Cup-product in Hom-Leibniz cohomology and Hom-Zinbiel algebras
aa r X i v : . [ m a t h . R A ] F e b CUP-PRODUCT IN HOM-LEIBNIZ COHOMOLOGY ANDHOM-ZINBIEL ALGEBRAS
RIPAN SAHAA bstract . We define a cup-product in Hom-Leibniz cohomology and showthat the cup-product satisfies the graded Hom-Zinbiel relation.
1. I ntroduction
The concept of Hom-Lie algebras was introduced by Hartwig, Larsson andSilvestrov in the study of algebraic structures describing some q-deformations ofthe Witt and the Virasoro algebras [4]. Hom-Leibniz algebras was introduced byMakhlouf and Silvestrov generalizing the notion of Leibniz algebras [7]. Hom-Leibniz algebra is obtained by twisting the Leibniz identity by a self linear map α . A Hom-Leibniz algebra is same as a Leibniz algebra when α is the identitymap.There is a cup-product on the graded Hochschild cohomology H ∗ ( A , A ) ofan associative algebra A and with this cup-product H ∗ ( A , A ) carries a Gersten-haber algebra structure [3]. A cup-product and Gerstenhaber algebra on Hom-associative algebras have been studied in [2]. Loday defined a cup-product onthe graded cohomology groups of the Leibniz algebras [6]. The cup-producton the cohomology satisfies the graded Zinbiel algebra relation. A cup-producton the equivariant cohomology of Leibniz algebra equipped with an action of afinite group has been studied in [9].Exploring the similarities and dissimilarities between the category of Leib-niz algebras and the larger category of Hom-Leibniz algebras is an interestingtheme in the literature. This paper contributes to the development of the non-associative algebra theory in this direction. The main purpose of this paper isto extend Loday’s result to Hom-Leibniz algebra case. We introduce cohomol-ogy for Hom-Leibniz algebra ( L , [ ., . ] , α ) with coe ffi cients in a Hom-associative,commutative algebra ( A , µ, α ) and define a cup-product on the graded cohomol-ogy HL ∗ α,α ( L ; A ) of the Hom-Leibniz algebra L ∪ : HL n α,α ( L ; A ) × HL m α,α ( L ; A ) → HL n + m α,α ( L ; A ) , which satisfies the following graded algebra relation: α ( a ∪ b ) ∪ c = a ∪ α ( b ∪ c ) + ( − | b || c | a ∪ α ( c ∪ b ) , where a ∈ HL n α,α ( L ; A ) , b ∈ HL m α,α ( L ; A ) , c ∈ HL r α,α ( L ; A ). Mathematics Subject Classification.
Key words and phrases.
Hom-Leibniz algebra, Cup-product, Hom-Zinbiel algebra.
2. P reliminaries
In this section, we recall some definitions and basic facts of Hom-algebras andshu ffl e algebras. In this paper, k always denotes any field. Definition 2.1.
A Hom-associative algebra over k is a triple ( A , µ, α ) consists ofa k -vector space A together with a k -bilinear map µ : A × A → A and a k -linearmap α : A → A satisfying α ( µ ( x , y )) = µ ( α ( x ) , α ( y )) and µ ( α ( x ) , µ ( y , z )) = µ ( µ ( x , y ) , α ( z )) , for all x , y , z ∈ A . Example 2.2.
Let A be a two dimensional vector space with basis { a , a } . De-fine a multiplication µ : A × A → A by µ ( a i , a j ) = a if ( i , j ) = (1 , a if ( i , j ) , (1 , . A linear map α : A → A is defined by α ( a ) = a − a and α ( a ) =
0. Thenthe triple ( A , µ, α ) defines a Hom-associative algebra structure. Definition 2.3.
A Hom-Lie algebra is a triple ( A , µ, α ) consisting of a linearspace A , a skew-bilinear map µ : V × V → V and a linear space homomorphism α : A → A satisfying the Hom-Jacobi identity (cid:9) x , y , z µ ( µ ( x , y ) , α ( z )) = , for all x , y , z in A , where (cid:9) x , y , z denotes summation over the cyclic permutationson x , y , z .A Hom-Lie algebra whose endomorphism α is the identity is a Lie algebra. Definition 2.4.
A Hom-Leibniz algebra is a k -linear vector space L equippedwith a k -bilinear map [ ., . ] : L × L → L and a k -linear map α : L → L satisfyingthe identity: [ α ( x ) , [ y , z ]] = [[ x , y ] , α ( z )] − [[ x , z ] , α ( y )] . We denote a Hom-Leibniz algebra as ( L , [ ., . ] , α ). A Hom-Leibniz algebra( L , [ ., . ] , α ) is called multiplicative if it satisfies [ α ( x ) , α ( y )] = α ([ x , y ]).A homomorphism between Hom-Leibniz algebras ( L , [ ., . ] , α ) and( L , [ ., . ] , α ) is a k -linear map φ : L → L which satisfies φ ([ x , y ] ) = [ φ ( x ) , φ ( y )] and φ ◦ α = α ◦ φ . Example 2.5.
Any Hom-Lie algebra is a Hom-Leibniz algebra as in the presenceof skew-symmetry Hom-Jacobi identity is same as Hom-Leibniz identity.
Example 2.6.
Given a Leibniz algebra ( L , [ ., . ]) and a Leibniz algebra homo-morphism α : L → L, one always get a Hom-Leibniz algebra ( L , [ ., . ] α , α ) , where [ x , y ] α = [ α ( x ) , α ( y )] . Example 2.7.
Let ( L , d ) be a di ff erential Lie algebra with the Lie bracket [ ., . ] and α : L → L is an endomorphism such that(1) α [ x , y ] = [ α ( x ) , α ( y )] , UP-PRODUCT IN HOM-LEIBNIZ COHOMOLOGY AND HOM-ZINBIEL ALGEBRAS 3 (2) α ◦ d = d ◦ α. Then L is a Hom-Leibniz algebra with the new bracket defined as [ x , y ] d ,α : = [ α ( x ) , d α ( y )] . Example 2.8.
Let L is a two-dimensional C -vector space with basis { e , e } . Wedefine a bracket as [ e , e ] = e and zero else where and the endomorphism isgiven by the matrix α = " . It is easy to check that ( L , [ ., . ] , α ) is a Hom-Leibniz algebra which is not Hom-Lie. Definition 2.9.
A Hom-Zinbiel algebra is a triple ( R , () , α ) consisting of a vectorspace R together with two k -linear maps() : R ⊗ R → R and α : R → R , satisfying the relation(( xy ) α ( z )) = ( α ( x )( yz )) + ( α ( x )( zy )) , (1)for all x , y , z ∈ R .For a graded vector space R , we can define a notion of a graded Hom-Zinbielalgebra, satisfying the following graded relation(( xy ) α ( z )) = ( α ( x )( yz )) + ( − | y || z | ( α ( x )( zy )) . (2)2.1. Shu ffl e algebra. Let S p be the permutation group of p elements 1 , . . . , p . A permutation σ ∈ S p is called a ( n , m )-shu ffl e if n + m = p and σ (1) < · · · < σ ( n ) and σ ( n + < · · · < σ ( n + m ) . In the group algebra k [ S p ] , let sh n , m be the element sh n , m : = X σ σ, where the summation is over all ( n , m )-shu ffl es . For any vector space V we let σ ∈ S p act on V ⊗ p by σ ( v . . . v p ) = ( v σ − (1) . . . v σ − ( p ) ) , where the generator v ⊗ · · · ⊗ v p of V ⊗ p is denoted by v . . . v p . Note that the linear map from k [ S p ] to itself induced by σ sgn( σ ) σ − for σ ∈ S p is an anti-homomorphism. Let us denote the image of α ∈ k [ S p ] underthis map by ˜ α. For any non-negative integers n and m we define ρ n , m : = ⊗ e sh n − , m = L ⊗ n + m → L ⊗ n + m , (3)is given by ρ n , m ( x , . . . , x n + m ) = X σ sgn( σ )( x , x σ (2) , . . . , x σ ( n + m ) ) , (4) RIPAN SAHA where the above sum is over all ( n − , m )-shu ffl es σ .For the generators b = b . . . b m , c = c . . . c r . Suppose τ m , r is a permutationsuch that τ r , m ( cb ) = bc . (5) Remark 2.10.
For non-negative integers n , m , r , under the anti-homomorphism α → ˜ α we have the following equality( ρ n , m ⊗ r ) ◦ ρ n + m , r = (1 n ⊗ ρ m , r + ( − rm ◦ τ r , m ◦ ρ r , m ) ◦ ρ n , m + r (6)(See Proposition (1.8) of [6])3. S ome results on H om -L eibniz and H om -Z inbiel algebras In this section, we prove some results regarding Hom-Leibniz and Hom-Zinbiel algebras along the same line as Leibniz and Zinbiel algebras.
Proposition 3.1.
Suppose ( L , [ ., . ] , α ) is a Hom-Leibniz algebra and ( R , () , α ) isa Hom-Zinbiel algebra. Then the tensor product L ⊗ R together with the bracketand k-linear map [ x ⊗ r , y ⊗ s ] = [ x , y ] ⊗ ( rs ) − [ y , x ] ⊗ ( sr ) ,α ⊗ α : L ⊗ R → L ⊗ R , ( α ⊗ α )( x ⊗ r ) = α ( x ) ⊗ α ( r ) , is a Hom-Lie algebra.Proof. Skew-symmetry part is trivial. Hom-Jacobi identity is same as the fol-lowing relation X σ ∈ S sgn ( σ ) σ [ α ( x ) ⊗ α ( r ) , [ y ⊗ s , z ⊗ t ]] = , here permutation group S is acting same way on both sets x , y , z and r , s , t . Wehave, X σ ∈ S sgn ( σ ) σ (cid:0) [ α ( x ) , [ y , z ]] ⊗ ( α ( r )( st )) − [[ y , z ] , α ( x )] ⊗ (cid:0) ( st ) α ( r )) − [ α ( x ) , [ z , y ]] ⊗ ( α ( r )( st )) + [[ z , y ] , α ( x )] ⊗ (( ts ) α ( r )) (cid:1) = . Using Hom-Leibniz relation on L this is same as X σ ∈ S sgn ( σ ) σ (cid:0) ([[ x , y ] , α ( z )] − [[ x , z ] , α ( y )]) ⊗ ( α ( r )( st )) − [[ y , z ] , α ( x )] ⊗ (cid:0) ( st ) α ( r )) − ([[ x , z ] , α ( y )] − [[ x , y ] , α ( z )]) ⊗ ( α ( r )( st )) + [[ z , y ] , α ( x )] ⊗ (( ts ) α ( r )) (cid:1) = . Note that S acting symmetrically on the sum. Thus, it is enough to prove thatthe coe ffi cient of ([ x , y ] , α ( z )]] is 0.( α ( r )( st )) + ( α ( r )( ts )) − (( rs ) α ( t )) + ( α ( r )( st )) + ( α ( r )( ts )) − (( rs ) α ( t ))2(( α ( r )( st )) + ( α ( r )( ts )) − (( rs ) α ( t ))) = (cid:3) UP-PRODUCT IN HOM-LEIBNIZ COHOMOLOGY AND HOM-ZINBIEL ALGEBRAS 5
Proposition 3.2.
Let ( R , α ) be a Hom-Zinbiel algebra. Then, ( R , α , ∗ ) is aHom-associative and commutative algebra, where ∗ : R ⊗ R → R , x ∗ y = xy + yx . Proof.
Commutativity is trivial from the definition. We only need to check theHom-associativity. α ( x ) ∗ ( y ∗ z ) = α ( x ) ∗ (cid:0) ( yz ) + ( zy ) (cid:1) = α ( x ) (cid:0) ( yz ) + ( zy ) (cid:1) + (cid:0) ( yz ) + ( zy ) (cid:1) α ( x ) = α ( x ) (cid:0) ( yz ) + ( zy ) (cid:1) + ( yz ) α ( x ) + ( zy ) α ( x ) = α ( x ) (cid:0) ( yz ) + ( zy ) (cid:1) + α ( y ) (cid:0) ( zx ) + ( xz ) (cid:1) + α ( z ) (cid:0) ( yx ) + ( xy ) (cid:1) . ( x ∗ y ) ∗ α ( z ) = (cid:0) ( xy ) + ( yx ) (cid:1) ∗ α ( z ) = (cid:0) ( xy ) + ( yx ) (cid:1) α ( z ) + α ( z ) (cid:0) ( xy ) + ( yx ) (cid:1) = ( xy ) α ( z ) + ( yx ) α ( z ) + α ( z ) (cid:0) ( xy ) + ( yx ) (cid:1) = α ( x ) (cid:0) ( yz ) + ( zy ) (cid:1) + α ( y ) (cid:0) ( xz ) + ( zx ) (cid:1) + α ( z ) (cid:0) ( xy ) + ( yx ) (cid:1) . Thus, ( R , α , ∗ ) is a Hom-associative, commutative algebra. (cid:3)
4. C ohomology of H om -L eibniz algebras In this section, we introduce a cohomology for Hom-Leibniz algebras withcoe ffi cients in Hom-associative and commutative algebras. To define cochaincomplex suitable for our work, we need to consider multiplicative Hom-Leibnizalgebras. From now on by Hom-Leibniz algebra we always mean a multiplica-tive Hom-Leibniz algebra.Suppose ( L , [ ., . ] , α ) is a Hom-Leibniz algebra. Denote CL n ( L ) = L ⊗ n anddefine d n : CL n ( L ) → CL n − ( L ) , where d n ( x , . . . , x n ) = X ≤ i < j ≤ n ( − j + (cid:0) α ( x ) , . . . , α ( x i − ) , [ x i , x j ] , α ( x i + ) , . . . , [ α ( x j ) , . . . , α ( x n ) (cid:1) . (7) Theorem 4.1.
For all n ≥ , d n − d n = .Proof. Proof follows from the Theorem (3.1) of [1]. (cid:3)
Thus, we have a chain complex CL ♯ ( L ) : · · · → L ⊗ n d n → L ⊗ ( n − d n − → · · · d → L ⊗ d → L . The homology groups of this chain complex are called homology groups of theHom-Leibniz algebra ( L , [ ., . ] , α ) and n th homology group is denoted by HL α n ( L ).Note that for α = id , HL α n ( L ) is same as the Leibniz homology of L defined in[5]. Suppose ( L , [ ., . ] , α ) is a Hom-Leibniz algebra and let ( A , µ, α ) be a Hom-associative and commutative algebra over k . RIPAN SAHA
Set CL n α,α ( L ; A ) = { f : L ⊗ n → A | α ◦ f = f ◦ α ⊗ n } , for all n ≥
1. We define δ n α : CL n α,α ( L ; A ) → CL n + α,α ( L ; A )by δ n α ( c ) = c ◦ d n + , c ∈ CL n α,α ( L ; A ) , where d n + : L ⊗ ( n + → L ⊗ n is the bound-ary map (7). Explicitly, for any c ∈ CL n α,α ( L ; A ) and ( x , . . . , x n + ) ∈ L ⊗ ( n + ,δ n α ( c )( x , . . . , x n + ) is given by the expression X ≤ i < j ≤ n ( − j + c (cid:0) α ( x ) , · · · , α ( x i − ) , [ x i , x j ] , α ( x i + ) , . . . , [ α ( x j ) , . . . , α ( x n + ) (cid:1) . (8)Note that δ n α ( c ) need to satisfy the condition α ◦ δ n α ( c ) = δ n α ( c ) ◦ α ⊗ n to be amember of CL n + α,α ( L ; A ). This is why we need to consider multiplicative Hom-Leibniz algebras to define cochain complex in this way.Clearly, δ α = d = , and therefore, ( CL ♯α,α ( L ; A ) , δ ) is a cochain com-plex. Its homology groups are called Hom-Leibniz cohomology groups of L with coe ffi cients in A and n th cohomology group is denoted by HL n α,α ( L ; A ) .
5. C up - product in H om -L eibniz cohomology Main aim of this section is to define a cup-product operation on the gradedcohomology of the Hom-Leibniz algebra and show that the cup-product satisfiesthe graded Hom-Zinbiel relation.
Proposition 5.1.
Given a Hom-associative and commutative algebra ( A , µ, α ) there is a induced map α : HL n α,α ( L ; A ) → HL n α,α ( L ; A ) , for all n ≥ .Proof. Using the twisting map α of A we define a map α : CL n α,α ( L ; A ) → CL n α,α ( L ; A ) , f α ◦ f = f ◦ α ⊗ n . Suppose f ∈ CL n α,α ( L ; A ) is a cocycle. This implies δ n α f =
0. We need to show δ n α ( α ◦ f ) =
0. For all ( x . . . , x n + ), we have, δ n α ( α ◦ f )( x . . . , x n + ) = X ≤ i < j ≤ n + ( α ◦ f )( α ( x ) , . . . , α ( x i − ) , [ x i , x j ] , α ( x i + ) , . . . , [ α ( x j ) , . . . , α ( x n + )) = α (cid:16) X ≤ i < j ≤ n + f ( α ( x ) , . . . , α ( x i − ) , [ x i , x j ] , α ( x i + ) , . . . , [ α ( x j ) , . . . , α ( x n + )) (cid:17) = α ( δ n α f ) = . Now suppose f = δ n − α g for some g ∈ CL n − α,α ( L ; A ). Using the linearity of α wehave α ◦ f = α ◦ ( δ n − α g ) = δ n − α ( α ◦ g ). UP-PRODUCT IN HOM-LEIBNIZ COHOMOLOGY AND HOM-ZINBIEL ALGEBRAS 7
Thus, under the image of α cocycles goes to cocycles and coboundaries goesto coboundaries. So α induces a map on the cohomology level α : HL n α,α ( L ; A ) → HL n α,α ( L ; A ) . Note that we have denoted the induced map on the cohomology level by thesame notation as for the twisting map α of A . (cid:3) Lemma 5.2.
Given a Hom-Leibniz algebra ( L , [ ., . ] , α ) and a Hom-associative,commutative algebra ( A , µ, α ) , there is a bilinear operation on the graded spaceCL ∗ α,α ( L ; A ) , ∪ : CL n α,α ( L ; A ) × CL m α,α ( L ; A ) → CL n + m α,α ( L ; A ) , defined as f ∪ g : = µ ◦ ( α m − f ⊗ α n − g ) ◦ ρ n , m (9) : = µ ◦ ( f ◦ ( α m − ) ⊗ n ⊗ g ◦ ( α n − ) ⊗ m ) ◦ ρ n , m , (10) for f ∈ CL n α ( L ; A ) and g ∈ CL m α ( L ; A ) and the operation ∪ satisfies δ α ( f ∪ g ) = δ α ( f ) ∪ g + ( − | f | f ∪ δ α ( g ) . (11) Proof.
To prove δ α ( f ∪ g ) = δ α ( f ) ∪ g + ( − | f | f ∪ δ α ( g ) . (12)It is enough to show the following (cid:0) ( α m − ) ⊗ n ⊗ ( α n − ) ⊗ m (cid:1) ◦ ρ n , m ◦ d n + m + = (cid:0) ( d n + ◦ ( α m − ) ⊗ n + ) ⊗ ( α n ) ⊗ m (cid:1) ◦ ρ n + , m (13) + ( − n (cid:0) ( α m ) ⊗ n ⊗ ( d m + ◦ ( α n − ) ⊗ m + ) (cid:1) ◦ ρ n , m + . We define δ i , j α : L ⊗ p → L ⊗ p − , for 1 ≤ i < j ≤ p as δ i , j α ( x , . . . , x p ) : = (cid:0) α ( x ) , . . . , α ( x i − ) , [ x i , x j ] , α ( x i + ) , . . . , [ α ( x j ) , . . . , α ( x n ) (cid:1) .Note that, d p = P ≤ i < j ≤ p ( − j δ i , j α .Consider (cid:0) ( δ k , l α ◦ ( α m − ) ⊗ n + ) ⊗ ( α n ) ⊗ m (cid:1) (1 ⊗ σ − ) , (14)where 1 ≤ k < l ≤ n + σ is a ( n , m )-shu ffl e acting on { , . . . , n + m + } .This operator is a part of (cid:0) ( d n + ◦ ( α m − ) ⊗ n + ) ⊗ ( α n ) ⊗ m (cid:1) ◦ ρ n + , m .As σ is a ( n , m )-shu ffl e, σ ( k ) < σ ( l ). The equation (14) is same as the follow-ing equation (cid:0) ( α m − ) ⊗ n ⊗ ( α n − ) ⊗ m (cid:1) (1 ⊗ ω − ) δ σ ( k ) ,σ ( l ) α (15)for some permutation ω . As l ≤ n + σ is a ( n , m )-shu ffl e, ω is a ( n − , m )-shu ffl e.Thus, equation (15) is a part of (cid:0) ( α m − ) ⊗ n ⊗ ( α n − ) ⊗ m (cid:1) ◦ ρ n , m ◦ d n + m + . RIPAN SAHA
Now consider, (cid:0) ( α m ) ⊗ n ⊗ ( δ k , l α ◦ ( α n − ) ⊗ m + ) (cid:1) (1 ⊗ σ − ) , (16)where n + ≤ k < l ≤ n + m + σ is a ( n − , m + ffl e acting on { , . . . , n + m + } . This operator is a part of (cid:0) ( α m ) ⊗ n ⊗ ( d m + ◦ ( α n − ) ⊗ m + ) (cid:1) ◦ ρ n , m + .As σ is a ( n − , m + ffl e, σ ( k ) < σ ( l ) . The equation (16) is same as thefollowing equation (cid:0) ( α m − ) ⊗ n ⊗ ( α n − ) ⊗ m (cid:1) (1 ⊗ ω − ) δ σ ( k ) ,σ ( l ) α (17)for some permutation ω . As k ≥ n + σ is a ( n , m )-shu ffl e, ω is a ( n − , m )-shu ffl e.Thus, equation (17) is a part of (cid:0) ( α m − ) ⊗ n ⊗ ( α n − ) ⊗ m (cid:1) ◦ ρ n , m ◦ d n + m + .Using the same argument from [6], it is easy to check that both sides of (13)contains the same number of elements. We just showed that terms in the righthand side belongs to terms in the left hand side. This prove the validity of theequation (13). Therefore, the cup product is well-defined. (cid:3) Theorem 5.3.
Let ( L , [ ., . ] , α ) and ( L , [ ., . ] , ¯ α ) be two Hom-Leibniz algebrasand ( A , µ, α ) be any Hom-associative, commutative algebra. The cup-productis functorial, that is, if φ : L → L is a homomorphism then the induced map φ ∗ : HL n ¯ α,α ( L ; A ) → HL n α,α ( L ; A ) , φ ∗ ( f ) f ◦ φ ⊗ n , commutes with ∪ , that is, φ ∗ ( f ∪ g ) = φ ∗ ( f ) ∪ φ ∗ ( g ) . Proof.
Let f ∈ CL n ¯ α,α ( L ; A ) and g ∈ CL m ¯ α,α ( L ; A ). To prove the theorem, it issu ffi cient to show φ ∗ ( f ∪ g ) = φ ∗ ( f ) ∪ φ ∗ ( g )on cochains. φ ∗ ( f ) ∪ φ ∗ ( g ) = µ ◦ ( α m − φ ∗ ( f ) ⊗ α n − φ ∗ ( g )) ◦ ρ n , m = µ ◦ ( α m − f ◦ φ ⊗ n ⊗ α n − g ◦ φ ⊗ m ) ◦ ρ n , m = µ ◦ ( α m − f ⊗ α n − g ) ◦ ρ n , m ◦ φ ⊗ n + m = φ ∗ ( f ∪ g ) . (cid:3) Corollary 5.4. If φ : L → L is an isomorphism then the induced map on thecohomology level φ ∗ : HL n ¯ α,α ( L ; A ) → HL n α,α ( L ; A ) , is also an isomorphism. Remark 5.5.
If f ∪ f = , that is, f is a square zero element for some f ∈ HL n ¯ α,α ( L ; A ) and f ∪ f , for any f ∈ HL n α,α ( L ; A ) . Then we can concludethat L and L can not be ismorphic as Hom-Leibniz algebras. In this way thecup-product can be used to distinguish Hom-Leibniz algebras. UP-PRODUCT IN HOM-LEIBNIZ COHOMOLOGY AND HOM-ZINBIEL ALGEBRAS 9
Now we show the cup-product satisfes the graded Hom-Zinbiel relation.
Theorem 5.6.
The operation ∪ induces a well defined bilinear product on thegraded cohomology, called the cup-product, ∪ : HL n α,α ( L ; A ) × HL m α,α ( L ; A ) → HL n + m α,α ( L ; A ) , and the cup product ∪ together with the linear map α : HL n α,α ( L ; A ) → HL n α,α ( L ; A ) , satisfies the following graded Hom-Zinbiel formula α ( a ∪ b ) ∪ c = a ∪ α ( b ∪ c ) + ( − | b || c | a ∪ α ( c ∪ b ) , where a ∈ HL n α,α ( L ; A ) , b ∈ HL m α,α ( L ; A ) , c ∈ HL r α,α ( L ; A ) .Proof. Suppose a , b , c is represented by cocycles f , g , h respectively. From theformula (11) of Lemma 5.2 it is clear that if f , g are cocycles then f ∪ g isalso a cocycle and if either f or g is a coboundary then f ∪ g is a coboundary.Thus, the cup-product defined on the cochain level induces a cup-product on thecohomology level.Since A is commutative, we have, µ ◦ ( h ◦ ( α m − ) ⊗ r ⊗ g ◦ ( α r − ) ⊗ m ) ◦ ρ r , m = µ ◦ ( g ◦ ( α r − ) ⊗ m ) ⊗ h ◦ ( α m − ) ⊗ r ) ◦ τ r , m ◦ ρ r , m . Where τ r , m is a permutaion defined in (5). Now, pre-composing both sides of (6)by µ ◦ (( f ( α m + r − ) ⊗ n ⊗ g ( α n + r − ) ⊗ m ) ⊗ h ( α m + n − ) ⊗ r ) µ ◦ (( f ( α m + r − ) ⊗ n ⊗ g ( α n + r − ) ⊗ m ) ⊗ h ( α m + n − ) ⊗ r ) ◦ ( ρ n , m ⊗ r ) ◦ ρ n + m , r = µ ◦ (( f ( α m + r − ) ⊗ n ⊗ g ( α n + r − ) ⊗ m ) ⊗ h ( α m + n − ) ⊗ r ) ◦ (1 n ⊗ ρ m , r ) ◦ ρ n , m + r + ( − rm µ ◦ (( f ( α m + r − ) ⊗ n ⊗ g ( α n + r − ) ⊗ m ) ⊗ h ( α m + n − ) ⊗ r ) ◦ (1 n ⊗ τ r , m ) ◦ ρ r , m ◦ ρ n , m + r . This is same as µ ◦ (( f ( α m − ) ⊗ n ⊗ g ( α n − ) ⊗ m )( α r ) ⊗ m + n ⊗ h ( α m + n − ) ⊗ r ) ◦ ( ρ n , m ⊗ r ) ◦ ρ n + m , r = µ ◦ ( f ( α m + r − ) ⊗ n ⊗ ( g ( α r − ) ⊗ m ⊗ h ( α m − ) ⊗ r )( α n ) ⊗ m + r ◦ (1 n ⊗ ρ m , r ) ◦ ρ n , m + r + ( − rm µ ◦ ( f ( α m + r − ) ⊗ n ⊗ ( g ( α r − ) ⊗ m ⊗ h ( α m − ) ⊗ r )( α n ) ⊗ m + r ◦ (1 n ⊗ τ r , m ) ◦ ρ r , m ◦ ρ n , m + r . This implies µ ◦ ( α ( f ( α m − ) ⊗ n ⊗ g ( α n − ) ⊗ m )( α r − ) ⊗ m + n ⊗ h ( α m + n − ) ⊗ r ) ◦ ( ρ n , m ⊗ r ) ◦ ρ n + m , r = µ ◦ ( f ( α m + r − ) ⊗ n ⊗ α ( g ( α r − ) ⊗ m ⊗ h ( α m − ) ⊗ r )( α n − ) ⊗ m + r ◦ (1 n ⊗ ρ m , r ) ◦ ρ n , m + r + ( − rm µ ◦ ( f ( α m + r − ) ⊗ n ⊗ α ( g ( α r − ) ⊗ m ⊗ h ( α m − ) ⊗ r )( α n − ) ⊗ m + r ◦ (1 n ⊗ τ r , m ) ◦ ρ r , m ◦ ρ n , m + r . Thus, α ( f ∪ g ) ∪ h = f ∪ α ( g ∪ h ) + ( − | g || h | f ∪ α ( h ∪ g ) . (cid:3)
6. A simple computation
We illustrate our theory by giving a simple computation which is based on theExample 2.8 with coe ffi cients in a Hom-associative and commutative algebragiven in the Example 2.2. Here we first recall Examples 2.8 and 2.2.Let L be a two dimensional C -vector space with basis { e , e } . Then L is aHom-Leibniz algebra with respect to the following bracket and endomorphisms:[ e , e ] = e and zero elsewhere.The endomorphism α is given by α ( e ) = e ,α ( e ) = e + e . Let A be a two dimensional vector space with basis { a , a } . The multiplication µ of the Hom-associative and commutative algebra A is defined as: µ ( a i , a j ) = a if ( i , j ) = (1 , a if ( i , j ) , (1 , . The endomorphism α : A → A is defined by α ( a ) = a − a and α ( a ) = CL ♯ ( L ) : · · · → L ⊗ n d n → L ⊗ ( n − d n − → · · · d → L ⊗ d → L . We write boundary maps d n for n ≤ d n = n ≤ , d ( x , x ) = [ x , x ] , d ( x , x , x ) = − ([ x , x ] , α ( x )) + ([ x , x ] , α ( x )) + ( α ( x ) , [ x , x ]) . To understand the boundary maps d , d , it is enough to write down boundarymaps on the basis elements. Here we write only non-zero cases. d ( e , e ) = [ e , e ] = e , d ( e , e , e ) = ( α ( e ) , [ e , e ]) = ( e , e ) , d ( e , e , e ) = − ([ e , e ] , α ( e )) = − ( e , e ) , d ( e , e , e ) = ( α ( e ) , [ e , e ]) = ( e + e , e ) . We define a map f : L → A as f ( e ) = a − a , f ( e ) = . We need to check that α ◦ f = f ◦ α .( α ◦ f )( e ) = α ( f ( e )) = α (0) = , ( f ◦ α )( e ) = f ( α ( e )) = f ( e ) = , ( α ◦ f )( e ) = α ( f ( e )) = α ( a − a ) = a − a , ( f ◦ α )( e ) = f ( α ( e )) = f ( e + e ) = f ( e ) = a − a . Thus, f ∈ CL α,α ( L , A ). Now, f ∪ f = µ ◦ ( f ⊗ f ) ◦ ρ , ∈ CL α,α ( L , A ) . UP-PRODUCT IN HOM-LEIBNIZ COHOMOLOGY AND HOM-ZINBIEL ALGEBRAS 11
Note that ρ , ( e , e ) = ρ , ( e , e ) = ,ρ , ( e , e ) = ( e , e ) ,ρ , ( e , e ) = ( e , e ) . One can check easily from the definition of the cup-product that( f ∪ f )( e , e ) = , ( f ∪ f )( e , e ) = , ( f ∪ f )( e , e ) = , ( f ∪ f )( e , e ) = . Thus, f ∪ f = . Remark 6.1.
It may be remarked that although cohomology operations are wellunderstood in topology and these operations are useful in distingushing spaces,however these operations have not been explored well in the context of cohomol-ogy of algebras. Having a cup-product is the first step towards it. In a futureproject we intend to address this problem in general, and hence, cup-product incohomology of Leibniz algebras or Hom Leibniz algebras are being studied atpresent. Further, any new algebraic structure on cohomology of a type of alge-bras helps in the classification problem of the algebras of a given type. This isone of the motivation behind introducing cup-product.
Acknowledgements:
The author express his gratitute to the esteemed refereefor her / his useful comments and suggestions on the earlier version of the man-uscript that have improved the exposition. The author would like to thank Prof.Goutam Mukherjee of Indian Statistical Institute, Kolkata, for his guidance andreading a draft version of this article. The author is also thankful to Apurba Dasand Surojit Ghosh for their help and useful comments on this article.R eferences [1] Y. S. Cheng and Y. C. Su, (Co)homology and universal central extensions of Hom-Leibniz algebras, Acta Math. Sin. (Engl. Ser.) (2011), no. 5, 813-830.[2] A. Das, Gerstenhaber algebra structure on the cohomology of a hom-associativealgebra, arXiv:1805.01207 [3] M. Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (1963) 267-288.[4] J.T.. Hartwig, D. Larsson and S. D. Silvestrov, Deformations of Lie algebras using σ -derivations, J. Algebra (2006), 314-361.[5] J.-L. Loday, Une version non commutative des alg` e bres de Lie: les alg` e bres deLeibniz, Ens. Math. (3-4) (1993) 269-293.[6] J.-L. Loday, Cup-product for Leibniz cohomology and dual Leibniz algebras, Math.Scand. (2) (1995) 189-196.[7] A. Makhlouf and S.D. Silvestrov. Hom-algebra structures, J. Gen. Lie Theory Appl. , (2008), 51-64. [8] A. Makhlouf, P. Zusmanovich, Hom-Lie structures on KacMoody algebras, Journalof Algebra , Volume 515, 2018, 278-297.[9] G. Mukherjee, R. Saha, Cup-product for equivariant Leibniz cohomology and Zin-biel algebras,
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