1609.07396.Pdf

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G Deﬁnition 2. A color n-ary algebra T is an n-ary -graded vector space T = g∈G Tg with a graded n-linear map [·,..., ·]: T × ... × T → T satisfying L G [Tθ1 ,...,Tθn ] ⊆ Tθ1+...+θn , θi ∈ . The main examples of color n-ary algebras are color Lie algebras [6], color Leibniz algebras [8], Filippov (n-Lie) superalgebras [4,5,33,35,36] and 3-Lie color algebras [42]. Let T = G Tg be a color algebra. An element x is called a homogeneous element of degree t ∈ G if x ∈ Tt. We denote Lg∈ this by hg(x)= t. A linear map D is homogeneous of degree t if D(Tg) ⊆ Tg+t for all g ∈ G. We denote this by hg(D)= t. If t =0 then D is said to be even. From now on, unless stated otherwise, we assume that all elements and maps are homogeneous. Let ǫ be a bicharacter on G. For two homogeneous elements a and b we set ǫ(a,b) := ǫ(hg(a),hg(b)). Let D1,D2 be linear maps on T and x1,...,xn ∈ T. We use the following notations:

(1) [D1,D2]= D1D2 − ǫ(D1,D2)D2D1,

Xi = hg(x1)+ ... + hg(xi).

In particular, we set X0 =0. By End(T ) we denote the set of all linear maps of T . Obviously, End(T )= g∈G Endg(T ) endowed with the color bracket (1) is a color Lie algebra over F. L 1.1. Binary Hom-algebras. In this subsection we give some definitions of Hom-algebras with a binary multiplication. It is known that for any family of polynomial identities Ω, every Ω-algebra is a Ω-superalgebra and every Ω-superalgebra is a color Ω-algebra. In what follows we only give the definitions of color Hom-Ω algebras. The definitions of Hom-Ω superalgebras and Hom-Ω algebras are particular cases of the former ones for G = Z2 and G = {0}, respectively. On the other hand, the definitions of Ω-algebras, Ω-superalgebras and color Ω-algebrascan be obtained from suitable Hom-definitions for α = id. G Definition 3. A color Hom-Lie algebra (T, [·, ·],ǫ,α) is a -graded vector space T = g∈G Tg with a bicharacter ǫ, an even bilinear map [·, ·]: T × T → T and an even linear map α : T → T satisfying L [x, y]= −ǫ(x, y)[y, x], ǫ(z, x)[α(x), [y,z]] + ǫ(x, y)[α(y), [z, x]] + ǫ(y,z)[α(z), [x, y]] = 0. G Definition 4. A color Hom-Jordan algebra (T, [·, ·],ǫ,α) is a -graded vector space T = g∈G Tg with a bicharacter ǫ, an even bilinear map [·, ·]: T × T → T and an even linear map α : T → T satisfying L [x, y]= ǫ(x, y)[y, x],

ǫ(z, x + w)asT ([x, y], α(w), α(z)) + ǫ(x, y + w)asT ([y,z], α(w), α(x)) + ǫ(y,z + w)asT ([z, x], α(w), α(y)) =0, where the trilinear map asT : T × T × T → T is the Hom-associator of T deﬁned by

asT (x,y,z) = [[x, y], α(z)] − [α(x), [y,z]]. 1.2. n-ary Hom-algebras. In this subsection we give some definitions related to multiplicative Hom-algebras with n-ary multiplication. In general (see, [7]), for the definition of n-ary Hom-algebras we must use a family of homomorphisms {αi}I , but αi = α and α([x1,...,xn]) = [α(x1),...,α(xn)] in the case of multiplicative n-ary Hom-algebras. G Definition 5. A multiplicative n-ary Hom-Lie color algebra (T, [·,..., ·],ǫ,α) is a -graded vector space T = g∈G Tg with an n-linear map [·,..., ·]: T × ... × T → T and an even linear map α : T → T satisfying L

[x1,...,xi, xi+1,...,xn]= −ǫ(xi, xi+1)[x1,...,xi+1, xi,...,xn], n [α(x ),...,α(x ), [y ,...,y ]] = ǫ(X , Y )[α(y ),...,α(y ), [x ,...,x ,y ], α(y ),...,α(y )]. 1 n−1 1 n X n−1 i−1 1 i−1 1 n−1 i i+1 n i=1 Deﬁnition 6. A multiplicative n-ary Hom-associative color algebra (T, [·,..., ·],ǫ,α) is a G-graded vector space T = G Tg with an n-linear map [·,..., ·]: T × ... × T → T and an even linear map α : T → T satisfying Lg∈ [α(x1),...,α(xi), [xi+1,...,xi+n], α(xi+n+1),...,α(x2n−1)] = [α(x1),...,α(xn−1), [xn,...,x2n−1]]. Now we deﬁne the notion of n-ary multiplicative Hom-Ω color algebra for arbitrary family of polynomial identities Ω.

Definition 7. For a (maybe n-ary) multilinear polynomial f(x1,...,xn) we fix the order of indices {i1,...,in} of one non- associative word [xi1 ...xin ]β from the polynomial f. Here, f = αβ,σ[xσ(i1 ) ...xσ(in)]β, where β is an arrangement β,σP∈Sn of brackets in the non-associative word. For the shift µi : {j1,...,jn} 7→ {j1,...ji+1, ji,...,jn} we define the element

ǫ(xji , xji+1 ). Now, for arbitrary non-associative word [xσ(i1 ) ...xσ(in)]β its order of indexes is a composition of suitable shifts

µi, and for this word we set ǫσ deﬁned as the product of corresponding ǫ(xji , xji+1 ). Now, for the multilinear polynomial f, we deﬁne the color multilinear polynomial f = α ǫ [x ...x ] . co X β,σ σ σ(i1 ) σ(in) β β,σ∈Sn 3

Hom To construct a color multiplicative multilinear Hom-polynomial fco from a color multilinear polynomial fco we use the algorithm from [7] and we are changing all free letters xj to α(xj ) in all words of the color multilinear polynomial fco.

Definition 8. Let Ω= {fi} be a family of n-ary multilinear polynimials. Then (1) A color n-ary Ω-algebra L is a color n-ary algebra satisfying the family of color multilinear polinomials Ωco = {(fi)co} . (2) A multiplicative n-ary Hom-Ω color algebra T is a color n-ary algebra satisfying the family of color multiplicative Hom Hom multilinear Hom-polynomials Ω = (fi) for a homomorphism α. co co 1.3. Linear maps on Hom-algebras. From now on, we denote a multiplicative n-ary Hom-Ω color algebra (T, [·,..., ·],ǫ,α) by T. Definition 9. A G-graded subspace M ⊆ T is a color Hom-subalgebra of T if α(M) ⊆ M and [M,...,M] ⊆ M. A G-graded subspace I ⊆ T is a color Hom-ideal of T if α(I) ⊆ I and [T,...,I,...,T ] ⊆ I. Definition 10. The vector subspace S of End(T ) is defined by S = {u ∈ End(T ) | uα = αu} with α˜ : S → S given by α˜(u)= αu. Notice that (S, [·, ·], ǫ, α˜), where [·, ·] is the color bracket in (1) and ǫ is a bicharacter, is a color Hom-Lie algebra over F. In what follows, we only consider linear maps from S. Let ξ ∈ G. k Definition 11. D ∈ Endξ(T) is a homogeneous α -derivation of T , with k ∈ N if it satisfies n D([x ,...,x ]) = ǫ(D,X )[αk(x ),...,αk(x ),D(x ), αk(x ),...,αk(x )]. 1 n X i−1 1 i−1 i i+1 n i=1 k We denote the set of all α -derivations of T by Derαk (T ). Notice that Der(T ) := k≥0 Derαk (T ) equipped with the color bracket defined by (1) and the even map α˜ : Der(T ) → Der(T ) defined by α˜(D)= DαL, is a color Hom-subalgebra of S, which is called the derivation algebra of T . k Definition 12. D ∈ Endξ(T) is a homogeneous generalized α -derivation of degree ξ of T if there exist endomorphisms (i) D ∈ Endξ(T ) satisfying n [D(x ), αk(x ),...,αk(x )] + ǫ(D,X )[αk(x ),...,αk(x ),D(i−1)(x ), αk(x ),...,αk(x )] 1 2 n X i−1 1 i−1 i i+1 n i=2 (n) = D ([x1,...,xn]). k ′ Definition 13. D ∈ Endξ(T) is a homogeneous α -quasiderivation of degree ξ of T if there exists D ∈ Endξ(T ) satisfying n ǫ(D,X )[αk(x ),...,αk(x ),D(x ), αk(x ),...,αk(x )] = D′([x ,...,x ]). X i−1 1 i−1 i i+1 n 1 n i=1 k k Let GDerαk (T ) and QDerαk (T ) be the sets of homogeneous generalized α -derivations and of homogeneous α - quasiderivations, respectively. We denote

GDer(T ) := GDerαk (T ), QDer(T ) := QDerαk (T ). Lk≥0 Lk≥0 k Definition 14. D ∈ Endξ(T ) is a homogeneous α -centroid of T , where k ∈ N if it satisfies k k k k D([x1,...,xn]) = ǫ(D,Xi−1)[α (x1),...,α (xi−1),D(xi), α (xi+1),...,α (xn)] for all i. k Definition 15. D ∈ Endξ(T ) is a homogeneous α -quasicentroid of degree ξ of T , where k ∈ N if it satisfies k k k k k k [D(x1), α (x2),...,α (xn)] = ǫ(D,Xi−1)[α (x1),...,α (xi−1),D(xi), α (xi+1),...,α (xn)] for all i. k k Denote the sets of homogeneous α -centroids and homogeneous α -quasicentroids by Cαk (T ) and QCαk (T ), respectively. Denote also

C(T ) := Cαk (T ), QC(T ) := QCαk (T ). Lk≥0 Lk≥0 k Deﬁnition 16. D ∈ Endξ(T) is a homogeneous α -center derivation of degree ξ of T if it satisﬁes

k k k k D([x1,...,xn]) = [α (x1),...,α (xi−1),D(xi), α (xi+1),...,α (xn)]=0. k Denote the set of homogeneous α -center derivations by ZDerαk (T ). We also denote

ZDer(T ) := ZDerαk (T ). Lk≥0 Notice that we have the following chain of inclusions: ZDer(T ) ⊆ Der(T ) ⊆ QDer(T ) ⊆ GDer(T ) ⊆ End(T ). 4

2. GENERALIZED DERIVATION ALGEBRAS AND THEIR COLOR Hom-SUBALGEBRAS In this section we present some basic properties of generalized derivations, quasiderivations and center derivations of a multiplicative n-ary Hom-Ω color algebra.

Lemma 17. Let T be a multiplicative n-ary Hom-Ω color algebra. Then the following statements hold: (1) GDer(T ), QDer(T ) and C(T ) are color Hom-subalgebras of S; (2) ZDer(T ) is a color Hom-ideal of Der(T ).

Proof. (1) Let Dξ ∈ GDerαk (T ),Dη ∈ GDerαs (T ), where k,s ∈ N. For arbitrary x1,...,xn ∈ T we have k+1 k+1 k+1 k+1 k k [˜α(Dξ)(x1), α (x2),...,α (xn)] = [(Dξα)(x1), α (x2),...,α (xn)] = α([Dξ(x1), α (x2),...,α (xn)]) n (n) k k (i−1) k k = α(Dξ ([x1, x2,...,xn]) − ǫ(Dξ,Xi−1)[α (x1),...,α (xi−1),Dξ (xi), α (xi+1),...,α (xn)]) iP=2 n (n) k+1 k+1 (i−1) k+1 k+1 =α ˜(Dξ )([x1, x2,...,xn]) − ǫ(Dξ,Xi−1)[α (x1),...,α (xi−1), α˜(Dξ )(xi), α (xi+1),...,α (xn)]. iP=2 N (i) For any i ∈ , α˜(Dξ ) belongs to Endξ(T ), thus α˜(Dξ) ∈ GDerαk+1 (T ) and is obviously of degree ξ. For arbitrary x1,...,xn ∈ T we obtain k+s k+s [DξDη(x1), α (x2),...,α (xn)] (n) s s = Dξ ([Dη(x1), α (x2),...,α (xn)]) n k k+s k+s (i−1) s k+s k+s − ǫ(Dξ,Dη + Xi−1)[α (Dη(x1)), α (x2),...,α (xi−1),Dξ (α (xi)), α (xi+1),...,α (xn)] iP=2 (n) (n) = Dξ Dη ([x1,...,xn]) n (n) s s (j−1) s s − ǫ(Dη,Xj−1)Dξ [α (x1),...,α (xj−1),Dη (xj ), α (xj+1),...,α (xn)] jP=2 n (n) k k (i−1) k k − ǫ(Dξ,Dη + Xi−1)Dη [α (x1),...,α (xi−1),Dξ (xi), α (xi+1),...,α (xn)] iP=2 n k+s (j−1) k (i−1) s k+s + ǫ(Dξ,Dη + Xi−1)ǫ(Dη,Xj−1)[α (x1),...,Dη (α (xj )),...,Dξ (α (xi)),...,α (xn)] i=2,jP=2,j

n s s s s (i−1) s s = Dξ([Dη(x1), α (x2),...,α (xn)]) + ǫ(Dη,Xi−1)Dξ([α (x1),...,α (xi−1),Dη (xi), α (xi+1),...,α (xn)]) = iP=2 0 and k+s k+s [[Dξ,Dη](x1), α (x2),...,α (xn)] k+s k+s k+s k+s = [DξDη(x1), α (x2),...,α (xn)] − ǫ(Dξ,Dη)[DηDξ(x1), α (x2),...,α (xn)] k+s k+s k k = −ǫ(Dξ,Dη)[Dη(Dξ(x1)), α (x2),...,α (xn)] = −ǫ(Dξ,Dη)Dη([Dξ(x1), α (x2),...,α (xn)]) n s k+s (i−1) k k+s + ǫ(Dξ,Dη)ǫ(Dη,Dξ + Xi−1)[α (Dξ(x1)), α (x2),...,Dη (α (xi)),...,α (xn)]=0. iP=2 Hence [Dξ,Dη] ∈ ZDerαk+s (T ) and is of degree ξ + η. Therefore, ZDer(T ) is a color Hom-ideal of Der(T ).

Lemma 18. Let T be a multiplicative n-ary Hom-Ω color algebra. Then (1) [Der(T ), C(T )] ⊆ C(T ); (2) [QDer(T ), QC(T )] ⊆ QC(T ); (3) C(T ) · Der(T ) ⊆ Der(T ); (4) C(T ) ⊆ QDer(T ); (5) [QC(T ), QC(T )] ⊆ QDer(T ); (6) QDer(T )+QC(T ) ⊆ GDer(T ).

Proof. (1) Let Dξ ∈ Derαk (T ),Dη ∈ Cαs (T ). For arbitrary x1,...,xn ∈ T we have k+s k+s [DξDη(x1), α (x2),...,α (xn)] n s s k k+s s k+s = Dξ([Dη(x1), α (x2),...,α (xn)]) − ǫ(Dξ,Dη + Xi−1)[Dη(α (x1)), α (x2),...,Dξ(α (xi)),...,α (xn)] iP=2 n k+s k+s = DξDη([x1, x2,...,xn]) − ǫ(Dξ,Dη)ǫ(Dξ + Dη,Xi−1)[α (x1),...,DηDξ(xi),...,α (xn)] iP=2 and k+s k+s [DηDξ(x1), α (x2),...,α (xn)] k k = Dη([Dξ(x1), α (x2),...,α (xn)]) n k k = DηDξ([x1, x2,...,xn]) − ǫ(Dξ,Xi−1)Dη([α (x1),...,Dξ(xi),...,α (xn)]) iP=2 n k+s k+s = DηDξ([x1, x2,...,xn]) − ǫ(Dξ + Dη,Xi−1)[α (x1),...,DηDξ(xi),...,α (xn)]. iP=2 Hence k+s k+s [[Dξ,Dη](x1), α (x2),...,α (xn)] k+s k+s k+s k+s = [DξDη(x1), α (x2),...,α (xn)] − ǫ(Dξ,Dη)[DηDξ(x1), α (x2),...,α (xn)] = [Dξ,Dη]([x1,...,xn]). Similarly, k+s k+s k+s k+s [[Dξ,Dη](x1), α (x2),...,α (xn)] = ǫ(Dξ + Dη,Xi−1)[α (x1),..., [Dξ,Dη](xi),...,α (xn)]. Thus, [Dξ,Dη] ∈ Cαk+s (T ), is of degree ξ + η and [Der(T ), C(T )] ⊆ C(T ). (2) Is similar to the proof of (1). (3) Let Dξ ∈ Cαk (T ),Dη ∈ Derαs (T ). For arbitrary x1,...,xn ∈ T we have DξDη([x1,...,xn]) n s s = Dξ( ǫ(Dη,Xi−1)[α (x1),...,Dη(xi),...,α (xn)]) iP=1 n k+s k+s = ǫ(Dξ + Dη,Xi−1)[α (x1),...,DξDη(xi),...,α (xn)]. iP=1 Therefore DξDη ∈ Derαk+s (T ) and is of degree ξ + η. Hence C(T ) · Der(T ) ⊆ Der(T ). (4) Let Dξ ∈ Cαk (T ). For arbitrary x1,...,xn ∈ T we have k k Dξ([x1,...,xn]) = ǫ(Dξ,Xi−1)[α (x1),...,Dξ(xi),...,α (xn)]. n k k Hence ǫ(Dξ,Xi−1)[α (x1),...,Dξ(xi),...,α (xn)] = nDξ[x1,...,xn]. iP=1 ′ Therefore Dξ ∈ QDerαk (T ) since D = nDξ ∈ Cαk (T ). (5) Let Dξ ∈ QCαk (T ),Dη ∈ QCαs (T ). For arbitrary x1,...,xn ∈ T we have k+s k+s [α (x1),..., [Dξ,Dη](xi),...,α (xn)] s k k+s k+s = ǫ(Xi−1,Dξ)ǫ(Xi−1 − X1,Dη)[Dξ(α (x1)),Dη(α (x2)),...,α (xi),...,α (xn)] s k k+s k+s −ǫ(Dξ,Dη)ǫ(Xi−1 − X1,Dη)ǫ(Xi−1 + Dη,Dξ)[Dξ(α (x1)),Dη(α (x2)),...,α (xi),...,α (xn)]=0. Hence n k+s k+s ǫ(Dξ + Dη,Xi−1)[α (x1),..., [Dξ,Dη](xi),...,α (xn)]=0. iP=1 therefore [Dξ,Dη] ∈ QDerαk+s (T ) and is of degree ξ + η. (6) Is obvious. ✷ 6

Lemma 19. If T is a multiplicative n-ary Hom-Ω color algebra, then QC(T ) + [QC(T ), QC(T )] is a color Hom-subalgebra of GDer(T ). Proof. By Lemma 18 (5) and (6), we have QC(T ) + [QC(T ), QC(T )] ⊆ GDer(T ) and [QC(T ) + [QC(T ), QC(T )], QC(T ) + [QC(T ), QC(T )]] ⊆ [QC(T ) + QDer(T ), QC(T ) + [QC(T ), QC(T )]] ⊆ [QC(T ), QC(T )] + [QC(T ), [QC(T ), QC(T )]] + [QDer(T ), QC(T )] + [QDer(T ), [QC(T ), QC(T )]] Using the color Hom-Jacobi identity and (2) of the previous lemma, it is easy to verify that [QDer(T ), [QC(T ), QC(T )]] ⊆ [QC(T ), QC(T )]. Thus, [QC(T ) + [QC(T ), QC(T )], QC(T ) + [QC(T ), QC(T )]] ⊆ QC(T ) + [QC(T ), QC(T )]. ✷ Deﬁnition 20. Let T is a multiplicative n-ary algebra. The annihilator of T is deﬁned by

Ann(T )= {x ∈ T | [x1,...,xi−1, x, xi+1,...,xn]=0 for all i}. Lemma 21. Let α be a surjective map. If T is a multiplicative n-ary Hom-Ω color algebra, then [C(T ), QC(T )] ⊆ End(T, Ann(T )). Moreover, if Ann(T )= {0}, then [C(T ), QC(T )] = {0}. ′ Proof. Assume that Dξ ∈ Cαk (T ),Dη ∈ QCαs (T ) and x ∈ T . As α is surjective, for any yi ∈ T, 1 ≤ i ≤ n, there exists ′ k+s yi ∈ T such that yi = α (yi). We have k+s k+s ǫ(Yi−1,Dξ + Dη)[α (y1),..., [Dξ,Dη](x),...,α (yn)] = k+s k+s ǫ(Yi−1,Dξ + Dη)[α (y1),..., (DξDη − ǫ(Dξ,Dη)DηDξ)(x),...,α (yn)] = s s s s s s Dξ([Dη(y1), α (y2),...,α (x),...,α (yn)] − [Dη(y1), α (y2),...,α (x),...,α (yn)]) = 0. Hence [Dξ,Dη](x) ∈ Ann(T ) and [Dξ,Dη] ∈ End(T, Ann(T )) as desired. Furthermore, if Ann(T )= {0}, it is clear that [C(T ), QC(T )] = {0}. ✷ Lemma 22. Let F be of characteristic 6=2 and (S, •, α˜) be a color Hom-algebra with multiplication 1 Dξ • Dη = 2 (DξDη + ǫ(Dξ,Dη)DηDξ) where Dξ,Dη are α-derivations of S. Then (1) (S, •, α) is a color Hom-Jordan algebra; (2) (QC(T ), •, α) is a color Hom-Jordan algebra.

Proof. (1) Let Dξ,Dη ∈ S. We have Dξ • Dη 1 = 2 (DξDη + ǫ(Dξ,Dη)DηDξ) 1 = 2 ǫ(Dξ,Dη)(DηDξ + ǫ(Dη,Dξ)DξDη) = ǫ(Dξ,Dη)Dη • Dξ. Consider now the second Hom-Jordan identity: 2 ((Dξ • Dη) • α(Dθ)) • α (Dγ ) 1 2 = 2 ((DξDη + ǫ(Dξ,Dη)DηDξ) • α(Dθ)) • α (Dγ) 1 2 = 4 ((DξDη + ǫ(Dξ,Dη)DηDξ)α(Dθ)+ ǫ(Dξ + Dη,Dθ)α(Dθ)(DξDη + ǫ(Dξ,Dη)DηDξ)) • α (Dγ ) 1 2 2 2 = 8 (DξDηα(Dθ)α (Dγ)+ ǫ(Dξ,Dη)DηDξα(Dθ)α (Dγ)+ ǫ(Dξ + Dη,Dθ)α(Dθ)DξDηα (Dγ) 2 2 +ǫ(Dξ + Dη,Dθ)ǫ(Dξ,Dη)α(Dθ)DηDξα (Dγ )+ ǫ(Dξ + Dη + Dθ,Dγ)α (Dγ )DξDηα(Dθ) 2 2 +ǫ(Dξ,Dη)ǫ(Dξ + Dη + Dθ,Dγ)α (Dγ )DηDξα(Dθ)+ ǫ(Dξ + Dη,Dθ)ǫ(Dξ + Dη + Dθ,Dγ)α (Dγ )α(Dθ)DξDη 2 +ǫ(Dξ + Dη,Dθ)ǫ(Dξ,Dη)ǫ(Dξ + Dη + Dθ,Dγ )α (Dγ)α(Dθ)DηDξ). On the other hand, we have α(Dξ • Dη) • (α(Dθ) • α(Dγ )) 1 = 4 α(DξDη + ǫ(Dξ,Dη)DηDξ) • (α(Dθ)α(Dγ )+ ǫ(Dθ,Dγ)α(Dγ )α(Dθ)) 1 = 8 (α(DξDη)α(Dθ)α(Dγ )+ ǫ(Dξ + Dη,Dθ + Dγ )α(Dθ)α(Dγ )α(DξDη) +ǫ(Dθ,Dγ)α(DξDη)α(Dγ )α(Dθ)+ ǫ(Dθ,Dγ)ǫ(Dξ + Dη,Dθ + Dγ)α(Dγ )α(Dθ)α(DξDη) +ǫ(Dξ,Dη)α(Dη Dξ)α(Dθ)α(Dγ )+ ǫ(Dξ,Dη)ǫ(Dξ + Dη,Dθ + Dγ)α(Dθ)α(Dγ )α(DηDξ) +ǫ(Dξ,Dη)ǫ(Dθ,Dγ)α(DηDξ)α(Dγ )α(Dθ)+ ǫ(Dξ,Dη)ǫ(Dθ,Dγ)ǫ(Dξ + Dη,Dγ + Dθ)α(Dγ )α(Dθ)α(Dη Dξ)). So, ǫ(Dγ,Dξ + Dθ)asα(Dξ • Dη, α(Dθ), α(Dγ )) 1 2 2 = 8 ǫ(Dγ ,Dξ + Dθ)(ǫ(Dξ + Dη,Dθ)α(Dθ)DξDηα (Dγ )+ ǫ(Dξ,Dη)ǫ(Dξ + Dη,Dθ)α(Dθ)DηDξα (Dγ) 2 2 +ǫ(Dξ + Dη + Dθ,Dγ)α (Dγ )DξDηα(Dθ)+ ǫ(Dξ,Dη)ǫ(Dξ + Dη + Dθ,Dγ)α (Dγ )DηDξα(Dθ) −ǫ(Dξ + Dη,Dθ + Dγ)α(Dθ)α(Dγ )α(DξDη) − ǫ(Dθ,Dγ)α(DξDη)α(Dγ )α(Dθ) −ǫ(Dξ,Dη)ǫ(Dξ + Dη,Dθ + Dγ)α(Dθ)α(Dγ )α(DηDξ) − ǫ(Dξ,Dη)ǫ(Dθ,Dγ)α(DηDξ)α(Dγ )α(Dθ)). Hence ǫ(Dγ,Dξ + Dθ)asα(Dξ • Dη, α(Dθ), α(Dγ )) + ǫ(Dξ,Dη + Dθ)asα(Dη • Dγ , α(Dθ), α(Dξ)) +ǫ(Dη,Dγ + Dθ)asα(Dγ • Dξ, α(Dθ), α(Dη))=0. 7

(2) We only need to show that for arbitrary Dξ,Dη ∈ QC(T ), Dξ • Dη ∈ QC(T ). We have k+s k+s [Dξ • Dη(x1), α (x2),...,α (xn)] 1 k+s k+s 1 k+s k+s = 2 [DξDη(x1), α (x2),...,α (xn)] + 2 ǫ(Dξ,Dη)[DηDξ(x1), α (x2),...,α (xn)] 1 k k+s k+s = 2 ǫ(Dξ,Dη + Xi−1)[Dη(α (x1)), α (x2),...,Dξ(xi),...,α (xn)] 1 s k+s k+s + 2 ǫ(Dη,Xi−1)[Dξ(α (x1)), α (x2),...,Dη(xi),...,α (xn)] 1 k+s k+s = 2 ǫ(Dξ,Dη + Xi−1)ǫ(Dη,Xi−1)[α (x1),...,DηDξ(xi),...,α (xn)] 1 k+s k+s + 2 ǫ(Dξ + Dη,Xi−1)[α (x1),...,DξDη(xi),...,α (xn)] k+s k+s = ǫ(Dξ + Dη,Xi−1)[α (x1),...,Dξ • Dη(xi),...,α (xn)]. Then Dξ • Dη ∈ QC(T ) and QC(T ) is a color Hom-Jordan algebra. ✷ Theorem 23. Let T be a multiplicative n-ary Hom-Ω color algebra. Then the following statements hold: (1) QC(T ) is a color Hom-Lie algebra with [Dξ,Dη]= DξDη − ǫ(Dξ,Dη)DηDξ if and only if QC(T ) is a color Hom- associative algebra with respect to the usual composition of operators; (2) If char F is not 2, α is a surjective map and Ann(T ) = {0}, then QC(T ) is a color Hom-Lie algebra if and only if [QC(T ), QC(T )] = {0}.

Proof. (1) (⇐) For arbitrary Dξ ∈ QCαk (T ),Dη ∈ QCαs (T ), we have DξDη ∈ QCαk+s (T ) and DηDξ ∈ QCαk+s (T ). So, [Dξ,Dη]= DξDη − ǫ(Dξ,Dη)DηDξ ∈ QCαk+s (T ). Hence, QC(T ) is a color Hom-Lie algebra. [Dξ,Dη ] (⇒) Note that DξDη = Dξ • Dη + 2 . By Lemma 22, we have Dξ • Dη ∈ QC(T ), [Dξ,Dη] ∈ QC(T ). It follows that DξDη ∈ QC(T ), as desired. (2) (⇒) Let Dξ ∈ QCαk (T ),Dη ∈ QCαs (T ). Now we have k+s k+s [[Dξ,Dη](x1), α (x2),...,α (xn)] k+s k+s = ǫ(Dξ,Dη + Xi−1)ǫ(Dη,Xi−1)[α (x1),...,DηDξ(xi),...,α (xn)] k+s k+s −ǫ(Dξ + Dη,Xi−1)[α (x1),...,DξDη(xi),...,α (xn)] = k+s k+s −ǫ(Dξ + Dη,Xi−1)[α (x1),..., [Dξ,Dη](xi),...,α (xn)]. On the other hand, since QC(T ) is a color Hom-Lie algebra, then, for arbitrary x1,...,xn ∈ T , we have k+s k+s k+s k+s [[Dξ,Dη](x1), α (x2),...,α (xn)] = ǫ(Dξ + Dη,Xi−1)[α (x1),..., [Dξ,Dη](xi),...,α (xn)]. Therefore k+s k+s ′ ′ [α (x1),..., [Dξ,Dη](xi),...,α (xn)]=0, hence [x1,..., [Dξ,Dη](xi),...,xn]=0.

Hence, [Dξ,Dη]=0. (⇐) Is clear. ✷

3. QUASIDERIVATIONS OF MULTIPLICATIVE n-ARY Hom-Ω COLOR ALGEBRAS In this section we investigate the quasiderivations of the multiplicative n-ary Hom-Ω color algebra T. We prove that QDer(T ) can be embedded in the derivation algebra of a larger multiplicative n-ary Hom-Ω color algebra T˘ for the same variety of polynomial Hom-identities Ω. Moreover, we conclude that Der(T˘) has a direct sum decomposition when Ann(T )= {0}.

Lemma 24. Let T be a multiplicative n-ary Hom-Ω color algebra over F and t be an indeterminate. Deﬁne T˘ := {Σ(x ⊗ t + y ⊗ tn) | x, y ∈ T } and α˘(T˘)= {Σ(α(x) ⊗ t + α(y) ⊗ tn) | x, y ∈ T }. Endow (T,˘ α˘) with the multiplication

i1 i2 in P ij [x1 ⊗ t , x2 ⊗ t ,...,xn ⊗ t ] = [x1, x2,...,xn] ⊗ t , k for i1,...,in ∈{1,n} (we put t =0 if k>n). Then (T,˘ α˘) is a multiplicative n-ary Hom-Ω color algebra.

Proof. Let the class of n-ary Hom-Ω color algebras be deﬁned by the family {fk} of color multilinear Hom-identities. Then, for arbitrary x1, x2,...,xm ∈ T and ij ∈{1,n} we have

i1 i2 im P il fj(x1 ⊗ t , x2 ⊗ t ,...,xm ⊗ t )= fj (x1, x2,...,xm) ⊗ t =0. Therefore T˘ is a n-ary Hom-Ω color algebra. ✷ For the sake of convenience, we write xt (xtn) instead of x ⊗ t (x ⊗ tn). If U is a G-graded subspace of T such that T = U ⊕ [T,...,T ], then T˘ = Tt + Ttn = Tt + Utn + [T,...,T ]tn. Now deﬁne a map ϕ : QDer(T ) → End(T˘) by ϕ(D)(at + utn + btn)= D(a)t + D′(b)tn, where D ∈ QDer(T),D′ is a map related to D by the deﬁnition of quasiderivation, a ∈ T,u ∈ U,b ∈ [T,...,T ]. 8

Theorem 25. Let T, T˘ , ϕ be as above. Then (1) ϕ is even; (2) ϕ is injective and ϕ(D) does not depend on the choice of D′; (3) ϕ(QDer(T )) ⊆ Der(T˘). Proof. (1) Follows from the deﬁnition of ϕ. (2) If ϕ(D1)= ϕ(D2), then for all a ∈ T,b ∈ [T,...,T ] and u ∈ U we have n n n n ϕ(D1)(at + ut + bt )= ϕ(D2)(at + ut + bt ), or, in terms of D1,D2, ′ n ′ n D1(a)t + D1(b)t = D2(a)t + D2(b)t , so D1(a)= D2(a). Hence D1 = D2, and ϕ is injective. Suppose that there exists D′′ such that ϕ(D)(at + utn + btn)= D(a)t + D′′(b)tn, and ǫ(D,X )[αk(x ),...,D(x ),...,αk(x )] = D′′([x ,...,x ]), X i−1 1 i n 1 n then we have ′ ′′ D ([x1,...,xn]) = D ([x1,...,xn]), thus D′(b)= D′′(b). Hence ϕ(D)(at + utn + btn)= D(a)t + D′(b)tn = D(a)t + D′′(b)tn, which implies that ϕ(D) is determined only by D. i1 in P ij (3) We have [x1t ,...,xnt ] = [x1,...,xn]t =0 for all ij ≥ n +1. Thus, to show that ϕ(D) ∈ Der(T˘), we only need to check that the following equality holds: P ϕ(D)([x t,...,x t]) = ǫ(D,X )[˘αk(x t),...,ϕ(D)(x t),..., α˘k(x t)]. 1 n X i−1 1 i n For arbitrary x1,...,xn ∈ T we have n ′ n ϕ(D)([x1t,...,xit,...,xnt]) = ϕ(D)([x1,...,xn]t )= D ([x1,...,xn])t = ǫ(D,X )[αk(x ),...,D(x ),...,αk(x )]tn X i−1 1 i n = ǫ(D,X )[αk(x t),...,D(x )t,...,αk(x t)] X i−1 1 i n = ǫ(D,X )[˘αk(x t),...,ϕ(D)(x t),..., α˘k(x t)]. X i−1 1 i n Therefore, for all D ∈ QDer(T ) we have ϕ(D) ∈ Der(T˘). ✷

Lemma 26. Let T be a multiplicative n-ary Hom-Ω color algebra such that Ann(T ) = {0} and let T˘ , ϕ be as previously defined. Then Der(T˘)= ϕ(QDer(T )) ⊕ ZDer(T˘). Proof. Since Ann(T )= {0}, we have Ann(T˘)= Ttn. For g ∈ Der(T˘) we have g(Ann(T˘)) ⊆ Ann(T˘), hence g(Utn) ⊆ g(Ann(T˘)) ⊆ Ann(T˘)= Ttn. Now define a map f : Tt + Utn + [T,...,T ]tn → Ttn by g(x) ∩ Ttn, x ∈ Tt;  n f(x)=  g(x), x ∈ Ut ; 0, x ∈ [T,...,T ]tn.  It is clear that f is linear. Note that f([T,...,˘ T˘]) = f([T,...,T ]tn)=0, [˘αk(T˘),...,f(T˘),..., α˘k(T˘)] ⊆ [αk(T )t + αk(T )tn,...,Ttn,...,αk(T )t + αk(T )tn]=0, hence f ∈ ZDer(T˘). Since (g − f)(Tt)= g(Tt) − g(Tt) ∩ Ttn = g(Tt) − Ttn ⊆ T t, (g − f)(Utn)=0, and (g − f)([T,...,T ]tn)= g([T,...,˘ T˘]) ⊆ [T,...,˘ T˘] = [T,...,T ]tn, there exist D,D′ ∈ End(T ) such that for all a ∈ T, b ∈ [T,...,T ], (g − f)(at)= D(a)t, (g − f)(btn)= D′(b)tn. Since g − f ∈ Der(T˘), by the definition of Der(T˘) we have ǫ(g − f, A )[˘αk(a t),..., (g − f)(a t),..., α˘k(a t)] = (g − f)([a t,...,a t]), X i−1 1 i n 1 n 9 for all a1,...,an ∈ T. Hence ǫ(D, A )[αk(a ),...,D(a ),...,αk(a )] = D′([a ,...,a ]). X i−1 1 i n 1 n Thus D ∈ QDer(T ). Therefore, g − f = ϕ(D) ∈ ϕ(QDer(T )), so Der(T˘) ⊆ ϕ(QDer(T )) + ZDer(T˘). By Lemma 25 (3) we have Der(T˘)= ϕ(QDer(T )) + ZDer(T˘). For any f ∈ ϕ(QDer(T )) ∩ ZDer(T˘) there exists an element D ∈ QDer(T ) such that f = ϕ(D). Then f(at + utn + btn)= ϕ(D)(at + utn + btn)= D(a)t + D′(b)tn, where a ∈ T,b ∈ [T,...,T ]. On the other hand, since f ∈ ZDer(T˘), we have f(at + btn + utn) ∈ Ann(T˘)= Ttn. That is, D(a)=0, for all a ∈ T and so D =0. Hence f =0. Therefore Der(T˘)= ϕ(QDer(T )) ⊕ ZDer(T˘) as desired. ✷ 10

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