Generalized Derivations of Multiplicative n-Ary Hom-Omega Color Algebras

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arXiv:1609.07396v1 [math.RA] 23 Sep 2016

Generalized derivations of multiplicative n-ary Hom-

_{Ω color algebras}

P. D. Beitesa, Ivan Kaygorodovb, Yury Popovc,d

a_{Universidade da Beira Interior, Covilh˜a, Portugal} b_{Universidade Federal do ABC, CMCC, Santo Andr´e, Brasil}

c_{Universidade Estadual de Campinas, IMECC-UNICAMP, Campinas, Brasil} d_{Novosibirsk State University, Novosibirsk, Russia}

E-mail addresses:

Patr´ıcia Damas Beites (pbeites@ubi.pt), Ivan Kaygorodov (kaygorodov.ivan@gmail.com), Yury Popov (yuri.ppv@gmail.com).

Abstract. We generalize the results of Leger and Luks, Zhang R. and Zhang Y.; Chen, Ma, Ni, Niu, Zhou and Fan;

Kay-gorodov and Popov (see, [6, 28, 31, 41, 44–49]) about generalized derivations of colorn-ary algebras to the case of n-ary Hom-Ω

color algebras. Particularly, we prove some properties of generalized derivations of multiplicativen-ary Hom-Ω color algebras.

Moreover, we prove that the quasiderivation algebra of any multiplicativen-ary Hom-Ω color algebra can be embedded into the

derivation algebra of a larger multiplicativen-ary Hom-Ω color algebra.

Keywords: generalized derivation, color algebra,Hom-algebra, Hom-Lie superalgebra, n-ary algebra.

INTRODUCTION

It is well known that the algebras of derivations and generalized derivations are very important in the study of Lie algebras and its generalizations. There are many generalizations of derivations (for example, Leibniz derivations [27] and Jordan derivations [15]). The notion of aδ-derivation appeared in the paper of Filippov [11], in which he studied δ-derivations of prime Lie and

Malcev algebras [12, 13]. After that,δ-derivations of Jordan and Lie superalgebras were studied in [18–21, 26, 43, 50] and many

other works. The notion of a generalized derivation is a generalization ofδ-derivation. The most important and systematic

research on the generalized derivations algebras of a Lie algebra and their subalgebras was due to Leger and Luks [31]. In their article, they studied properties of generalized derivation algebras and their subalgebras, for example, the quasiderivation algebras. They have determined the structure of algebras of quasiderivations and generalized derivations and proved that the quasiderivation algebra of a Lie algebra can be embedded into the derivation algebra of a larger Lie algebra. Their results were generalized by many authors. For example, Zhang and Zhang [41] generalized the above results to the case of Lie superalgebras; Chen, Ma, Ni, Zhou and Fan considered generalized derivations of color Lie algebras,Hom-Lie superalgebras, Lie triple systems, Hom-Lie

triple systems and n-Hom Lie superalgebras [6, 29, 44–49]. Generalized derivations of color n-aryΩ-algebras were studied

in [28].

Generalized derivations of simple algebras and superalgebras were investigated in [16, 30, 38, 39]. P´erez-Izquierdo and Jim´enez-Gestal used generalized derivations to study non-associative algebras [17, 32]. Derivations and generalized deriva-tions ofn-ary algebras were considered in [3, 5, 22–25, 34, 40] and other works. For example, Williams proved that, unlike the

case of binary algebras, for anyn≥ 3 there exist a non-nilpotent n-Lie algebra with invertible derivation [40], Kaygorodov

de-scribed(n+ 1)-ary derivations of simple n-ary Malcev algebras [24] and generalized derivations algebras of semisimple Filippov

algebras over an algebraically closed field of characteristic zero [25].

The study ofHom-structures was started in the classic paper of Hartwig, Larsson and Silvestrov [14]. After Hom-Lie

algebras were studied some questions related toHom-Lie bialgebras [37], Hom-Lie color algebras [1], Hom-alternative and Hom-Malcev algebras [9], ternary Hom-Nambu-Lie algebras [2], Hom-Sabinin algebras [7] and many others.

The main purpose of our work is to generalize the results of Leger and Luks [31]; Zhang R. and Zhang Y. [41]; Chen, Ma, Ni, Niu, Zhou and Fan [6, 44–49]; Kaygorodov and Popov [28] to the case of multiplicativen-ary Hom-Ω color algebras for an

arbitrary variety ofHom-identities Hom-Ω. Particularly, we prove some properties of generalized derivations of multiplicative n-ary Hom-Ω color algebras; prove that the quasiderivation algebra of a multiplicative n-ary Hom-Ω color algebra can be

embedded into the derivation algebra of a larger multiplicativen-ary Hom-Ω color algebra.

1. PRELIMINARIES

In this section we consider some well knownHom-algebraic structures that are twisted versions of the original algebraic

structures. In fact, they are (n-ary) algebras where the identities defining the structure are twisted by homomorphisms, usually

called twisting maps. All of the followingHom-algebras are particular cases of n-ary Hom-Ω color algebras that we consider

in the present work.

Definition 1. Let F be a field and G be an abelian group. A mapǫ: G × G → F∗_{is called a bicharacter on G if the following}

relations hold for allf, g, h∈ G : (1)ǫ(f, g + h) = ǫ(f, g)ǫ(f, h); (2)ǫ(g + h, f ) = ǫ(g, f )ǫ(h, f ); (3)ǫ(g, h)ǫ(h, g) = 1.

1

_{The authors were supported by Fundação para a Ciência e a Tecnologia (Portugal), project PEst-OE/MAT/UI0212/2015 of CMA-UBI and by Ministerio}

de Econom´ıa y Competitividad (Spain), project MTM2013-45588-C3-1-P and by RFBR 16-31-00096.

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Definition 2. A colorn-ary algebra T is an n-ary G-graded vector space T =L

g∈GTgwith a gradedn-linear map[·, . . . , ·] :

T× . . . × T → T satisfying

[Tθ1, . . . , Tθn] ⊆ Tθ1+...+θn, θi∈ G.

The main examples of colorn-ary algebras are color Lie algebras [6], color Leibniz algebras [8], Filippov (n-Lie)

superalge-bras [4, 5, 33, 35, 36] and3-Lie color algebras [42].

LetT =L

g∈GTgbe a color algebra. An elementx is called a homogeneous element of degree t∈ G if x ∈ Tt. We denote

this byhg(x) = t. A linear map D is homogeneous of degree t if D(Tg) ⊆ Tg+tfor allg∈ 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.

Letǫ be a bicharacter on G. For two homogeneous elements a and b we set ǫ(a, b) := ǫ(hg(a), hg(b)).

LetD1, D2be linear maps onT and x1, . . . , xn ∈ T. We use the following notations:

[D1, D2] = D1D2− ǫ(D1, D2)D2D1,

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

In particular, we setX0= 0.

ByEnd(T ) we denote the set of all linear maps of T . Obviously, End(T ) =L

g∈GEndg(T ) endowed with the color bracket

(1) is a color Lie algebra over F.

1.1. BinaryHom-algebras. In this subsection we give some definitions of Hom-algebras with a binary multiplication. It is

known that for any family of polynomial identitiesΩ, 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

andHom-Ω algebras are particular cases of the former ones for G = Z2 and G = {0}, respectively. On the other hand, the

definitions ofΩ-algebras, Ω-superalgebras and color Ω-algebrascan be obtained from suitable Hom-definitions for α = id.

Definition 3. A colorHom-Lie algebra(T, [·, ·], ǫ, α) is a G-graded vector space T =L

g∈GTgwith a bicharacterǫ, an even

bilinear map[·, ·] : T × T → T and an even linear map α : T → T satisfying [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.

Definition 4. A colorHom-Jordan algebra(T, [·, ·], ǫ, α) is a G-graded vector space T =L

g∈GTgwith a bicharacterǫ, an

even bilinear map[·, ·] : T × T → T and an even linear map α : T → T satisfying [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 mapasT : 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

mul-tiplication. In general (see, [7]), for the definition ofn-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.

Definition 5. A multiplicativen-ary Hom-Lie color algebra(T, [·, . . . , ·], ǫ, α) is a G-graded vector space T =L

g∈GTgwith

ann-linear map[·, . . . , ·] : T × . . . × T → T and an even linear map α : T → T satisfying [x1, . . . , xi, xi+1, . . . , xn] = −ǫ(xi, xi+1)[x1, . . . , xi+1, xi, . . . , xn],

[α(x1), . . . , α(xn−1), [y1, . . . , yn]] = n

X

i=1

ǫ(Xn−1, Yi−1)[α(y1), . . . , α(yi−1), [x1, . . . , xn−1, yi], α(yi+1), . . . , α(yn)].

Definition 6. A multiplicative n-ary Hom-associative color algebra (T, [·, . . . , ·], ǫ, α) is a G-graded vector space T = L

g∈GTgwith ann-linear map[·, . . . , ·] : T × . . . × T → T and an even linear map α : T → T satisfying

[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 ofn-ary multiplicative Hom-Ω color algebra for arbitrary family of polynomial identities Ω.

Definition 7. For a (mayben-ary) multilinear polynomial f(x1, . . . , xn) we fix the order of indices {i1, . . . , in} of one

non-associative word[xi1. . . xin]β from the polynomialf . Here, f =

P

β,σ∈Sn

αβ,σ[xσ(i1). . . xσ(in)]β, where β is an arrangement

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)]βits order of indexes is a composition of suitable shifts

µi, and for this word we setǫσdefined as the product of correspondingǫ(xji, xji+1). Now, for the multilinear polynomial f , we

define the color multilinear polynomial

fco=

X

β,σ∈Sn

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To construct a color multiplicative multilinearHom-polynomial fHom

co from a color multilinear polynomialfco we use the

algorithm from [7] and we are changing all free lettersxjtoα(xj) in all words of the color multilinear polynomial fco. Definition 8. LetΩ = {fi} be a family of n-ary multilinear polynimials. Then

(1) A colorn-aryΩ-algebra L is a color n-ary algebra satisfying the family of color multilinear polinomials Ωco= {(fi)co} .

(2) A multiplicativen-ary Hom-Ω color algebra T is a color n-ary algebra satisfying the family of color multiplicative multilinearHom-polynomialsΩHom

co =(fi)Homco for a homomorphism α.

1.3. Linear maps onHom-algebras. From now on, we denote a multiplicative n-ary Hom-Ω color algebra (T, [·, . . . , ·], ǫ, α)

byT.

Definition 9. A G-graded subspaceM ⊆ T is a color Hom-subalgebra of T if α(M ) ⊆ M and [M, . . . , M ] ⊆ M. A G-graded subspaceI⊆ T is a color Hom-ideal of T if α(I) ⊆ I and [T, . . . , I, . . . , T ] ⊆ I.

Definition 10. The vector subspaceS ofEnd(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. Letξ∈ G.

Definition 11. D∈ Endξ(T) is a homogeneous αk-derivation ofT , with k∈ N if it satisfies

D([x1, . . . , xn]) = n

X

i=1

ǫ(D, Xi−1)[αk(x1), . . . , αk(xi−1), D(xi), αk(xi+1), . . . , αk(xn)].

We denote the set of allαk_{-derivations of}_{T by}_Der

αk(T ). Notice that Der(T ) :=L_k≥0Der_αk(T ) equipped with the color

bracket defined by (1) and the even mapα˜: Der(T ) → Der(T ) defined by ˜α(D) = Dα, is a color Hom-subalgebra of S, which

is called the derivation algebra ofT .

Definition 12. D ∈ Endξ(T) is a homogeneous generalized αk-derivation of degree ξ of T if there exist endomorphisms

D(i) ∈ Endξ(T ) satisfying

[D(x1), αk(x2), . . . , αk(xn)] + n

X

i=2

ǫ(D, Xi−1)[αk(x1), . . . , αk(xi−1), D(i−1)(xi), αk(xi+1), . . . , αk(xn)]

= D(n)([x1, . . . , xn]).

Definition 13. D∈ Endξ(T) is a homogeneous αk-quasiderivation of degreeξ of T if there exists D′∈ Endξ(T ) satisfying n

X

i=1

ǫ(D, Xi−1)[αk(x1), . . . , αk(xi−1), D(xi), αk(xi+1), . . . , αk(xn)] = D′([x1, . . . , xn]).

Let GDerαk(T ) and QDer_αk(T ) be the sets of homogeneous generalized αk-derivations and of homogeneous αk

-quasiderivations, respectively. We denote

GDer(T ) :=L

k≥0GDerαk(T ), QDer(T ) :=L_k≥0QDer_αk(T ).

Definition 14. D∈ Endξ(T ) is a homogeneous αk-centroid ofT , where k∈ N if it satisfies

D([x1, . . . , xn]) = ǫ(D, Xi−1)[αk(x1), . . . , αk(xi−1), D(xi), αk(xi+1), . . . , αk(xn)]

for alli.

Definition 15. D∈ Endξ(T ) is a homogeneous αk-quasicentroid of degreeξ of T , where k∈ N if it satisfies

[D(x1), αk(x2), . . . , αk(xn)] = ǫ(D, Xi−1)[αk(x1), . . . , αk(xi−1), D(xi), αk(xi+1), . . . , αk(xn)]

for alli.

Denote the sets of homogeneousαk_{-centroids and homogeneous}_αk_{-quasicentroids by}_C

αk(T ) and QC_αk(T ), respectively.

Denote also

C(T ) :=L

k≥0Cαk(T ), QC(T ) :=L_k≥0QC_αk(T ).

Definition 16. D∈ Endξ(T) is a homogeneous αk-center derivation of degreeξ of T if it satisfies

D([x1, . . . , xn]) = [αk(x1), . . . , αk(xi−1), D(xi), αk(xi+1), . . . , αk(xn)] = 0.

Denote the set of homogeneousαk_{-center derivations by}_ZDer

αk(T ). We also denote

ZDer(T ) :=L

k≥0ZDerαk(T ).

Notice that we have the following chain of inclusions:

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2. GENERALIZED DERIVATION ALGEBRAS AND THEIR COLORHom-SUBALGEBRAS

In this section we present some basic properties of generalized derivations, quasiderivations and center derivations of a multi-plicativen-ary Hom-Ω color algebra.

Lemma 17. LetT 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 ), D_η ∈ GDer_αs(T ), where k, s ∈ N. For arbitrary x₁, . . . , x_n∈ T we have

[˜α(Dξ)(x1), αk+1(x2), . . . , αk+1(xn)] = [(Dξα)(x1), αk+1(x2), . . . , αk+1(xn)] = α([Dξ(x1), αk(x2), . . . , αk(xn)])

= α(D(n)_ξ ([x1, x2, . . . , xn]) − n

P

i=2

ǫ(Dξ, Xi−1)[αk(x1), . . . , αk(xi−1), D(i−1)_ξ (xi), αk(xi+1), . . . , αk(xn)])

= ˜α(D_ξ(n))([x1, x2, . . . , xn]) − n

P

i=2

ǫ(Dξ, Xi−1)[αk+1(x1), . . . , αk+1(xi−1), ˜α(D(i−1)_ξ )(xi), αk+1(xi+1), . . . , αk+1(xn)].

For anyi ∈ N, ˜α(D(i)_ξ ) belongs to Endξ(T ), thus ˜α(Dξ) ∈ GDerαk+1(T ) and is obviously of degree ξ. For arbitrary

x1, . . . , xn∈ T we obtain [DξDη(x1), αk+s(x2), . . . , αk+s(xn)] = D(n)_ξ ([Dη(x1), αs(x2), . . . , αs(xn)]) − n P i=2

ǫ(Dξ, Dη+ Xi−1)[αk(Dη(x1)), αk+s(x2), . . . , αk+s(xi−1), D_ξ(i−1)(αs(xi)), αk+s(xi+1), . . . , αk+s(xn)]

= D(n)_ξ D(n)η ([x1, . . . , xn]) − n P j=2 ǫ(Dη, Xj−1)D_ξ(n)[αs(x1), . . . , αs(xj−1), D(j−1)η (xj), αs(xj+1), . . . , αs(xn)] −Pn i=2

ǫ(Dξ, Dη+ Xi−1)D(n)η [αk(x1), . . . , αk(xi−1), Dξ(i−1)(xi), αk(xi+1), . . . , αk(xn)]

+ Pn i=2,j=2,j<i ǫ(Dξ, Dη+ Xi−1)ǫ(Dη, Xj−1)[αk+s(x1), . . . , Dη(j−1)(αk(xj)), . . . , D(i−1)_ξ (αs(xi)), . . . , αk+s(xn)] + n P i=2,j=2,i<j ǫ(Dξ, Xi−1)ǫ(Dη, Xj−1)[αk+s(x1), . . . , D(i−1)_ξ (αs(xi)), . . . , D (j−1) η (αk(xj)), . . . , αk+s(xn)] +Pn i=2

ǫ(Dξ, Dη+ Xi−1)ǫ(Dη, Xi−1)[αk+s(x1), . . . , αk+s(xi−1), Dη(i−1)D(i−1)ξ (xi), αk+s(xi+1), . . . , αk+s(xn)].

Thus, for arbitraryx1, . . . , xn ∈ T , we have

[[Dξ, Dη](x1), αk+s(x2), . . . , αk+s(xn)] = [(DξDη− ǫ(Dξ, Dη)DηDξ)(x1), αk+s(x2), . . . , αk+s(xn)]

= [D(n)_ξ , D(n)η ]([x1, . . . , xn]) − n

P

i=2

ǫ(Dξ+ Dη, Xi−1)[αk+s(x1), . . . , [D_ξ(i−1), Dη(i−1)](xi), . . . , αk+s(xn)].

For all 2 ≤ i ≤ n, [D(i−1)_ξ , D(i−1)η ] ∈ Endξ+η(T ). Therefore [D(n)_ξ , D(n)η ] ∈ GDerαk+s(T ) and GDer(T ) is a color

Hom-subalgebra of S.

Similarly,QDer(T ) is a color Hom-subalgebra of S.

LetDξ∈ Cαk(T ), D_η∈ C_αs(T ). For arbitrary x₁, . . . , x_n ∈ T we have

˜

α(Dξ)([x1, . . . , xn]) = αDξ([x1, . . . , xn]) = α([Dξ(x1), αk(x2), . . . , αk(xn)]) = [˜α(Dξ)(x1), αk+1(x2), . . . , αk+1(xn)]

and

[˜α(Dξ)(x1), αk+1(x2), . . . , αk+1(xn)] = α([Dξ(x1), αk(x2), . . . , αk(xn)])

= ǫ(Dξ, Xi−1)α([αk(x1), . . . , αk(xi−1), Dξ(xi), αk(xi+1), . . . , αk(xn)])

= ǫ(Dξ, Xi−1)[αk+1(x1), . . . , αk+1(xi−1), ˜α(Dξ)(xi), αk+1(xi+1), . . . , αk+1(xn)].

Thus,α(D˜ ξ) ∈ Cαk+1(T ). Notice that [[Dξ, Dη](x1), αk+s(x2), . . . , αk+s(xn)] = [DξDη(x1), αk+s(x2), . . . , αk+s(xn)] − ǫ(Dξ, Dη)[DηDξ(x1), αk+s(x2), . . . , αk+s(xn)] = DξDη([x1, x2, . . . , xn]) − ǫ(Dξ, Dη)DηDξ([x1, x2, . . . , xn]) = [Dξ, Dη]([x1, x2, . . . , xn]). Similarly, we have ǫ(Dξ+ Dη, Xi−1)[αk+s(x1), . . . , [Dξ, Dη](xi), . . . , αk+s(xn)] = [Dξ, Dη]([x1, x2, . . . , xn]).

Thus,[Dξ, Dη] ∈ Cαk+s(T ) of degree ξ + η and C(T ) is a color Hom-subalgebra of S.

(2) LetDξ ∈ ZDerαk(T ), D_η∈ Der_αs(T ). For arbitrary x₁, . . . , x_n ∈ T we have

[˜α(Dξ)(x1), αk+1(x2), . . . , αk+1(xn)] = [(αDξ)(x1), αk+1(x2), . . . , αk+1(xn)]

= α([Dξ(x1), αk(x2), . . . , αk(xn)]) = αDξ([x1, x2, . . . , xn]) = ˜α(Dξ)([x1, x2, . . . , xn]) = 0.

So,α(D˜ ξ) ∈ ZDerαk+1(T ).

Observe that

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= Dξ([Dη(x1), αs(x2), . . . , αs(xn)]) + n

P

i=2

ǫ(Dη, Xi−1)Dξ([αs(x1), . . . , αs(xi−1), Dη(i−1)(xi), αs(xi+1), . . . , αs(xn)]) =

0 and [[Dξ, Dη](x1), αk+s(x2), . . . , αk+s(xn)] = [DξDη(x1), αk+s(x2), . . . , αk+s(xn)] − ǫ(Dξ, Dη)[DηDξ(x1), αk+s(x2), . . . , αk+s(xn)] = −ǫ(Dξ, Dη)[Dη(Dξ(x1)), αk+s(x2), . . . , αk+s(xn)] = −ǫ(Dξ, Dη)Dη([Dξ(x1), αk(x2), . . . , αk(xn)]) + n P i=2 ǫ(Dξ, Dη)ǫ(Dη, Dξ+ Xi−1)[αs(Dξ(x1)), αk+s(x2), . . . , D(i−1)η (αk(xi)), . . . , αk+s(xn)] = 0.

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. LetT 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 ), D_η∈ C_αs(T ). For arbitrary x₁, . . . , x_n ∈ T we have

[DξDη(x1), αk+s(x2), . . . , αk+s(xn)] = Dξ([Dη(x1), αs(x2), . . . , αs(xn)]) − n P i=2 ǫ(Dξ, Dη+ Xi−1)[Dη(αk(x1)), αk+s(x2), . . . , Dξ(αs(xi)), . . . , αk+s(xn)] = DξDη([x1, x2, . . . , xn]) − n P i=2 ǫ(Dξ, Dη)ǫ(Dξ+ Dη, Xi−1)[αk+s(x1), . . . , DηDξ(xi), . . . , αk+s(xn)] and [DηDξ(x1), αk+s(x2), . . . , αk+s(xn)] = Dη([Dξ(x1), αk(x2), . . . , αk(xn)]) = DηDξ([x1, x2, . . . , xn]) − n P i=2 ǫ(Dξ, Xi−1)Dη([αk(x1), . . . , Dξ(xi), . . . , αk(xn)]) = DηDξ([x1, x2, . . . , xn]) − n P i=2 ǫ(Dξ+ Dη, Xi−1)[αk+s(x1), . . . , DηDξ(xi), . . . , αk+s(xn)]. Hence [[Dξ, Dη](x1), αk+s(x2), . . . , αk+s(xn)] = [DξDη(x1), αk+s(x2), . . . , αk+s(xn)] − ǫ(Dξ, Dη)[DηDξ(x1), αk+s(x2), . . . , αk+s(xn)] = [Dξ, Dη]([x1, . . . , xn]). Similarly, [[Dξ, Dη](x1), αk+s(x2), . . . , αk+s(xn)] = ǫ(Dξ+ Dη, Xi−1)[αk+s(x1), . . . , [Dξ, Dη](xi), . . . , αk+s(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 ), D_η ∈ Der_αs(T ). For arbitrary x₁, . . . , x_n∈ T we have

DξDη([x1, . . . , xn]) = Dξ( n P i=1 ǫ(Dη, Xi−1)[αs(x1), . . . , Dη(xi), . . . , αs(xn)]) = n P i=1 ǫ(Dξ+ Dη, Xi−1)[αk+s(x1), . . . , DξDη(xi), . . . , αk+s(xn)].

ThereforeDξDη∈ Derαk+s(T ) and is of degree ξ + η. Hence C(T ) · Der(T ) ⊆ Der(T ).

(4) Let Dξ∈ Cαk(T ). For arbitrary x₁, . . . , x_n ∈ T we have

Dξ([x1, . . . , xn]) = ǫ(Dξ, Xi−1)[αk(x1), . . . , Dξ(xi), . . . , αk(xn)]. Hence n P i=1 ǫ(Dξ, Xi−1)[αk(x1), . . . , Dξ(xi), . . . , αk(xn)] = nDξ[x1, . . . , xn].

ThereforeDξ∈ QDerαk(T ) since D ′

= nDξ∈ Cαk(T ).

(5) Let Dξ∈ QCαk(T ), D_η ∈ QC_αs(T ). For arbitrary x1, . . . , xn∈ T we have

[αk+s_(x 1), . . . , [Dξ, Dη](xi), . . . , αk+s(xn)] = ǫ(Xi−1, Dξ)ǫ(Xi−1− X1, Dη)[Dξ(αs(x1)), Dη(αk(x2)), . . . , αk+s(xi), . . . , αk+s(xn)] −ǫ(Dξ, Dη)ǫ(Xi−1− X1, Dη)ǫ(Xi−1+ Dη, Dξ)[Dξ(αs(x1)), Dη(αk(x2)), . . . , αk+s(xi), . . . , αk+s(xn)] = 0. Hence n P i=1 ǫ(Dξ+ Dη, Xi−1)[αk+s(x1), . . . , [Dξ, Dη](xi), . . . , αk+s(xn)] = 0.

therefore[Dξ, Dη] ∈ QDerαk+s(T ) and is of degree ξ + η.

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Lemma 19. IfT is a multiplicative n-ary Hom-Ω color algebra, then QC(T ) + [QC(T ), QC(T )] is a color Hom-subalgebra ofGDer(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 colorHom-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. LetT 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 thatDξ ∈ Cαk(T ), D_η ∈ QC_αs(T ) and x ∈ T . As α is surjective, for any y′_i ∈ T, 1 ≤ i ≤ n, there exists

yi∈ T such that yi′= αk+s(yi). We have

ǫ(Yi−1, Dξ+ Dη)[αk+s(y1), . . . , [Dξ, Dη](x), . . . , αk+s(yn)] =

ǫ(Yi−1, Dξ+ Dη)[αk+s(y1), . . . , (DξDη− ǫ(Dξ, Dη)DηDξ)(x), . . . , αk+s(yn)] =

Dξ([Dη(y1), αs(y2), . . . , αs(x), . . . , αs(yn)] − [Dη(y1), αs(y2), . . . , αs(x), . . . , αs(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 characteristic6= 2 and (S, •, ˜α) be a color Hom-algebra with multiplication Dξ• Dη= 12(DξDη+ ǫ(Dξ, Dη)DηDξ)

whereDξ, Dηareα-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)

=1₂ǫ(Dξ, Dη)(DηDξ+ ǫ(Dη, Dξ)DξDη)

= ǫ(Dξ, Dη)Dη• Dξ.

Consider now the secondHom-Jordan identity: ((Dξ• Dη) • α(Dθ)) • α2(Dγ) =1₂((DξDη+ ǫ(Dξ, Dη)DηDξ) • α(Dθ)) • α2(Dγ) =1 4((DξDη+ ǫ(Dξ, Dη)DηDξ)α(Dθ) + ǫ(Dξ+ Dη, Dθ)α(Dθ)(DξDη+ ǫ(Dξ, Dη)DηDξ)) • α 2_(D γ) =1 8(DξDηα(Dθ)α 2_(D γ) + ǫ(Dξ, Dη)DηDξα(Dθ)α2(Dγ) + ǫ(Dξ+ Dη, Dθ)α(Dθ)DξDηα2(Dγ) +ǫ(Dξ+ Dη, Dθ)ǫ(Dξ, Dη)α(Dθ)DηDξα2(Dγ) + ǫ(Dξ+ Dη+ Dθ, Dγ)α2(Dγ)DξDηα(Dθ) +ǫ(Dξ, Dη)ǫ(Dξ+ Dη+ Dθ, Dγ)α2(Dγ)DηDξα(Dθ) + ǫ(Dξ+ Dη, Dθ)ǫ(Dξ+ Dη+ Dθ, Dγ)α2(Dγ)α(Dθ)DξDη +ǫ(Dξ+ Dη, Dθ)ǫ(Dξ, Dη)ǫ(Dξ+ Dη+ Dθ, Dγ)α2(Dγ)α(Dθ)DηDξ).

On the other hand, we have

α(Dξ• Dη) • (α(Dθ) • α(Dγ)) =1 4α(DξDη+ ǫ(Dξ, Dη)DηDξ) • (α(Dθ)α(Dγ) + ǫ(Dθ, Dγ)α(Dγ)α(Dθ)) =1₈(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(DηDξ)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(DηDξ)). So, ǫ(Dγ, Dξ+ Dθ)asα(Dξ• Dη, α(Dθ), α(Dγ)) =1₈ǫ(Dγ, Dξ+ Dθ)(ǫ(Dξ+ Dη, Dθ)α(Dθ)DξDηα2(Dγ) + ǫ(Dξ, Dη)ǫ(Dξ+ Dη, Dθ)α(Dθ)DηDξα2(Dγ) +ǫ(Dξ+ Dη+ Dθ, Dγ)α2(Dγ)DξDηα(Dθ) + ǫ(Dξ, Dη)ǫ(Dξ+ Dη+ Dθ, Dγ)α2(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.

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(2) We only need to show that for arbitrary Dξ, Dη∈ QC(T ), Dξ• Dη∈ QC(T ). We have [Dξ• Dη(x1), αk+s(x2), . . . , αk+s(xn)] =1₂[DξDη(x1), αk+s(x2), . . . , αk+s(xn)] +21ǫ(Dξ, Dη)[DηDξ(x1), αk+s(x2), . . . , αk+s(xn)] =1 2ǫ(Dξ, Dη+ Xi−1)[Dη(αk(x1)), α k+s_(x 2), . . . , Dξ(xi), . . . , αk+s(xn)] +1 2ǫ(Dη, Xi−1)[Dξ(αs(x1)), α k+s_(x 2), . . . , Dη(xi), . . . , αk+s(xn)] =1₂ǫ(Dξ, Dη+ Xi−1)ǫ(Dη, Xi−1)[αk+s(x1), . . . , DηDξ(xi), . . . , αk+s(xn)] +1₂ǫ(Dξ+ Dη, Xi−1)[αk+s(x1), . . . , DξDη(xi), . . . , αk+s(xn)] = ǫ(Dξ+ Dη, Xi−1)[αk+s(x1), . . . , Dξ• Dη(xi), . . . , αk+s(xn)].

ThenDξ• Dη∈ QC(T ) and QC(T ) is a color Hom-Jordan algebra. ✷ Theorem 23. LetT 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)DηDξif and only ifQC(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 ), D_η ∈ QC_αs(T ), we have D_ξD_η ∈ QC_αk+s(T ) and D_ηD_ξ ∈ QC_αk+s(T ).

So,[Dξ, Dη] = DξDη− ǫ(Dξ, Dη)DηDξ ∈ QCαk+s(T ). Hence, QC(T ) is a color Hom-Lie algebra.

(⇒) Note that DξDη= Dξ• Dη+[Dξ₂,Dη]. By Lemma 22, we haveDξ• 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

[[Dξ, Dη](x1), αk+s(x2), . . . , αk+s(xn)]

= ǫ(Dξ, Dη+ Xi−1)ǫ(Dη, Xi−1)[αk+s(x1), . . . , DηDξ(xi), . . . , αk+s(xn)]

−ǫ(Dξ+ Dη, Xi−1)[αk+s(x1), . . . , DξDη(xi), . . . , αk+s(xn)] =

−ǫ(Dξ+ Dη, Xi−1)[αk+s(x1), . . . , [Dξ, Dη](xi), . . . , αk+s(xn)].

On the other hand, sinceQC(T ) is a color Hom-Lie algebra, then, for arbitrary x1, . . . , xn∈ T , we have

[[Dξ, Dη](x1), αk+s(x2), . . . , αk+s(xn)] = ǫ(Dξ+ Dη, Xi−1)[αk+s(x1), . . . , [Dξ, Dη](xi), . . . , αk+s(xn)].

Therefore

[αk+s(x1), . . . , [Dξ, Dη](xi), . . . , αk+s(xn)] = 0, hence [x′1, . . . ,[Dξ, Dη](xi), . . . , x′n] = 0.

Hence,[Dξ, Dη] = 0.

(⇐) Is clear. ✷

3. QUASIDERIVATIONS OF MULTIPLICATIVEn-ARYHom-ΩCOLOR ALGEBRAS

In this section we investigate the quasiderivations of the multiplicativen-ary Hom-Ω color algebra T. We prove that QDer(T )

can be embedded in the derivation algebra of a larger multiplicativen-ary Hom-Ω color algebra ˘T for the same variety of

polynomialHom-identitiesΩ. Moreover, we conclude that Der( ˘T) has a direct sum decomposition when Ann(T ) = {0}.

Lemma 24. LetT 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) ⊗ t}n_{) | x, y ∈ T }.}

Endow( ˘T ,α) with the multiplication˘

[x1⊗ ti1, x2⊗ ti2, . . . , xn⊗ tin] = [x1, x2, . . . , xn] ⊗ t P

ij_,

fori1, . . . , in∈ {1, n} (we put tk= 0 if k > n).

Then( ˘T ,α) is a multiplicative n-ary Hom-Ω color algebra.˘

Proof. Let the class ofn-ary Hom-Ω color algebras be defined by the family {fk} of color multilinear Hom-identities.

Then, for arbitraryx1, x2, . . . , xm∈ T and ij∈ {1, n} we have

fj(x1⊗ ti1, x2⊗ ti2, . . . , xm⊗ tim) = fj(x1, x2, . . . , xm) ⊗ t P

il _{= 0.}

Therefore ˘T is a n-ary Hom-Ω color algebra. ✷

For the sake of convenience, we writext(xtn_{) instead of x ⊗ t (x ⊗ t}n_).

IfU is a G-graded subspace of T such that T = U ⊕ [T, . . . , T ], then ˘

T = T t + T tn= T t + U tn+ [T, . . . , T ]tn.

Now define a mapϕ: QDer(T ) → End( ˘T) by

ϕ(D)(at + utn_{+ bt}n_{) = D(a)t + D}′_(b)tn_,

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Theorem 25. LetT, ˘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 ofϕ.

(2) Ifϕ(D1) = ϕ(D2), then for all a ∈ T, b ∈ [T, . . . , T ] and u ∈ U we have

ϕ(D1)(at + utn+ btn) = ϕ(D2)(at + utn+ btn),

or, in terms ofD1, D2,

D1(a)t + D1′(b)tn= D2(a)t + D′2(b)tn,

soD1(a) = D2(a). Hence D1= D2, and ϕ is injective.

Suppose that there existsD′′_{such that}

ϕ(D)(at + utn_{+ bt}n_{) = D(a)t + D}′′_(b)tn_, and X ǫ(D, Xi−1)[αk(x1), . . . , D(xi), . . . , αk(xn)] = D′′([x1, . . . , xn]), then we have D′([x1, . . . , xn]) = D′′([x1, . . . , xn]), thusD′_{(b) = D}′′_{(b). Hence}

ϕ(D)(at + utn+ btn) = D(a)t + D′(b)tn= D(a)t + D′′(b)tn,

which implies thatϕ(D) is determined only by D.

(3) We have[x1ti1, . . . , xntin] = [x1, . . . , xn]t P

ij _{= 0 for all}_{P i}

j≥ n + 1. Thus, to show that ϕ(D) ∈ Der( ˘T), we only

need to check that the following equality holds:

ϕ(D)([x1t, . . . , xnt]) =

X

ǫ(D, Xi−1)[˘αk(x1t), . . . , ϕ(D)(xit), . . . , ˘αk(xnt)].

For arbitraryx1, . . . , xn∈ T we have

ϕ(D)([x1t, . . . , xit, . . . , xnt]) = ϕ(D)([x1, . . . , xn]tn) = D′([x1, . . . , xn])tn

=Xǫ(D, Xi−1)[αk(x1), . . . , D(xi), . . . , αk(xn)]tn

=Xǫ(D, Xi−1)[αk(x1t), . . . , D(xi)t, . . . , αk(xnt)]

=Xǫ(D, Xi−1)[˘αk(x1t), . . . , ϕ(D)(xit), . . . , ˘αk(xnt)].

Therefore, for allD∈ QDer(T ) we have ϕ(D) ∈ Der( ˘T). ✷

Lemma 26. LetT 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. SinceAnn(T ) = {0}, we have Ann( ˘T) = T tn_{. For g}_{∈ Der( ˘}_T_{) we have g(Ann( ˘}_T_{)) ⊆ Ann( ˘}_T_{), hence g(U t}n_{) ⊆}

g(Ann( ˘T)) ⊆ Ann( ˘T) = T tn_{. Now define a map f}_{: T t + U t}n_{+ [T, . . . , T ]t}n _{→ T t}n_by

f(x) =    g(x) ∩ T tn_, _x_{∈ T t;} g(x), x∈ U tn_; 0, x∈ [T, . . . , T ]tn_.

It is clear thatf is linear. Note that

f([ ˘T , . . . , ˘T]) = f ([T, . . . , T ]tn_{) = 0,}

[˘αk( ˘T), . . . , f ( ˘T), . . . , ˘αk( ˘T)] ⊆ [αk_{(T )t + α}k_{(T )t}n_{, . . . , T t}n_{, . . . , α}k_{(T )t + α}k_{(T )t}n_{] = 0,}

hencef ∈ ZDer( ˘T). Since

(g − f )(T t) = g(T t) − g(T t) ∩ T tn= g(T t) − T tn⊆ T t, (g − f )(U tn) = 0,

and

(g − f )([T, . . . , T ]tn) = g([ ˘T , . . . , ˘T]) ⊆ [ ˘T , . . . , ˘T] = [T, . . . , T ]tn,

there existD, D′_{∈ End(T ) such that for all a ∈ T, b ∈ [T, . . . , T ],}

(g − f )(at) = D(a)t, (g − f )(btn_{) = D}′_(b)tn_.

Sinceg− f ∈ Der( ˘T), by the definition of Der( ˘T) we have X

(9)

for alla1, . . . , an ∈ T. Hence

X

ǫ(D, Ai−1)[αk(a1), . . . , D(ai), . . . , αk(an)] = D′([a1, . . . , an]).

ThusD∈ QDer(T ). Therefore, g − f = ϕ(D) ∈ ϕ(QDer(T )), so Der( ˘T) ⊆ ϕ(QDer(T )) + ZDer( ˘T). By Lemma 25 (3) we

haveDer( ˘T) = ϕ(QDer(T )) + ZDer( ˘T).

For anyf ∈ ϕ(QDer(T )) ∩ ZDer( ˘T) there exists an element D ∈ QDer(T ) such that f = ϕ(D). Then f(at + utn_{+ bt}n_{) = ϕ(D)(at + ut}n_{+ bt}n_{) = D(a)t + D}′_(b)tn_,

wherea∈ T, b ∈ [T, . . . , T ].

On the other hand, sincef ∈ ZDer( ˘T), we have

f(at + btn+ utn) ∈ Ann( ˘T) = T tn.

That is,D(a) = 0, for all a ∈ T and so D = 0. Hence f = 0.

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