ) , [ y, z ] i ] s − i = 0or X s ≥ t s (cid:9) x,y,z X i,s ≥ ε ( z, x )[ α ( x ) , [ y, z ] i ] s − i = 0which is equivalent to the following infinite system(4.46) (cid:9) x,y,z X i,k ≥ ε ( z, x )[ α ( x ) , [ y, z ] i ] s − i = 0 , ∀ s = 0 , , · · · In particular, for s = 0, we have (cid:9) x,y,z ε ( z, x )[ α ( x ) , [ y, z ] ] = 0, which is the ε -Hom-Jacobi condition of A .The equation for s = 1, leads to δ ([ ., . ] )( x, y, z ) = 0. Then [ ., . ] is a 2-cocycle.For s ≥ 2, the identity (4 . 46) is equivalent to δ ([ ., . ] s )( x, y, z ) = − X p + q = s (cid:9) x,y,z ε ( z, x )[ α ( x ) , [ y, z ] q ] p = 0 . Equivalent and trivial deformations.Definition 4.2. Let ( A , [ ., . ] , ε, α ) be a multiplicative color Hom-Lie algebra. Given two deformations A t =( A , [ ., . ] t , ε, α ) and A ′ t = ( A , [ ., . ] ′ t , ε, α ′ ) of A where [ ., . ] t = k P i ≥ t i [ ., . ] i and [ ., . ] ′ t = k P i ≥ t i [ ., . ] ′ i with [ ., . ] =[ ., . ] ′ = [ ., . ] . We say that A t and A ′ t are equivalent if there exists a formal automorphism φ t : A [[ t ]] −→ A [[ t ]] that may be written in the form φ t = P i ≥ φ i t i , where φ i ∈ End ( A ) and φ = Id such that φ t ([ x, y ] t ) = [ φ t ( x ) , φ t ( y )] ′ t (4.47) φ t ( α ( x )) = α ′ ( φ t ( x )) . A deformation A t of A is said to be trivial if and only if A t is equivalent to A . Viewed as an algebra on A [[ t ]]. Definition 4.3. Let ( A , [ ., . ] , ε, α ) be a color Hom-Lie algebra and [ ., . ] ∈ Z ( A , A ) .The -cocycle [ ., . ] is said to be integrable if there exists a family ([ ., . ] i ) i ≥ such that [ ., . ] t = P i ≥ t i [ ., . ] i definesa formal deformation A t = ( A , [ ., . ] t , ε, α ) of A . Theorem 4.1. Let ( A , [ ., . ] , ε, α ) be a color Hom-Lie algebra and A t = ( A , [ ., . ] t , ε, α ) be a one parameterformal deformation of A , where [ ., . ] t = P i ≥ t i [ ., . ] i . Then (1) The first term [ ., . ] is a -cocycle with respect to the cohomology of ( A , [ ., . ] , ε, α ) . ONSTRUCTIONS AND COHOMOLOGY OF COLOR HOM-LIE ALGEBRAS 19 (2) there exists an equivalent deformation A ′ t = ( A , [ ., . ] ′ t , ε, α ′ ) , where [ ., . ] ′ t = P i ≥ t i [ ., . ] ′ i such that [ ., . ] ′ ∈ Z ( A , A ) and [ ., . ] ′ B ( A , A ) .Moreover, if H ( A , A ) = 0 , then every formal deformation is trivial. Deformation by composition. In the sequel, we give a procedure of deforming color Lie algebrasinto color Hom-Lie algebras using the following Proposition: Proposition 4.1. Let ( A , [ ., . ] , ε ) be a color Lie algebra and α t be an even algebra endomorphism of theform α t = α + p P i ≥ t i α i , where α i are linear maps on A , t is a parameter in K and p is an integer. Let [ ., . ] t = α t ◦ [ ., . ] , then ( A , [ ., . ] t , ε, α t ) is a color Hom-Lie algebra which is a deformation of the color Liealgebra viewed as a color Hom-Lie algebra ( A , [ ., . ] , ε, Id ) .Moreover, the n th derived Hom-algebra A nt = ( A , [ ., . ] ( n ) t = α n − t ◦ [ ., . ] t , ε, α n t ) is a deformation of ( A , [ ., . ] , ε, Id ) .Proof. The first assertion follows from Theorem 1 . 1. In particular for an infinitesimal deformation of theidentity α t = Id + tα , we have [ ., . ] t = [ ., . ] + tα ◦ [ ., . ].The proof of the ε -Hom-Jacobi condition of the nth derived Hom-algebra follows from Theorem 1 . 1. In case n = 1 and α t = Id + tα the bracket is[ ., . ] (1) = Id + tα ◦ Id + tα ◦ [ ., . ]= [ ., . ] + 2 tα ◦ [ ., . ] + t α ◦ [ ., . ]and the twist map is α t = ( Id + tα ) = Id + 2 tα + t α . Therefore we get another deformation of the colorLie algebra viewed as a color Hom-Lie algebra ( A , [ ., . ] , ε, Id ). The proof in the general case is similar. (cid:3) Remark 4.2. More generally if ( A , [ ., . ] , ε, α ) is a multiplicative color Hom-Lie algebra where α may bewritten of the form α = Id + tα , then the nth derived Hom-algebra A nt = ( A , [ ., . ] ( n ) = α n ◦ [ ., . ] , ε, α n +1 ) gives a one parameter formal deformation of ( A , [ ., . ] , ε, α ) . But for any α one obtains just new color Hom-Liealgebra. Generalized α k -derivations of color Hom-Lie algebras The purpose of this section is to study the homogeneous α k -generalized derivations and homogeneous α k -centroid of color Hom-Lie algebras generalizing the homogeneous generalized derivations discussed in [8].In Proposition 5 . α -derivation of color Hom-Lie algebras gives rise to a Hom-Jordan coloralgebras.We need the following definitions: Definition 5.1. Let P l γ ( A ) = { D ∈ Hom ( A , A ) : D ( A γ ) ⊂ A γ + µ for all γ, µ ∈ Γ } .Then (cid:16) P l ( A ) = L γ ∈ Γ P l γ ( A ) , [ ., . ] , α (cid:17) is a color Hom-Lie algebra with the color Lie bracket [ D γ , D µ ] = D γ ◦ D µ − ε ( γ, µ ) D µ ◦ D γ for all D γ , D µ ∈ H ( P l ( A )) and with α : A −→ A is an even homomorphism.A homogeneous α k -derivation of degree γ of A is an endomorphism D ∈ P l γ ( A ) such that [ D, α ] = 0 ,D ([ x, y ]) = [ D ( x ) , α k ( y )] + ε ( γ, x )[ α k ( x ) , D ( y )] for all x, y in A .We denote the set of all homogeneous α k -derivations of degree γ of A by Der γα k ( A ) . The space Der ( A ) = M k ≥ Der α k ( A ) provided with the color-commutator is a color Lie algebra. Indeed, the fact that Der α k ( A ) is Γ -graded impliesthat Der ( A ) is Γ -graded ( Der ( A )) γ = M k ≥ ( Der α k ( A )) γ , ∀ γ ∈ Γ . Definition 5.2. (1) An endomorphism D ∈ P l γ ( A ) is said to be a homogeneous generalized α k -derivation of degree γ of A , if there exist two endomorphisms D ′ , D ′′ ∈ P l γ ( A ) such that [ D, α ] = 0 , [ D ′ , α ] = 0 , [ D ′′ , α ] = 0(5.48) D ′′ ([ x, y ]) = [ D ( x ) , α k ( y )] + ε ( γ, x )[ α k ( x ) , D ′ ( y )] for all x, y in A .We denote the set of all homogeneous generalized α k -derivations of degree γ of A by GDer γα k ( A ) .The space GDer ( A ) = M k ≥ GDer α k ( A ) . (2) We call D ∈ P l γ ( A ) a homogeneous α k -quasi-derivation of degree γ of A , if there exists an endo-morphism D ′ ∈ P l γ ( A ) such that [ D, α ] = 0 , [ D ′ , α ] = 0(5.49) D ′ ([ x, y ]) = [ D ( x ) , α k ( y )] + ε ( γ, x )[ α k ( x ) , D ( y )] for all x, y ∈ A .We denote the set of all homogeneous α k -quasi-derivations of degree γ of A by QDer γα k ( A ) . Thespace QDer ( A ) = M k ≥ QDer α k ( A ) . (3) If C ( A ) = L k ≥ C γα k ( A ) , ∀ γ ∈ Γ , with C γα k ( A ) consisting of D ∈ P l γ ( A ) satisfying (5.50) D ([ x, y ]) = [ D ( x ) , α k ( y )] = ε ( γ, x )[ α k ( x ) , D ( y )] for all x, y in A , then C ( A ) is called the α k -centroid of A .We denote the set of all homogeneous α k -centroid of degree γ of A by C γα k ( A ) . Proposition 5.1. Let ( A , [ ., . ] , ε, α ) be a multiplicative color Hom-Lie algebra.If D γ ∈ GDer α k ( A ) and ∆ γ ′ ∈ C α k ′ ( A ) , then ∆ γ ′ D γ ∈ GDer α k + k ′ ( A ) is of degree ( γ + γ ′ ) .Proof. Let D γ ∈ GDer α k ( A ). Then for all x, y ∈ H ( A ), there exist D ′ γ , D ′′ γ ∈ P l γ ( A ) such that D ′′ γ ([ x, y ]) = [ D γ ( x ) , α k ( y )] + ε ( γ, x )[ α k ( x ) , D ′ γ ( y )] . Now, let ∆ γ ′ ∈ C α k ′ ( A ) then we have:∆ γ ′ D ′′ γ ([ x, y ]) = ∆ γ ′ ([ D γ ( x ) , α k ( y )] + ε ( γ, x )[ α k ( x ) , D ′ γ ( y )])= [∆ γ ′ D γ ( x ) , α k + k ′ ( y )] + ε ( γ + γ ′ , x )[ α k + k ′ ( x ) , ∆ γ ′ D ′ γ ( y )] . Then ∆ γ ′ D γ ∈ GDer α k + k ′ ( A ) of degree ( γ + γ ′ ). (cid:3) Proposition 5.2. Let D γ ∈ C α k ( A ) , then D γ is an α k -Quasi-derivation of A .Proof. Let x, y ∈ H ( A ), we have[ D γ ( x ) , α k ( y )] + ε ( γ, x )[ α k ( x ) , D γ ( y )] = [ D γ ( x ) , α k ( y )] + [ D γ ( x ) , α k ( y )]= 2 D γ ([ x, y ])= D ” γ ([ x, y ]) . Then D γ is an α k -Quasi-derivation of degree γ of A . (cid:3) ONSTRUCTIONS AND COHOMOLOGY OF COLOR HOM-LIE ALGEBRAS 21 Definition 5.3. If QC ( A ) = L γ ∈ Γ QC γα k ( A ) and QC γα k ( A ) consisting of D ∈ P l γ ( A ) such that for all x, y in A [ D ( x ) , α k ( y )] = ε ( γ, x )[ α k ( x ) , D ( y )] , then QC ( A ) is called the α k -quasi-centroid of A . Proposition 5.3. Let D γ ∈ H ( QC α k ( A )) and D µ ∈ H ( QC α k ′ ( A )) . Then [ D γ , D µ ] is an α k + k ′ -generalizedderivation of degree ( γ + µ ) . Proof. Assume that D γ ∈ H ( QC α k ( A )) , D µ ∈ H ( QC α k ′ ( A )). then for all x, y ∈ H ( A ), we have[ D γ ( x ) , α k ( y )] = ε ( γ, x )[ α k ( x ) , D γ ( y )]and [ D µ ( x ) , α k ′ ( y )] = ε ( µ, x )[ α k ′ ( x ) , D µ ( y )] . Hence[[ D γ , D µ ]( x ) , α k + k ′ ( y )] = [( D γ ◦ D µ − ε ( γ, µ ) D µ ◦ D γ )( x ) , α k + k ′ ( y )]= [ D γ ◦ D µ ( x ) , α k + k ′ ( y )] − ε ( γ, µ )[ D µ ◦ D γ ( x ) , α k + k ′ ( y )]= ε ( γ + µ, x )[ α k + k ′ ( x ) , D γ ◦ D µ ( y )] − ε ( γ, µ ) ε ( γ + µ, x )[ α k + k ′ ( x ) , D µ ◦ D γ ( y )]= ε ( γ + µ, x )[ α k + k ′ ( x ) , [ D γ , D µ ]( y )] + [[ D γ , D µ ]( x ) , α k + k ′ ( y )] , which implies that [[ D γ , D µ ]( x ) , α k + k ′ ( y )] + [[ D γ , D µ ]( x ) , α k + k ′ ( y )] = 0 . Then [ D γ , D µ ] ∈ GDer α k + k ′ ( A ) and is of degree ( γ + µ ) . (cid:3) Hom-Jordan color algebras and Derivations.Definition 5.4. Let ( A , µ, α ) be a Hom-color algebra. (1) The Hom-associator of A is the trilinear map as α : A × A × A −→ A defined as (5.51) as α = µ ◦ ( µ ⊗ α − α ⊗ µ ) . In terms of elements, the map as α is given by as α ( x, y, z ) = µ ( µ ( x, y ) , α ( z )) − µ ( α ( x ) , µ ( y, z )) for all x, y, z in H ( A ) . (2) Let A be a Hom-algebra over a field K of characteristic = 2 with an even bilinear multiplication ◦ .If A is graded by the abelian group Γ , ε : Γ × Γ −→ K ∗ and α : A −→ A be an even linear map, then ( A , ◦ , ε, α ) is a Hom-Jordan color algebra if the identities ( HCJ 1) : x ◦ y = ε ( x, y ) y ◦ x ( HCJ 2) : ε ( w, x + z ) as α ( x ◦ y, α ( z ) , α ( w )) + ε ( x, y + z ) as α ( y ◦ w, α ( z ) , α ( x ))+ ε ( y, w + z ) as α ( w ◦ x, α ( z ) , α ( y )) = 0 are satisfied for all x, y, z and w in H ( A ) . The identity ( HCJ 2) is called the color Hom-Jordan identity .Observe that when α = Id , the color Hom-Jordan identity ( HCJ 2) reduces to the usual color Jordan identity. Proposition 5.4. Let ( A , [ ., . ] , ε, α ) be a multiplicative color Hom-Lie algebra, with the operation D γ • D µ = D γ ◦ D µ − ε ( γ, µ ) D γ ◦ D µ for all α -derivations D γ , D µ ∈ H ( P l ( A )) , the triple ( P l ( A ) , • , ε, α ) is a Hom-Jordancolor algebra.Proof. Assume that D λ , D θ , D µ , D γ ∈ H ( P l ( A )), we have D λ • D θ = D λ ◦ D θ − ε ( λ, θ ) D θ ◦ D λ = ε ( λ, θ )( D θ ◦ D λ − ε ( λ, θ ) D λ ◦ D θ )= ε ( λ, θ ) D θ • D λ . Since (( D λ • Dθ ) • α ( D µ )) • α ( D γ )= D λ D θ α ( D µ ) α ( D γ ) + ε ( λ + θ + µ, γ ) α ( D γ ) D λ D θ α ( D µ )+ ε ( λ + θ, µ ) α ( D µ ) D λ D θ α ( D γ ) + ε ( λ + θ, µ ) ε ( λ + θ + µ, γ ) α ( D γ ) α ( D µ ) D λ D θ + ε ( λ, θ ) D θD λ α ( D µ ) α ( D γ ) + ε ( λ, θ ) ε ( λ + θ + µ, γ ) α ( D γ ) D θ D λ α ( D µ )+ ε ( λ, θ ) ε ( λ + θ, µ ) α ( D µ ) D θ D λ α ( D γ ) + ε ( λ, θ ) ε ( λ + θ, µ ) ε ( λ + θ + µ, γ ) α ( D γ ) α ( D µ ) D θ D λ and α ( D λ • D θ ) • ( α ( D µ ) • α ( D γ )) = α ( D λ D θ ) α ( D µ ) α ( D γ ) + ε ( λ + θ, µ + γ ) α ( D µ ) α ( D γ ) α ( D λ D θ )+ ε ( µ, γ ) α ( D λ D θ ) α ( D γ ) α ( D µ )+ ε ( µ, γ ) ε ( λ + θ, µ + γ ) α ( D γ ) α ( D µ ) α ( D λ D θ )+ ε ( λ, θ ) α ( D θ D λ ) α ( D µ ) α ( D γ )+ ε ( λ, θ ) ε ( λ + θ, µ + γ ) α ( D µ ) α ( D γ ) α ( D θ D λ )+ ε ( λ, θ ) ε ( µ, θ ) α ( D θ D λ ) α ( D γ ) α ( D µ )+ ε ( λ, θ ) ε ( µ, γ ) ε ( λ + θ, µ + γ ) α ( D γ ) α ( D µ ) α ( D θ D λ ) . We have ε ( γ, λ + µ ) as α ( D λ • D θ , α ( D µ ) , α ( D γ ))= ε ( γ, λ + µ ) (cid:16) ε ( λ + θ + µ, γ ) α ( D γ ) D λ D θ α ( D µ ) + ε ( λ + θ, µ ) α ( D µ ) D λ D θ α ( D γ )+ ε ( λ, θ ) ε ( λ + θ + µ, γ ) α ( D γ ) D θ D λ α ( D µ ) + ε ( λ, θ ) ε ( λ + θ, µ ) α ( D µ ) D θ D λ α ( D γ ) − ε ( λ + θ, µ + γ ) α ( D µ ) α ( D γ ) α ( D θ D λ ) − ε ( µ, γ ) α ( D λ D θ ) α ( D γ ) α ( D µ ) − ε ( λ, θ ) ε ( λ + θ, µ + γ ) α ( D µ ) α ( D γ ) α ( D θ D λ ) − ε ( λ, θ ) ε ( µ, θ ) α ( D θ D λ ) α ( D γ ) α ( D µ ) (cid:17) . Therefore, we get ε ( γ, λ + µ ) as α ( D λ • D θ , α ( D µ ) , α ( D γ )) + ε ( λ, θ + µ ) as α ( D θ • D γ , α ( D µ ) , α ( D γ ))+ ε ( θ, γ + µ ) as α ( D γ • D λ , α ( D µ ) , α ( D θ )) = 0 , and so the statement holds. (cid:3) Corollary 5.1. 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