Cohomology of Hom-Leibniz algebras via beta-NiJenhuis-Richardson bracket
aa r X i v : . [ m a t h . R A ] J a n Representation and cohomology ofHom–Leibniz algebra via β -Nijenhuis-Richardson bracket Nejib Saadaoui ∗ February 2, 2021
Abstract
In this paper we define the left and right β -Richardson-Nijenhuisbracket. With this bracket we define a right, left and symmetric 2 p -Hom-Leibniz algebras and their cohomology. Moreover with the β -RNbracket we study the extensions and deformations of Hom-Leibnizalgebra. Introduction
In 1993, J.L. Loday introduced Leibniz algebras which are a generalizationof Lie algebras [19, 20]. They are defined by a bilinear bracket which is nolonger skew-symmetric. More precisely, A left Leibniz algebra over a a field K is a K -vector space A with a K -bilinear map [ · , · ] : A × A → A satisfying (cid:2) a, [ b, c ] (cid:3) = (cid:2) [ a, b ] , c (cid:3) + (cid:2) b, [ a, c ] (cid:3) , (0.1)for all a, b, c ∈ A .Similarly, a right Leibniz algebra is defined by the identity : (cid:2) [ a, b ] , c (cid:3) = (cid:2) [ a, c ] , b (cid:3) + (cid:2) a, [ b, c ] (cid:3) . (0.2)A left or right Leibniz algebra in which bracket [ · , · ] is skew-symmetric (al-ternating, if K is of characteristic 2) is a Lie algebra. Notice from (1) that a ∗ Universit´e de Gab`es, Institut Sup´erieur de l’Informatique Medenine, Rue Djerba km3,B.P 283 Medenine 4100.Tunisie. , [email protected] A satisfies, for all a, x, y ∈ A [ a, [ x, y ]] = − [ a, [ y, x ]] , anddually, a right Leibniz algebra satisfies [[ x, y ] , a ] = − [[ y, x ] , a ]. SymmetricLeibniz algebras satisfy [ a, [ x, y ]] = − [[ x, y ] , a ] . (0.3)A representation of right-Leibniz algebras A is a K -module V equipped withtwo actions of A satisfying the following three axioms.[ v, [ x, y ]] = [[ v, x ] , y ] − [[ v, y ] , x ][ x, [ v, y ]] = [[ x, v ] , y ] − [[ x, y ] , v ][ x, [ y, v ]] = [[ x, y ] , v ] − [[ x, v ] , y ]Let V a representation of the right-Leibniz algebra L . The Hochschild coho-mology of the Leibniz algebra ( L, [ · · · , · · · ]) is given by the coboundary map δ : C k ( A, V ) → C k ( A, V ) defined by δ ( f )( a , · · · , a k +1 ) = [ a , f ( a , · · · , a k +1 ) , a s ] + k +1 X s =2 ( − s [ f ( a , · · · , b a s , · · · , a k +1 ) , a s ] , + X ≤ s Im δ k is the space of k -cobord, Z k ( L, V ) = ker δ k is the space of k -cocycle and the quotient H k ( L, V ) = Z k ( L, V ) /B k ( L, V ) is the k -cohomology group of L .An abelian extension of L by V is a short exact sequence of Leibnizalgebras 0 −→ V −→ M −→ L −→ , such that the sequence is split over K (see [21]). Likewise, the cohomologiesof Lie algebra are defined (see [18, 10]).In [23], Nijenhuis and Richardson gives a new approach to the cohomol-ogy of a Lie algebra. The principle is as follows:Suppose that ξ is the set of Lie algebras structures on a vector space M . Thenthere exists a Z -graded Lie algebra (cid:0) Alt ( M ) = ⊕ k ≥ Alt k ( M ) , [ · , · ] (cid:1) such that d ∈ ξ if and only if d ∈ Alt ( M ) such that [ d, d ] = 0. Moreover, if M isequipped with such a d , the Chevalley-Eilenberg coboundary operator of theadjoint operator δ p of the adjoint representation of ( M, d ) is just the action2f d on Alt ( M ) up to a sign. In [2], using this approach, Fialowski andpenkava defined and studied the representation, extensions, deformation ofLie algebras.Makhlouf and Silvestrov introduced the notion of a Hom-Leibniz algebra( L, [ · , · ] , α ) in [22] , which is a natural generalization of Leibniz algebras andHom-Lie algebras. Later, Yong Sheng Cheng [9] studied the representationand the cohomology of right Hom-Leibniz algebras. In this paper we extendedsome results of Lie algebras in [2] to Hom-Leibniz case.The structure of this paper is as follows. After some preliminary defini-tions and notations in Section 2, we define the left (resp. right) β -Richardson-Nijenhuis bracket. With this bracket we redefine the left (resp.right) Hom-Leibniz algebra and their cohomology and we define the symmetric Hom-Leibniz and their cohomology. In section 3 we define and study the de-formation of Hom-Leibniz and Hom-Lie algebras with the β -Richardson-Nijenhuis bracket. In section 4, we will recall classical definition of a Hom-Leibniz algebra extension and we recognize it and the representation, cocy-cles, cobords...with the action of the β -Richardson-Nijenhuis bracket to astructure d ∈ C ( M ) and we study the split extension of Hom-algebras.In this paper, K always denotes any field of characteristic = 2. In this section we briefly recall some of the definitions on Hom-Leibniz andwe give the notations that we will need in the remainder of the paper. Definition 1.1. A Hom-algebra is a triple ( L, [ · , · ] , α ) in which L is a vectorspace, [ · , · ] a bilinear map on L and α : L → L (the twisting map).A Hom-algebra ( L, [ · , · ] , α ) is said to be multiplicative if α ([ x, y ]) = [ α ( x ) , α ( y )] for all x, y in L .By a homomorphism of Hom algebras f : ( L, [ · , · ] , α ) → ( L ′ , [ · , · ] ′ , α ′ ) wemean a linear map from L to L ′ such that f ◦ α = α ′ ◦ f ; (1.1) f ([ x, y ]) = [ f ( x ) , f ( y )] ′ . (1.2) If f is bijective we say that L and L ′ are isomorphic. Definition 1.2. (i) The Hom-algebra ( L, [ · , · ] , α ) is called a left Hom-Leibnizif it satisfy the equation [ α ( x ) , [ y, z ]] = [[ x, y ] , α ( z )] + [ α ( y ) , [ x, z ]] , or all x, y, z ∈ L . (see [3] ).(ii) The Hom-algebra ( L, [ · , · ] , α ) is called a right Hom-Leibniz algebra if itsatisfy the equation [ α ( x ) , [ y, z ]] = [[ x, y ] , α ( z )] − [[ x, z ] , α ( y )] . (1.3) for all x, y, z ∈ L . (see [22] ).(iii) A triple ( L, [ · , · ] , α ) is called a symmetric Hom-Leibniz algebra if it is aleft and a right Hom-Leibniz algebra. Definition 1.3. [17, 4, 22] A Hom-Lie algebra is a triple ( L, [ · , · ] , α ) con-sisting of a K vector space L , a bilinear map [ · , · ] : L × L → L and a K -linear map α : L → L satisfying [ x, y ] = − [ y, x ] , (skew-symmetry) (1.4) (cid:9) x,y,z (cid:2) α ( x ) , [ y, z ] (cid:3) = 0 , (Hom–Jacobi identity) (1.5) for all x, y, z ∈ L . Clearly Hom-Lie algebras and Leibniz algebras are examples of Hom-Leibniz algebras.In this paper we mean by Hom-algebra a left or right or symmetric Hom-Leibniz algebra or a Hom-Lie algebra. In general, a Hom-algebra algebra isdenoted ( M, d, β ) or ( L, δ, α ). Definition 1.4. [9] A representation (module) of the right Hom-Leibniz al-gebra ( L, [ · , · ] , α ) on the vector space V with respect to α V ∈ g l ( V ) , is definedwith two actions (left and right) on L . These actions are denoted by thefollowing brackets as well [ · , · ] V : L × V → V ; and [ · , · ] V : V × L → V satisfying the following five axioms, α V ([ x, v ] V ) = [ α ( x ) , α V ( v )] V ,α V ([ v, x ] V ) = [ α V ( v ) , α ( x )] V . [ α ( x ) , [ y, v ] V ] V = [[ x, y ] , β ( v )] V + [ α ( y ) , [ x, v ] V ] V [ α V ( v ) , [ x, y ]] V = [[ v, x ] V , α ( y )] V + [ α ( x ) , [ v, y ] V ] V [ α ( x ) , [ v, y ] V ] V = [[ x, v ] V , α ( y )] V + [ α V ( v ) , [ x, y ]] V , for any v ∈ V and x, y ∈ L. 4n this paper, [ · , · ] V : L × V → V (resp. [ · , · ] V : V × L → V ) is denoted λ l (resp. λ r ). Definition 1.5. [25, 9] Let V a representation of Hom-Lie (resp. Leibniz)algebra L . The set of k -cochains on L with values in V , which we denoteby Alt k ( L, V ) (resp. C k ( L, V ) ), is the set of skew-symmetric k -linear mapsfrom L × L · · · × L | {z } k − times to V .A k -Hom-cochain on L with values in V is defined to be a k -cochain f ∈ Alt k ( L, V ) satisfies α V ( f ( x · · · , x k )) = f ( α ( x ) , · · · , α ( x k )) (1.6) for all x , · · · , x k ∈ L. We denote by Alt kα ( L, V ) (resp. C kα ( L, V ) ) the set ofall k -Hom-cochains of L . The set of Hom-cochain of symmetric Hom-Leibnizalgebra L is denoted by C ′ kα ( L, V ) .We denote by χ α ( L, V ) the set of Hom-cochains of Hom- algebra L . Hence,if L is a left (or right) Hom-Leibniz algebra χ α ( L, V ) = C kα ( L, V ) . if L isa symmetrc Hom-Leibniz algebra, χ α ( L, V ) = C ′ α ( L, V ) . If L is a Hom-Liealgebra χ α ( L, V ) = Alt kα ( L, V ) .In this paper, the equation (1.6) is denoted α V ◦ f = f ⋆ α . Definition 1.6. [9] Define a K -Linear map δ k : C kα ( L, V ) → C k +1 α ( L, V ) by δ k ( f )( x , . . . , x k +1 ) = X ≤ s V, µ, α V ) is a Hom-algebra and the extension depends on µ and α V . β -Richardson-Nijenhuis bracket In this section, we define the space C β ( M ) of β -cohains (or Hom-cochain),we show that if ( C β ( M ) , [ · , · ]) is a graded Lie algebra we can define a typeof Hom-algebras and their cohomolgie. We define the left and right β -RNbracket as example of graded Lie algebra ( C β ( M ) , [ · , · ]).Recall that the set of n -cochains Alt n ( M ) of a Lie algebra M with val-ues in the adjoint representation, is defined to be the vector space of allalternating n -linear map of M into self and Alt ( M ) denotes the direct sum ⊕ n ≥ Alt n ( M ). For f ∈ Alt k ( M ), g ∈ Alt l ( M ) and n = k + l − 1, we define f ◦ g ∈ Alt n ( M ) by f ◦ g ( v , · · · , v n ) = X σ ∈ Sh ( l,k − ǫ ( σ ) f (cid:16) g ( v σ (1) , · · · , v σ ( l ) ) , v σ ( l +1) , · · · , v σ ( n ) (cid:17) (2.1)where Sh ( l, k − 1) is the sum of all permutation σ of { , · · · , n } such that σ (1) < σ (2) < · · · < σ ( l ) and σ ( l + 1) < · · · < σ ( n ). We now define theRichardson-Nijenhuis bracket [ f, g ] ∈ C n ( M ) by[ f, g ] = f ◦ g − ( − ( k − l − g ◦ f. (2.2)Then ( Alt ( M ) , [ · , · ]) is a graded Lie-algebra . Moreover if d is a 2-cochainand [ d, d ] = 0, then ( M, d ) is a Lie algebra and their cohomology is gives bythe cobondary δ = [ d, · ] (see[1]).Let β : M → M a linear map and let C k ( M ) a subspace of Hom ( M k , M ).In this section we define the β -cochain and the left and right β -Richardson-Nijenhuis bracket and their applications.6 efinition 2.1. A k - β -cochain is a map f ∈ C k ( M ) (resp. Alt k ( M ) )satisfying f ⋆ β = β ◦ f where f ⋆ β ( a , · · · , a k ) = f ( β ( a ) , · · · , β ( a m )) for all a , · · · , a k ∈ M. We denote by C β ( M ) = ⊕ k ≥ C kβ ( M ) (resp. Alt β ( M ) = ⊕ k ≥ Alt kβ ( M ) )where C kβ ( M ) (resp. Alt β ( M ) = ⊕ k ≥ Alt kβ ( M ) ) is the set of all k - β cochain of M .The space C β ( M ) (resp. Alt β ( M ) ) has a Z -grading, where the degree of anelement in C kβ ( M ) is k − .The following result is a Hom version of that obtained in [24]. Definition 2.2. Let [ · , · ] : C β ( M ) × C β ( M ) → C β ( M ) be a bilinear map. Acouple ( C β ( M ) , [ · , · ]) is a graded Lie algebra (or a color Lie algebra) if for any f ∈ C mβ ( M ) , g ∈ C nβ ( M ) and h ∈ C pβ ( M ) (i.e deg ( f ) = m − , deg ( g ) = n − , deg ( h ) = p − ), we have(i) deg ([ f, g ]) = deg ( f ) + deg ( g ) ;(ii) [ f, g ] = − ( − deg ( f ) deg ( g ) [ g, f ] ;(iii) ( − deg ( f ) deg ( h ) [ f, [ g, h ]]+( − deg ( f ) deg ( g ) [ g, [ h, f ]]+( − deg ( g ) deg ( h ) [ h, [ f, g ]] =0 (graded Jacobi indentity). Proposition 2.3. If the space ( C β ( M ) , ◦ ) is a graded pre-Lie algebra, i.e. ( f ◦ g ) ◦ h − f ◦ ( g ◦ h ) = ( − ( n − p − (( f ◦ h ) ◦ g − f ◦ ( h ◦ g )) (2.3) for all f ∈ C mβ ( M ) , g ∈ C nβ ( M ) , h ∈ C pβ ( M ) . The bracket [ f, g ] = f ◦ g − ( − ( m − n − g ◦ f. (2.4) make C β ( M ) a graded Lie algebra.Proof. Similar to [23, 24]. (cid:4) Moreover, C β ( M ) has a Z -grading, where the parity of an element in C nβ ( M ) is even if n is odd, and odd if n is even . Definition 2.4. Let ( C β ( M ) , [ · , · ]) be a graded Lie algebra and let d ∈ C nβ ( M ) be an odd p -Hom-cochain. Then, the triple ( M, d, β ) is called a p -Homalgebra if [ d, d ] = 0 . et ( M, d, β ) a p -Hom-algebra. Define D k : C kβ ( M ) → C k +1 β ( M ) by set-ting D k ( f ) = [ d, f ] l . Proposition 2.5. The map D k is a k -coboundary operator, i.e D k +1 ◦ D k = 0 .Thus, we have a well-defined cohomological complex ( M k> C kβ ( M ) , D k ) .Proof. Let f ∈ C kβ ( M ). We have( − ( n − k − D k +1 ◦ D k ( f ) = ( − ( n − k − [ d, [ d, f ]] . Then, by the graded Jacobi identity on C β ( M ) we have( − ( n − k − D k +1 ◦ D k ( f ) = − ( − ( n − [ d, [ f, d ]] − ( − ( k − n − [ f, [ d, d ]]Thus, using [ d, d ] = 0, n is even and (ii) in Definition 2.2 we obtain D k +1 ◦ D k ( f ) = − [ d, [ d, f ]] = − D k +1 ◦ D k ( f ) . Then, 2 D k +1 ◦ D k ( f ) = 0. (cid:4) The conclusion is for each even integer p , any graded Lie algebra makea type of p -Hom-algebra and their cohomolgy.A k - β -cochain f ∈ C kβ ( M ) is called a k -cocycle if D k ( f ) = 0 . k - β -cochain f ∈ C kβ ( M ) is called a k -coboundary if f = D k ( g ) for some g ∈ C k − β ( M ) .Denote by Z kβ ( M ) and B kβ ( M ) the sets of k -cocycles and k -coboundaries re-spectively. We define the pth cohomology group H kβ ( M ) to be Z kβ ( M ) /B kβ ( M ) .The aim of the rest of this section is give an example of graded lie alge-bra and use it for determines some type of Hom-Leibniz algebras and theircohomologies. Define a map l : Sh ( p, q ) → { , · · · , q − } by l ( σ ) = 1 if σ ( p ) < σ ( p + 1) and l ( σ ) = i if σ ( p +1) < · · · < σ ( p + i − < σ ( p ) < σ ( p + i ) < · · · < σ ( p + q ) .Let M be a vector space and let β : M → M a linear map. For f ∈ Hom ( M m , M ) and g ∈ Hom ( M n , M ) , we define f ◦ l g ∈ Hom ( M m + n − M, M ) by f ◦ l g ( a , · · · , a k + l − ) = X σ ∈ Sh ( n,m − ( − i − ǫ ( σ ) f (cid:16) β n − ( a σ ( n +1) ) , · · · ,β n − ( a σ ( n + i ) ) , g ( a σ (1) , · · · , a σ ( n ) ) , β n − ( a σ ( n + i +1) ) , · · · , β n − ( a σ ( n + m − ) (cid:17) ( i = l ( σ ))(2.5)8 nd [ f, g ] l ∈ C m + n − ( M ) by [ f, g ] l = f ◦ l g − ( − ( m − n − g ◦ l f. (2.6)[ · , · ] l is called the left β -Richardson-Nijenhuis bracket ( RN for short). Proposition 2.6. Let n an even integer and d be a n -Hom-cochain. Then,if [ d, d ] l = 0 , ( M, d, β ) is a left n -Hom-Leibniz algebra.Proof. First we show that ( M, [ · , · ] l ) is a graded Lie algebra. We use Sh ( p, q, r )to denote the set of all ( p, q, r )- unshuffles, that is, an element σ in the permu-tation group S p,q,r satisfying σ (1) < σ (2) < · · · < σ ( p ); σ ( p + 1) < σ ( p + 2) < · · · < σ ( p + q ) and σ ( p + q + 1) < σ ( p + q + 2) < · · · < σ ( p + q + s ).We define the maps l ′ : Sh ( p, q, r ) → { , · · · p + q − } by l ′ ( σ ) = 1 if σ ( p + 1) < σ ( p ); l ′ ( σ ) = i if σ ( p + 1) < · · · < σ ( p + i − < σ ( p ) < σ ( p + i );and l ” : Sh ( p, q, r ) → { , · · · p − } by l ”( σ ) = 1 if σ ( p + q ) < σ ( p + q + 1); l ”( σ ) = j if σ ( p + q + 1) < · · · < σ ( p + q + j − < σ ( p + q ) < σ ( p + q + j ).We compute f ∈ C k ( M ), g ∈ C l ( M ) and h ∈ C r ( M ):(( f ◦ g ) ◦ h − f ◦ ( g ◦ h )) ( a , · · · , a k + r + l − )= X σ ∈ Sh ( r,l,k − ≤ i 2) such that i < j (resp. i > j ), define σ ′ ∈ Sh ( r, l, k − 2) as σ ′ = σ ◦ (cid:18) · · · l r + 1 · · · r + lr + 1 · · · r + l · · · r (cid:19) . (resp. σ ′ = σ ◦ (cid:18) · · · r r + 1 · · · r + ll + 1 · · · r + l · · · l (cid:19) . ) Then ǫ ( σ ) = ( − rl ǫ ( σ ′ ) and( − ( r − l − (2.9) = X σ ∈ Sh ( l,r,k − ≤ i Define D k : C kβ ( M ) → C k +1 β ( M ) by setting D k ( f ) = [ d, f ] l .Then the map D k is a k -coboundary operator, i.e D k +1 ◦ D k = 0 . Thus, wehave a well-defined cohomological complex ( M k> C kβ ( M ) , D k ) . Remark 2.8. For n = 2 we have 12 [ d, d ] l ( a, b, c ) = d ( d ( a, b ) , β ( c )) + d ( β ( b ) , d ( a, c )) − d ( β ( a ) , d ( b, c )) Hence, [ d, d ] l = 0 gives the usual left Hom-Leibniz algebra. Now, we give the explicit description of the k -coboundary operator D k . D k ( f )( a , · · · , a k +1 ) = [ d, f ] l ( a , · · · , a k +1 )= k X s =1 ( − s d ( β k − ( a s ) , f ( a , · · · , b a s , · · · , a k +1 )+ d ( f ( a , · · · , a k ) , β k − ( a k +1 )) (2.11)+ X ≤ s Proposition 2.9. The cohomology of a right p -Hom-Leibniz algebra is de-termined by D k : C kβ ( M ) → C k +1 β ( M ) , given by D k ( f ) = [ d, f ] r where d isthe right Hom-Leibniz structure, interpreted as an element of C pβ ( M ) . In this section, by the left β -NR bracket, we define the symmetric Hom-Leibniz algebras generalizing the well known Leibniz algebras given in [5, 16].If f ∈ C mβ and g ∈ C nβ and σ ∈ Sh ( n, m − 1) we define f ◦ l ( σ ) g by f ◦ r ( σ ) g ( a , · · · , a n + m − )= f (cid:0) β n − ( a σ ( n +1) ) , · · · , β n − ( a σ ( n + r ( σ ) − ) , g ( a σ (1) , · · · , a σ ( n ) ) , · · · , β n − ( a σ ( m + n − ) (cid:1) and f ◦ l ( σ ) g ( a , · · · , a n + m − )= f (cid:0) β n − ( a σ ( n +1) ) , · · · , β n − ( a σ ( n + l ( σ ) − ) , g ( a σ (1) , · · · , a σ ( n ) ) , · · · , β n − ( a σ ( m + n − ) (cid:1) . A ( m, n )- β -cochain is a pair ( f, g ) where f ∈ C mβ and g ∈ C nβ by f ◦ r ( σ ) g = ( − l ( σ ) − r ( σ ) f ◦ l ( σ ) g. for all σ ∈ Sh ( n, m − C m,nβ ( M ) the set of all ( m, n )- β -cochain. We denote C n,nβ ( M ) = C ′ nβ ( M ).12 efinition 2.10. Let d a (2 , -beta-cochain. The triple ( M, d, β ) is called asymmetric Hom-Leibniz algebra if [ d, d ] l = 0 . i.e d ( β ( a ) , d ( b, c )) = − d ( d ( b, c ) , β ( a )) d ( β ( a ) , d ( b, c )) = d ( d ( a, b ) , β ( c )) + d ( β ( b ) , d ( a, c )) for all a, b, c ∈ M . Clearly, the triple ( M, d, β ) is symmetric Hom-Leibniz algebra if and onlyif it is a left and right Hom-Leibniz algebra. Definition 2.11. A k -Hom cochain of the symmetric Hom-Leibniz algebra ( M, d, β ) is a k -linear map f such that the pair ( f, d ) is a ( k, - β -cochain.We denote by C ′ kβ ( M ) the set of all k -Hom cochain of M . Define D ′ k : C ′ kβ ( M ) → C ′ k +1 β ( M ) by setting D ′ k ( f )( a , · · · , a k +1 ) = k +1 X s =1 ( − k − s d ( β k − ( a s ) , f ( a , · · · , b a s , · · · , a k +1 )+ X ≤ s The map D ′ k is a coboundary operator, i.e. D k +1 ◦ D k =0 .Proof. Since f is a k -Hom cochain of the symmetric Hom-Leibniz algebra( M, d, β ), we have d ( f ( a , · · · , a k ) , β k − ( a k +1 )) = − d ( β k − ( a k +1 ) , f ( a , · · · , a k ))Thus, D ′ k ( f )( a , · · · , a k +1 ) = k X s =1 ( − k − s d ( β k − ( a s ) , f ( a , · · · , b a s , · · · , a k +1 )+ d ( β k − ( a k +1 ) , f ( a , · · · , a k ))+ X ≤ s Recall that we mean by a Hom-algebra a left or right or symmetric Leibnizalgebra or Hom-Lie algebra. Let ( M, d , β ) be a Hom-algebra and [ · , · ] the β -NR bracket. A one-parameter formal Hom-algebra deformation of M isgiven by the K [[ t ]]-bilinear d t = X i ≥ t i d i (where each d i ∈ χ ( M )) satisfies[ d t , d t ] = 0 which is equivalent to the follwing infinite system X ≤ i ≤ s [ d i , d s − i ] = 0 , for s = 0 , , , · · · (3.1)In particular, For s = 0 we have [ d , d ] wich is the Hom-Jacobie identity of M .The equation for s = 1, leads to [ d , d ] = 0 . (3.2)Then d is a 2-cocycle.For s = 2, we obtain [ d , d ] + 12 [ d , d ] = 0 . (3.3)Hence [ d , d ] is a 3-cobord (i.e [ d , d ] ∈ B ( M )). So it is a trivial 3-cocycle.For s > a s = 2[ d , d s ] + Ψ( d , · · · , d s − ) = 0where Ψ( d , · · · , d s − ) = s − X i =1 [ d i , d s − i ] . Clearly Ψ( d , · · · , d s − ) is a 3-cobordon M .Conversely, let d t = X i ≥ t i d i . Hence [ d t , d t ] = X i ≥ a s t s where a = 2[ d , d ]and a s = 2[ d , d s ] + Ψ( d , · · · , d s − ) , where s ≥ . We assume that Ψ( d , · · · , d s − ) is a 3-coboundary, then there a exists d ′ s ∈ χ ( M ) such that Ψ( d , · · · , d s − ) = [ d , d ′ s ]. Hence a s = [ d , d s + d ′ s ] for all s > Proposition 3.1. Let ( d i ) i ∈ N a set of -Hom-cohain in ( M, d , β ) . Then d t = X i ≥ t i d i is a one-parameter formal Hom-Leibniz algebra deformation of M if and only if the following statements are satisfies: i) d is a -cocycle on M ;(ii) for all s > it exists a d ′ s ∈ χ ( M ) such that Ψ( d , · · · , d s − ) = [ d , d ′ s ] and d s + d ′ s is a -cocycle on M . Definition 3.2. The deformation is said to be of order k if d t = k X i =0 d i t i . Now,we assume that the deformation is of order k − d ′ t = d t + d k t k and [ d ′ t , d ′ t ] = k − X i =0 a i t i . Then a k = 2[ d , d k ]+Ψ( d , · · · , d k − ). Hence, if a k = 0,Ψ( d , · · · , d k − ) is a 3-cobord in M .Conversely, we assume that Ψ( d , · · · , d k − ) is a 3-cobord in M . Then thereexists a d k ∈ χ ( M ) satisfies Ψ( d , · · · , d k − ) = − d , d k ]. Hence a k = 0 . Then we get the following result. Proposition 3.3. Let ( M, d , β ) be a Hom-algebra and M t = ( M, d t , β ) bea k − -order one-parameter formal deformation of M , where d t = k − X i =0 d i t i .Then Ψ( d , · · · , d k − ) = k − X i =1 [ d i , d k − i ] . is a -cocycle of M . Therefore the deformation extends to a deformation oforder k if and only if Ψ( d , · · · , d k − ) is a -coboundary. Definition 3.4. Let ( L, d , β ) be a Hom-algebra. Given two deformation M t = ( M, d t , β ) and M ′ t = ( M, d ′ t , β ) of M where d t = X i ≥ t i d i , d ′ t = X i ≥ t i d ′ i .We say that they are equivalent if there exists a formal automorphism φ t =exp( tφ ) where φ ∈ End ( M ) such that φ t ◦ d t = d ′ t ⋆ φ t (3.4) and φ t ◦ β = β ′ ◦ φ t . (3.5) A deformation M t is said to be trivial if and only if M t is equivalent to M (viewed as an algebra on M [[ t ]] ). X i,j ≥ φ i ( d j ) t i + j = X i,j,k ≥ d ′ i ⋆ ( φ j , φ k ) t i + j + k (3.6)By identification of coefficients, one obtain that the constant coefficientsare identical, i.e. d = d ′ because Φ = Id. For coefficients of t one has d ′ ( a, b ) = d ( a, b ) + Φ ( d ( a, b )) − d (Φ ( a ) , b ) − d ( a, Φ ( b )) . (3.7)Hence d ′ = d + [ d , φ ].The condition on homomorphisms (3.5) implies that β = β ′ mod t and thatΦ ◦ β = β ′ ◦ Φ mod t . (3.8)Then β = β ′ .Consequently the following Proposition holds. Proposition 3.5. Let ( M, d , β ) be a Hom-Leibniz algebra. There is, over K [[ t ]] /t , a one-to-one correspondence between the elements of H ( M ) andthe infinitesimal deformation defined by d t = d + d t. Hence, if the Hom-Leibniz algebra M is rigid then every formal deformationis equivalent to a trivial deformation. Remark 3.6. If ( M, d , β ) is a symmetric Hom-Leibniz algebra. Then, χ ( M ) = C ′ β . Hence, the condition d t ∈ χ ( M ) is equivalent to d i ( β ( a ) , d i ( b, c )) = − d i ( d i ( b, c ) , β ( a )) for all a, b, c ∈ M . Remark 3.7. If ( M, d , β ) is a Hom-Lie algebra. Then, χ ( M ) = Alt ( M ) .Hence, the condition d t ∈ χ ( M ) is equivalent to d i ( a, b ) = − d i ( b, a ) for all a, b ∈ M . The principal aim of this section is to extend the notions and results aboutextension of Lie algebras [2] to more generalized cases: Hom-Leibniz algebrasand Hom-Lie algebras. 16 efinition 4.1. Let ( L, δ, α ) , ( V, µ, α V ) , and ( M, d, α M ) be Hom-algebrasand i : V → M , π : M → L be morphisms of Hom-algebras. The followingsequence of Hom- algebra is a short exact sequence if Im ( i ) = ker( π ) , i isinjective and π is surjective: −→ ( V, µ, α V ) i −→ ( M, d, α M ) π −→ ( L, δ, α ) −→ , where α M ◦ i = i ◦ α V and α ◦ π = π ◦ α M . (4.1) In this case, we call ( M, d, α M ) an extension of ( L, δ, α ) by ( V, µ, α V ) .An extension −→ ( V, µ, α V ) i −→ ( M, d, α M ) π −→ ( L, δ, α ) −→ is called:(1) trivial if there exists an ideal I complementary to ker π ,(2) split if there exists a Hom-subalgebra S ⊂ M complementary to ker π ,(3) central if the ker π is contained in the center Z ( M ) of L . That is d ( i ( V ) , M ) = 0 . (4) abelian if µ = 0 .The standard split extension E of L by V is given by E : 0 −→ ( V, µ, α V ) i −→ ( L ⊕ V, d, α + α V ) π −→ ( L, δ, α ) −→ , where i ( v ) = v , π ( x + v ) = x . In Section 2 we have defined the cohomology of a Hom-algebra on it self(adjoint representation). In this section we define, starting with a split ex-tension of Hom- algebra a representation and cohomology of Hom-algebras L on a Hom- algebras V .Let ( L, δ, α ) and ( V, µ, α V ) be two Left Hom-algebras over K such that0 −→ ( V, µ, α V ) i −→ ( L ⊕ V, d, α + α V ) π −→ ( L, δ, α ) −→ L by V . For convenience, we introduce the following notation for certainspaces of cochains on M = L ⊕ V : C n ( M ) = Hom ( M n , M ) , C n ( L ) = Hom ( L n , L ) and C k,l ( LV, V ) = Hom ( L k V l , V ) , L k V l is the subspace of C k + l ( M ) consisting by products of k elementsfrom L and l elements from V ordered by σ ∈ Sh ( l, k − β = α + α V ,[ · , · ] be a β -NR bracket (left or right) and d ∈ C ( M ). We want to determinethe conditions under which ( L ⊕ V, d, β ) is a Hom-Leibniz algebra and V isan ideal of M = L ⊕ V . For all f ∈ C β ( M ) , u, v ∈ C β ( M ), denote f ⋆ ( u, v )( a, b ) = f ( u ( a ) , v ( b ))and if u = v we denote f ⋆ u ( a, b ) = f ( u ( a ) , u ( b )) for all a, b ∈ M .If V is an ideal of Hom-Leibniz algebra L ⊕ V , then [ d, d ] = 0 and there exists δ ∈ C ( L ) , λ l ∈ C , ( LV, V ), λ r ∈ C , ( V L, V ), θ ∈ C , ( L, V ) , µ ∈ C ( V )such that d = δ + λ r + λ l + µ + θ . Hence[ δ + λ r + λ l + µ + θ, δ + λ r + λ l + µ + θ ] = 0Therefore[ δ, δ ] |{z} =0 + 2[ δ, λ r + λ l ] + [ λ r + λ l , λ r + λ l ] + 2[ µ, θ ] | {z } ∈ C , + 2[ µ, λ r + λ l ] | {z } ∈ C , + [ δ + λ r + λ l , θ ] | {z } ∈ C , + [ µ, µ ] | {z } =0 = 0 . Then we can deduce that2[ δ, λ r + λ l ] + [ λ r + λ l , λ r + λ l ] + 2[ µ, θ ] : The ”module” or ”representation” relation(4.2)[ µ, λ r + λ l ] = 0 : The module-algebra structures are compatible (4.3)[ δ + λ r + λ l , θ ] = 0 : θ is a 2-cocycle with values in V. (4.4)By d is a 2- β cochain, we have( δ + λ r + λ l + µ + θ ) ⋆ ( α + α V ) = ( α + α V ) ◦ ( δ + λ r + λ l + µ + θ ) . Then δ ⋆ α = α ◦ δ (4.5) λ r ⋆ ( α V , α ) = α V ◦ λ r (4.6) λ l ⋆ ( α, α V ) = α V ◦ λ l (4.7) µ ⋆ α V = α V ◦ µ (4.8) θ ⋆ α = α V ◦ θ. (4.9)By (4.6),(4.7) and (4.2) we have: 18 efinition 4.2. A representation (module ) of the Hom-Leibniz algebra ( L, δ, α ) is a Hom-module ( V, µ, α V ) equiped with two L -actions (left and right) λ l : L × V → V , λ r : V × L → V and a bilinear map θ ∈ C , satisfying the followingaxioms, λ r ⋆ ( α V , α ) = α V ◦ λ r λ l ⋆ ( α, α V ) = α V ◦ λ l [ δ, λ r + λ l ] + 12 [ λ r + λ l , λ r + λ l ] + [ µ, θ ] = 0For the case of left Leibniz algebra, the previous definition can be writtenas follows: Definition 4.3. A representation (module ) of the left Hom-Leibniz algebra ( L, δ, α ) is a Hom-module ( V, µ, α V ) equiped with two L -actions (left andright) λ l : L × V → V , λ r : V × L → V and a bilinear map θ ∈ C , satisfyingthe following axioms, λ r ( α V ( v ) , α ( x )) = α V ( λ r ( v, x )) λ l ( α ( x ) , α V ( v )) = α V ( λ l ( x, v )) λ l ( α ( x ) , λ l ( y, v )) = λ l ( δ ( x, y ) , α V ( v )) + λ l ( α ( y ) , λ l ( x, v )) + µ ( θ ( x, y ) , α V ( v )) ; λ l ( α ( x ) , λ r ( v, y )) = λ r ( λ l ( x, v ) , α ( y )) + λ r ( α V ( v ) , δ ( x, y )) + µ ( α V ( v ) , θ ( x, y )) λ r ( α V ( v ) , δ ( x, y )) = λ r ( λ r ( v, x ) , α ( y )) + λ l ( α ( x ) , λ r ( v, y )) − µ ( α V ( v ) , θ ( x, y )) . for any x, y ∈ L and v ∈ V . Remark 4.4. (1) If the extension is trivial (i.e L is also an ideal of M )then d = δ + µ .(2) If L and V be right Hom-Leibniz algebras and µ = 0 (or θ = 0 ), λ = λ r + λ l is the representation of L on V defined in (1.4) .(3) If L and V be Hom-Lie algebras. Then Alt k ( M ) is the set of k -cochains.Hence, by d ∈ Alt ( M ) we obtain λ l ( x, v ) = − λ r ( v, x ) . Therefore λ = λ r + λ l ∈ Alt ( M ) and λ is the representation of L on V defined in [25].(4) If L and V be symmetric Hom-Leibniz algebras. Then C ′ kβ ( M ) is the set f k -cochains. Hence, by d ∈ C ′ β ( M ) we have λ l ( α ( x ) , λ l ( y, v )) = − λ r ( λ l ( y, v ) , α ( x )) λ l ( α ( x ) , λ r ( v, y )) = − λ r ( λ r ( v, y ) , α ( x )) λ r ( α V ( v ) , δ ( x, y )) = − λ l ( δ ( x, y ) , α V ( v )) µ ( α V ( u ) , λ r ( v, x )) = − µ ( λ r ( v, x ) , α V ( u )) µ ( α V ( u ) , λ l ( x, v )) = − µ ( λ l ( x, v ) , α V ( u )) λ l ( α ( x ) , µ ( u, v )) = − λ r ( µ ( u, v ) , α ( x )) for all x, y ∈ L , u, v ∈ V . That is λ, ( λ, δ ) , ( µ, λ ) , ( λ, µ ) ∈ C ′ β ( M ) . (4.10)Now, by (4.1) and (4.9) we define the 2 cocycle as follow. Definition 4.5. Let ( L, δ, α ) be a Hom-Leibniz algebra and ( V, µ, α V ) be arepresentation of L . If the bilinear map θ satisfies θ ⋆ α = α V ◦ θ (4.11)[ δ + λ r + λ l , θ ] = 0 . (4.12) Then θ is called a - cocycle of the left Hom-Leibniz algebra ( L, δ, α ) relatedto the representation ( V, µ, α V ) , or simply a -cocycle of L on V . The set ofall -cocycles on V is denoted Z ( L, V ) . For example for the left Hom-Leibniz case, the equalities (4.11) and (4.12)are equivalent to θ ( α ( x ) , α ( y )) = α V ( θ ( x, y )) ; θ ( δ ( x, y ) , α ( z )) + θ ( α ( y ) , δ ( x, z )) − θ ( α ( x ) , δ ( y, z ))+ λ r ( θ ( x, y ) , α ( z )) + λ l ( α ( y ) , θ ( x, z )) − λ l ( α ( x ) , θ ( y, z )) = 0(i.e [ d, θ ] = 0 ).Since [ µ, µ ] = 0, µ ⋆ ( α + α V ) = ( α + α V ) ◦ µ and [ µ, λ r + λ l ] = 0, then bysection 2, λ r + λ l is a 2-cocycle of Hom-Leibniz algebra ( M, µ, α + α V ).We summarize the main facts in the theorem below.20 heorem 4.6. −→ ( V, µ, α V ) i −→ ( L ⊕ V, d, α + α V ) π −→ ( L, δ, α ) −→ is asplit extension of L by V if and only if d = δ + λ r + λ l + µ + θ , where λ = λ r + λ l is a representation of L on V also it is a representation of ( M, µ, α + α V ) and θ is a -cocycle of L on V . Precisely when the six conditions below hold: λ r ⋆ ( α V , α ) = α V ◦ λ r (4.13) λ l ⋆ ( α, α V ) = α V ◦ λ l (4.14) θ ⋆ α = α V ◦ θ ; (4.15)[ δ, λ r + λ l ] + 12 [ λ r + λ l , λ r + λ l ] + [ µ, θ ] = 0 (4.16)[ δ + λ r + λ l , θ ] = 0; (4.17)[ µ, λ r + λ l ] = 0 . (4.18)Similarly to the symmetric Hom-Leibniz case, we have Theorem 4.7. Let ( L, δ, α ) and ( V, µ, α V ) be symmetric Hom-Leibniz alge-bras. Then, −→ ( V, µ, α V ) i −→ ( L ⊕ V, d, α + α V ) π −→ ( L, δ, α ) −→ is asplit extension of L by V if and only if there exists d = δ + λ r + λ l + µ + θ ,where λ r , λ l and θ satisfies the equations (4.10) , (4.13) , (4.14) , (4.15) , (4.16) , (4.17) and (4.18) . Definition 4.8. Let ( L, δ, α ) be a Hom-Leibniz algebra, and let ( V, µ, α V ) be a ( λ, θ ) -representation of ( L, δ, α ) . Then for every linear map h : L → V satisfying h ◦ α = α V ◦ h , the bilinear map [ d, h ] l : L × L → V is called a -coboundary on V . The set of all -coboundaries on V is denoted B ( L, V ) .The quotient vector space H ( L, V ) = Z ( L, V ) /B ( L, V ) is called the secondcohomology group of L on V . Let ( L ⊕ V, d, α + β ) an extension of Hom-Leibniz algebra ( L, δ, α ). De-fine the operator D k : C k ( L, V ) → C k +1 ( L, V ) by D k ( f )(( x , . . . , x k +1 )) = [ d, g ]( x , . . . , x k +1 )= X ≤ s The map D is a coboundary operator, i.e. D = 0 .Proof. This is a direct result of Proposition 2.7 (2.9 for the right case, 2.12for the symmetric case). (cid:4) .2 Equivalence of extensions of Hom-algebras Definition 4.10. Two extensions E : 0 −→ ( V , µ , α V ) i −→ ( L ⊕ V , d , α + α V ) π −→ ( L , δ , α ) −→ ,E : 0 −→ ( V , µ , α V ) i −→ ( L ⊕ V , d , α + α V ) π −→ ( L , δ , α ) −→ , are equivalent if there is an isomorphism Φ : ( L , d , α ) → ( L , d , α ) such that Φ ◦ i = i ◦ Φ and π ◦ Φ = π ◦ Φ . In the following theorem, we show that any split extension M of L by V is is equivalent to L ⊕ V . Theorem 4.11. Let E : 0 −→ ( V, µ, α V ) i −→ ( M, d ′ , α M ) π −→ ( L, δ, α ) −→ , be a split extension of L by V . Then there exist bilinear map d = δ + λ l + λ r + θ + µ such that ( λ l , λ r ) is a representation of L , θ is a -cocycle and theshort exact sequence E : 0 −→ ( V, µ, α V ) i −→ ( L ⊕ V, d, α + α V ) π −→ ( L, δ, α ) −→ , is an extension equivalent to E .Proof. Let 0 −→ ( V, µ, α V ) i −→ ( M, d ′ , α M ) π −→ ( L, δ, α ) −→ , be a split extension of L . Then there exist a Hom-subalgebra H ⊂ M complementary to ker π . Since Im v = ker π , we have M = H ⊕ i ( V ).The map π /H : H → L (resp k : V → i ( V )) defined by π /H ( x ) = π ( x )(resp. k ( v ) = i ( v )) is bijective, its inverse s (resp. l ). Considering the mapΦ : L ⊕ V → M defined by Φ( x + v ) = s ( x ) + i ( v ) , it is easy to verify thatΦ is an isomorphism, i ( V ) is an ideal of M and M = s ( L ) ⊕ i ( V ). Then(by the provious section ), d ′ = δ ′ + λ ′ r + λ ′ l + θ ′ + µ ′ where λ ′ = λ ′ r + λ ′ l isa representation of s ( L ) on i ( V ) also it is a representation of ( M, µ, α + α V )and θ is a 2-cocycle of s ( L ) on i ( V ).Define a bilinear map d : L ⊕ V → L ⊕ V by d ( x + v, y + w ) = Φ − ( d ′ ( s ( x ) + i ( v ) , s ( y ) + i ( w ))) . Then d ′ (Φ( x + v ) , Φ( y + w )) = d ′ ( s ( x ) + i ( v ) , s ( y ) + i ( w )) = Φ ( d ( x + v, y + w ))(4.19)22learly ( L ⊕ V, d, α + α V ) is a Hom-algebra and E : 0 −→ ( V, µ, α V ) i −→ ( L ⊕ V, d, α + α V ) π −→ ( L, δ, α ) −→ , be a split extension of L by V equivalent to E . By (4.19) we have δ = π ◦ ( δ ′ ⋆s ) , θ = k ◦ ( θ ′ ⋆s ) , λ l = k ◦ ( λ ′ l ⋆ ( s, i )) , λ r = k ◦ ( λ ′ r ⋆ ( i, s )) , µ = k ◦ ( µ⋆i )and d = δ + θ + λ l + λ r + µ . (cid:4) We deduce from the last theorem that any split extension of M by V isequivalent to0 −→ ( V, µ, α V ) i −→ ( L ⊕ V, d, α + α V ) π −→ ( L, δ, α ) −→ , where i ( v ) = v , π ( x + v ) = x .Let ( M, d, β ) an extension of a Hom-algebra ( L, δ, α ) by ( V, µ, α V ). Thenthere it equivalent to ( L ⊕ V, d, α + α V ) where d = δ + λ r + λ l + θ + µ .This extension is trivial if and only if λ r + λ l + θ = 0, central if and only if λ r + λ l + θ + π = 0. Now, we extend the definition of the semidirect sum ofLie-algebras given in [14] to Hom-Leibniz case. Definition 4.12. The split extension ( M, d, β ) is called semidirect sum ofHom-algebras L and V if and only if θ = 0 . Define the following split extension by E : 0 −→ ( V, µ, α V ) i −→ ( L ⊕ V, d, α + α V ) π −→ ( L, δ, α ) −→ ,E ′ : 0 −→ ( V, µ ′ , α V ′ ) i ′ −→ ( L ′ ⊕ V ′ , d ′ , α ′ + α V ′ ) π ′ −→ ( L ′ , δ ′ , α ′ ) −→ , where i ( v ) = v , π ( x + v ) = x . i ′ ( v ′ ) = v ′ , π ′ ( x ′ + v ′ ) = x ′ .We assume that E and E ′ are equivalents. Then there exist an isomorphismof Hom-algebraΦ : ( L ⊕ V, d, α + α V ) → ( L ′ ⊕ V ′ , d ′ , α ′ + α ′ V )Satisfies Φ ◦ i = i ′ ◦ Φ and π ′ ◦ Φ = Φ ◦ π .We set Φ = s + i and s = s + i , i = s + i , where s : L → L ′ ⊕ V ′ , s : L → L ′ , i : L → V ′ , i : V → L ′ ⊕ V ′ , s : V → L ′ , i : V → V ′ . We denote Φ( v ) = v ′ and Φ( x ) = x ′ .23e have i ( v ) = Φ( v ) = Φ( i ( v )) = i ′ (Φ( v )) = i ′ ( v ′ ) = v ′ and x ′ = Φ( x ) = Φ( π ( x )) = π ′ (Φ( x )) = π ′ ( s ( x )) = π ′ ( s ( x ) + i ( x )) = s ( x )Hence i ( x ) = s ( x ) − x ′ , i ( v ) = i ( v ) = v ′ , s = 0.We have d ′ ⋆ ( s + i, s + i )= δ ′ ⋆ ( s , s ) + θ ′ ⋆ ( s , s ) + λ ′ l ⋆ ( s , i + i ) + λ ′ r ⋆ ( i + i, s ) + µ ′ ⋆ ( i + i, i + i )Moreover, d ′ ⋆ ( s + i, s + i ) = Φ( d ( x + v, y + w ))= s ◦ δ + i ◦ δ + i ◦ θ + i ◦ λ l + i ◦ λ r + i ◦ µ. So, δ ′ ⋆ ( s , s ) = s ◦ δ , θ ′ ⋆ ( s , s ) = i ◦ δ + i ◦ θ , λ ′ l ⋆ ( s , i + i ) = i ◦ λ l , λ ′ r ⋆ ( i + i, s ) = i ◦ λ r and µ ′ ⋆ ( i + i, i + i ) = i ◦ µ .Define the linear map h : L ′ → V ′ by h ( s ( x )) = − i ( x ). We have[ d ′ , h ] ⋆ ( s + i, s + i ) = λ ′ r ⋆ ( i , s ) + µ ′ ⋆ ( i , i + i ) + λ ′ l ⋆ ( s , i ) + µ ′ ⋆ ( i + i, i ) + i ◦ δ.d ′ ⋆ ( s + i, s + i ) = s ◦ δ + i ◦ λ l + i ◦ λ r + i ◦ θ + i ◦ µ + [ d ′ , h ] ⋆ ( s + i, s + i ) . since s , i , and i are bijective, we can identify s ◦ δ and δ , i ◦ λ l and λ l , i ◦ λ r and λ r , i ◦ θ and θ , i ◦ µ and µ . Hence, we get the following result. Theorem 4.13. Two extensions ( L ⊕ V, d, α + α V ) and ( L ⊕ V, d ′ , α + α V ) are equivalent if and only if there exists a -Hom-cochain h : L → V suchthat d ′ = d + [ d, · ] . Corollary 4.14. The set Ext ( L, V ) of equivalence classes of central split ex-tensions of ( L, δ, α ) by ( V, µ, α V ) is one-to-one correspondence with Z ( L, V ) /B ( L, V ) ,that is Ext ( L, V ) ∼ = Z ( L, V ) /B ( L, V ) . References [1] Albert, N., Richardson, J.R , Deformation of Lie algebra structures. [2] Alice, F., Michael, P. Extensions of (super) Lie algebras . Communica-tions in Contemporary Mathematics. (5) (2009) 709 − Some caracterizations of Hom-Leibniz algebras Interna-tional electronic journal of algebra. (2013) 1 − Hom–Lie algebras with symmetric invari-ant nondegenerate bilinear forms, Journal of Geometry and Physics. (2014), 38 − . [5] Benayadi, S., and Hidri, S., Quadratic Leibniz Algebras, Journal of LieTheory Volume (2014) 737–759.[6] Casas, J. M., M. Ladra, B. A. Omirove, and I. A. Karimjanov, Classifica-tion of solvable Leibniz algebras with naturally graded filiform nilradical , Linear Algebra Appl. (2013), 2973–3000.[7] Casas J. M., Insua M.A., On universal central extensions of Hom-Leibnizalgebras. Journal of Algebra and Its Applications. (2014), 8.[8] Yan, C., Liangyun, C., Bing,S., On split regular Hom-Leibniz algebras .Journal of Algebra and Its Applications. (2018), 10.[9] Cheng, Y.S, (Co)Homology and Universal Central Extension of Hom-Leibniz Algebras . Acta Mathematica Sinica, English Series. (27) − Cohomology theory of Lie groups and Liealgebras . Transactions of the American Mathematical Society. (1948),85 − The local integration of Leibniz algebras, Annales de l’InstitutFourier, (2013) no. 1, 1 − Alg`ebres de Leibnitz: D´efinition, propri´et´ees, Annales scien-tifiques de l’E.N.S. (4e s´erie) (1994) , On some structures of Leibnizalgebras , contemporary Mathematics, Extensions of LieAlgebras . Studies in Mathematical Physics, Chapter 18, 1997, 5 − (2013), 63–77.[16] Geoffrey, M., and Y., Gaywalee, Leibniz algebras and Lie algebras , Sym-metry, Integrability and Geometry: Methods and Applications, SIGMA9 4(2013)4, . 2517] Hartwig J. T., Larsson D. and Silvestrov S.D., Deformations of Liealgebras using σ -derivations, Journal of Algebra, (2006), 314 − Cohomology of Lie algebra , Annals of Math-ematics, (3) (1953).[19] Loday, J.-L., Une version non-commutative des alg`ebres de Lie: Lesalg`ebres de Leibniz , Ens. Math. (1993), 269–293.[20] Loday, J.-L., Cyclic Homology , Springer-Verlag,(1992)[21] Loday, J.L., Pirashvili,T., Universal enveloping algebras of Leibniz alge-bras and (co)homology , Mathematische Annalen, (1993) 139 − . [22] Makhlouf,A., and Silvestrov, S., Hom-algebra structures , Journal of Gen-eralized Lie Theory and Applications . (2008) 51 − Cohomology and deformation in gradedLie algebras. Bulletin of the American Mathematical Society. (1)(1966), 1 − Deformations of Lie algebras structures. Journal of Mathematics and Mechanics, , (1) (1967), 89 − . [25] Sheng Y., Representations of Hom-Lie Algebras . Algebras and Repre-sentation Theory,15