Generalized Hom Γ -Derivation of n-BiHom Γ -Lie algebra

The purpose of this paper, is to introduce a new concepts which are induced n-Bi-Hom Γ - Lie algebra, Γ -Center, ( 𝜃 1𝑠 , 𝜃 2𝑟 ) Γ -Center, ( 𝜃 1𝑠 , 𝜃 2𝑟 ) Hom Γ - derivation, ( 𝜃 1𝑠 , 𝜃 2𝑟 ) Q-Hom 𝐷𝑒𝑟 Γ , ( 𝜃 1𝑠 , 𝜃 2𝑟 ) Central Hom Γ - derivation, ( 𝜃 1𝑠 , 𝜃 2𝑟 ) Hom Γ - Centroid and give the condition to construct induced n-Bi-Hom Γ -Lie algebra, studied Generalized Hom Γ -derivations on direct sum of ideals and we studied the relation between Hom 𝐷𝑒𝑟 𝜆 (𝑔) , Hom 𝐶𝑒𝑛 𝜆 (𝑔) and Q Hom 𝐷𝑒𝑟 𝜆 (𝑔) , Q Hom 𝐶𝑒𝑛 𝜆 (𝑔) , G Hom 𝐷𝑒𝑟 𝜆 (𝑔) .


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Amine, in [1], introduce n-Bi-Hom Lie algebra and custom to studying a   on an n-Bi-Hom  .For several years an algebras of  and   has been topic about the study by many ℎ.Leger and Luks, in [2], introduced research is more important on the algebras of   of   and those sub algebras, where a writers studied the  and features on an algebras on  , Q Cen of limited dimensional  .The result of Leger and Luks where Generalized by more other researchers on algebras.For instance, Chen and Li, in [3], lesson the   of color- .Zhou and Fan, in [4,5], cases are considered on Hom Lie Color algebras and n-Hom Lie super algebras.Zhou, Niu and Chen, in [6], investigated   on Hom- .Kygorodov and Popov, in [7], find they out   of  n-ary -algebras.For more of a   algebras, which is going to be find in [8,9,10,11,12].Rezaei and Davvaz, in [13], define -algebra. A. Al-Zaiadi and R. Shaheen, in [14] studied more result on -Lie algebra.The purpose of this paper, is to define n-Bi-Hom -Lie algebra, ( 1  , 2  ) Hom -derivation and generalized Hom -derivation on n-BiHom -Lie algebra, ( 1  , 2  ) Q Hom -derivation, ( 1  , 2  ) Central Hom -derivation and ( 1  , 2  ) Centroid Hom -derivation on n-Bi-Hom,.We also reached some results, [    () ,     () ]  ⊆ Q Hom   (g), Studied Generalized derivations on direct sum of ideals.Now, we will recall the followings concepts which are necessary in this paper.

Definition 1.1:-[1] (n-BiHom Lie-algebra)
An n-Bi-Hom-  be a   g equipped a linear-function [.,…,.] linear-functions and such that = for all and A subset is a called sub algebra of if and and , and S is an ideal if S

Definition 1.3:-[1]
The center of is the set of such that For all .A center is ideal on g which symbolize by .

Definition 1.4:-[1]
The center of is the set .
For any
Assume  is the associative  −  on a field .Therefore, for all ƛ ∈  one can create the  −    ƛ ( ).Like a  ,  ƛ ( ) be a same  .A Lie Γ-arch of 2-elements on  ƛ ( ) be defined to be them