aa r X i v : . [ m a t h . R A ] J a n O n B i H om - associative dialgebras A hmed Z ahari * Universit´e de Haute Alsace,IRIMAS-D´epartement de Math´ematiques,6, rue des Fr`eres Lumi`ere F-68093 Mulhouse, France. I brahima BAKAYOKO † D´epartement de Math´ematiques, Universit´e de N’Z´er´ekor´e,BP 50 N’Z´er´ekor´e, Guin´ee.
Abstract.
The aim of this paper is to introduce and study BiHom-associative dialgebras. We givevarious constructions and study its connections with BiHom-Poisson dialgebras and BiHom-Leibnizalgebras. Next we discuss the central extensions of BiHom-diassociative and we describe the clas-sification of n -dimensional BiHom-diassociative algebras for n ≤
4. Finally, we discuss their deriva-tions.
AMS Subject Classification: . Keywords : BiHom-associative dialgebra, BiHom-Poisson dialgebra, centroid, averaging operator,Nijenhuis operator, Rota-Baxter operator, Extension, Derivation, Classification.
The associative dialgebras (also known as diassociative algebras) has been introduced by Lodayin 1990 (see [6] and references therein) as a generalization of associative algebras. They are ageneralization of associative algebras in the sens that they possess two associative multiplicationsand obey to three other conditions; when the two associative low are equal we recover associativealgebra. One of his motivation were to find an algebra whose commutator give rises to Leibnizalgebra as it is the case in the relation between Lie and associative algebra. Another motivationcome from the research of an obstruction to the periodicity in algebraic K-theory. Now, thesealgebras found their applications in classical geometry, non-commutative geometry and physics.The centroid plays an important role in understanding forms of an algebra. It is an element inthe classification of associative and diassociative algebras. They occurs naturally is in the studyof derivations of an algebra. The centroid and averaging operators are used in the deformation ofalgebra in order to generate another algebraic structure. The Nijenhuis operator on an associativealgebra was introduced in [16] to study quantum bi-Hamiltonian systems while the notion Nijenhuis * e-mail address: [email protected] † e-mail address: [email protected] { σ, τ } -Rota-Baxter operators, infinitesimalHom-bialgebras and the associative (Bi)Hom-Yang-Baxter equation [22], The construction and de-formation of BiHom-Novikov algebras [27], On n-ary Generalization of BiHom-Lie algebras andBiHom-Associative Algebras [3], Rota-Baxter operators on BiHom-associative algebras and relatedstructures [19].The goal of this paper is to introduce, classify and study structures, central extensions andderivations of BiHom-associative algebras. The paper is organized as follows. In section 2, wedefine BiHom-associative dialgebras, give some constructions using twisting, direct sum, elementsof centroid, averaging operator, Nijenhuis operator and Rota-Baxter relation. We give a connec-tion between BiHom-associative dialgebras and BiHom-Leibniz algebras. We introduce actionof a BiHom-Leibniz algebra onto another and give a Leibniz structure on the semidirect struc-ture. Then, we show that the semidirect sum of BiHom-Leibniz algebras associated to BiHom-associative dialgebras is the same that the BiHom-Leibniz algebra associated to the semidirect ofBiHom-associative dialgebras. Finally, we introduce BiHom-associative dialgebras and show thatany BiHom-associative dialgebra carries a structure of BiHom-Poisson dialgebra. In section 3, weintroduce the notion of central extension of BiHom-associative dialgebras and define 2-cocycles and2-coboundaries of BiHom-associative dialgebras with coe ffi cients in a trivial BiHom-module. Thenwe establish relationship between 2-cocycles and central extensions. Section 4, is devoted to theclassification of n -dimensional BiHom-associative dialgebras for n ≤
4. We dedicated Section 5 tothe derivations of BiHom-associative dialgebras.
Definition 2.1.
A BiHom-associative dialgebras is a 5-truple ( A , ⊣ , ⊢ , α, β ) consisting of a linearspace A linear maps ⊣ , ⊢ , : A × A −→ A and α, β : A −→ A satisfying, for all x , y , z ∈ A the followingconditions : α ◦ β = β ◦ α, (2.1)( x ⊣ y ) ⊣ β ( z ) = α ( x ) ⊣ ( y ⊣ z ) , (2.2)( x ⊣ y ) ⊣ β ( z ) = α ( x ) ⊣ ( y ⊢ z ) , (2.3)( x ⊢ y ) ⊣ β ( z ) = α ( x ) ⊢ ( y ⊣ z ) , (2.4)( x ⊣ y ) ⊢ β ( z ) = α ( x ) ⊢ ( y ⊢ z ) , (2.5)( x ⊢ y ) ⊢ β ( z ) = α ( x ) ⊢ ( y ⊢ z ) . (2.6)We called α and β ( in this order ) the structure maps of A. Example 2.2.
Any Hom-associative dialgebra [10] or any associative dialgebra is a BiHom-associativedialgebra by setting β = α or α = β = id. hmed Zahari and Ibrahima Bakayoko 3 Example 2.3.
Let ( A , ⊣ , ⊢ , α, β ) a BiHom-associative dialgebra. Consider the module of n × n-matrices M n ( D ) = M n ( K ) ⊗ D with the linear maps α ( A ) = ( α ( a i j )) , β ( A ) = ( β ( a i j ) for all A ∈ M n ( D ) and the products ( a ⊳ b ) i j = P k a ik ⊣ b k j and ( a ⊲ b ) i j = P k a ik ⊢ b k j . Then, ( M n ( D ) , ⊳, ⊲, α, β ) is aBiHom-associative dialgebra. Definition 2.4.
A morphism f : ( D , ⊣ , ⊢ , α, β ) and ( D ′ , ⊣ ′ , ⊢ ′ , α ′ , β ′ ) be a BiHom-associative dial-gebras is a linear map f : D → D ′ such that α ′ ◦ f = f ◦ α, β ′ ◦ f = f ◦ β and f ( x ⊣ y ) = f ( x ) ⊣ ′ f ( y ) , f ( x ⊢ y ) = f ( x ) ⊢ ′ f ( y ), for all x , y ∈ D . Definition 2.5.
A BiHom-associative dialgebra ( A , ⊣ , ⊢ , α, β ) in which α and β are morphism is saidto be a multiplicative BiHom-associative dialgebra.If moreover, α and β are bijective (i.e. automorphisms), then ( A , ⊣ , ⊢ , α, β ) is said to be a regularBiHom-associative dialgebra.We prove in the following proposition that any BiHom-associative dialgebra turn to another onevia morphisms. Theorem 2.6.
Let ( D , ⊣ , ⊢ , α, β ) be a BiHom-associative dialgebra and α ′ , β ′ : D → D two morphismsof BiHom-associative dialgebras such that the maps α, α ′ , β, β ′ commute pairewise. ThenD ( α ′ ,β ′ ) = ( D , ⊳ : = ⊣ ( α ′ ⊗ β ′ ) , ⊲ : = ⊢ ( α ′ ⊗ β ′ ) , αα ′ , ββ ′ ) is a BiHom-associative dialgebra.Proof. We prove only one axiom and leave the rest to the reader. For any x , y , z ∈ D ,( x ⊳ y ) ⊳ ββ ′ ( z ) − αα ′ ( x ) ⊳ ( y ⊲ z ) = α ′ ( α ′ ( x ) ⊣ β ′ ( y )) ⊣ β ′ ββ ′ ( z ) − α ′ αα ′ ( x ) ⊣ β ′ ( α ′ ( y ) ⊢ β ′ ( z )) = ( α ′ α ′ ( x ) ⊣ α ′ β ′ ( y )) ⊣ ββ ′ β ′ ( z ) − αα ′ α ′ ( x ) ⊣ ( α ′ β ′ ( y ) ⊢ β ′ β ′ ( z )) . The left hand side vanishes by (2.3). And, this ends the proof. (cid:3)
Corollary 2.7.
Let ( D , ⊣ , ⊢ , α, β ) be a multiplicative BiHom-associative dialgebra. Then ( D , ⊣ ◦ ( α n ⊗ β n ) , ⊢ ◦ ( α n ⊗ β n ) , α n + , β n + ) is also a multiplicative BiHom-associative dialgebra.Proof. It su ffi ses to take α ′ = α n and β ′ = β n in Theorem 2.6. (cid:3) Corollary 2.8.
Let ( D , ⊣ , ⊢ , α ) be a multiplicative Hom-associative dialgebra and β : D → D anendomorphism of D. Then ( D , ⊣ ◦ ( α ⊗ β ) , ⊢ ◦ ( α ⊗ β ) , α , β ) is also a Hom-associative dialgebra.Proof. It su ffi ses to take α ′ = α and replace β by Id D , and β ′ by β in Theorem 2.6. (cid:3) Any regular Hom-associative dialgebra give rises to associative dialgebra as stated in the nextcorollary.
Corollary 2.9. If ( D , ⊣ , ⊢ , α, β ) is a regular BiHom-associative dialgebra, then ( D , ⊣ ◦ ( α − ⊗ β − ) , ⊢ ◦ ( α − ⊗ β − )) is an associative dialgebra. Proof.
We have to take α ′ = α − and β ′ = β − in Theorem 2.6. (cid:3) Corollary 2.10.
Let ( D , ⊣ , ⊢ ) be an associative dialgebra and α : D → D and β : D → D a pair ofcommuting endomorphisms of D. Then ( D , ⊣ ◦ ( α ⊗ β ) , ⊢ ◦ ( α ⊗ β ) , α, β ) is a BiHom-associative dialgebra.Proof. We have to take α = β = Id D and replace α ′ by α , and β ′ by β in Theorem 2.6. (cid:3) Definition 2.11.
Let ( D , ⊣ , ⊢ , α, β ) be a BiHom-associative dialgebra. For any integers k , l , an evenlinear map θ : D → D is called an element of ( α k , β l )-centroid on D if α ◦ θ = θ ◦ α, β ◦ θ = θ ◦ β, (2.7) θ ( x ) ⊣ α k β l ( y ) = θ ( x ) ⊣ θ ( y ) = α k β l ( x ) ⊣ θ ( y ) , (2.8) θ ( x ) ⊢ α k β l ( y ) = θ ( x ) ⊢ θ ( y ) = α k β l ( x ) ⊢ θ ( y ) , (2.9)for all x , y ∈ D .The set of elements of centroid is called centroid. Proposition 2.12.
Let ( A , ⊣ , ⊢ , α, β ) a BiHom-associative dialgebra and φ : A → A and ψ : A → A bea paire of commuting elements of cenroid. Let us definedx ⊳ y : = φ ( x ) ⊣ y and x ⊲ y : = ψ ( x ) ⊢ y . Then, ( A , ⊳, ⊲, α, β ) is a BiHom-associative dialgebra if and only ifIm ( φ − ψ ) ∈ Z ⊣ ( A ) : = { x ∈ A / x ⊣ y = , ∀ y ∈ A } and Im ( φ − ψ ) ∈ Z ⊢ ( A ) : = { x ∈ A / y ⊢ x = , ∀ y ∈ A } .Proof. We only prove axioms (2.3) and (2.5), the three other comes from BiHom-associativity. Sofor any x , y , z ∈ A ,( x ⊳ y ) ⊳ β ( z ) − α ( x ) ⊳ ( y ⊲ z ) = ( φ ( x ) ⊣ y ) ⊣ φβ ( z ) − φα ( x ) ⊣ ( y ⊢ ψ ( z )) = ( φ ( x ) ⊣ y ) ⊣ βφ ( z ) − αφ ( x ) ⊣ ( y ⊢ ψ ( z )) = ( φ ( x ) ⊣ y ) ⊢ βφ ( z ) − ( φ ( x ) ⊣ y ) ⊢ βψ ( z )) = ( φ ( x ) ⊣ y ) ⊢ β ( φ − ψ )( z ) = αφ ( x ) ⊣ ( y ⊢ ( φ − ψ )( z )) . and ( x ⊳ y ) ⊲ β ( z ) − α ( x ) ⊲ ( y ⊲ z ) = ( φ ( x ) ⊣ y ) ⊢ βψ ( z ) − ψα ( x ) ⊢ ( y ⊢ ψ ( z )) = ( φ ( x ) ⊣ y ) ⊢ βψ ( z ) − αψ ( x ) ⊢ ( y ψ ( z )) = ( φ ( x ) ⊣ y ) ⊢ βψ ( z ) − ( ψ ( x ) ⊣ y ) ⊢ βψ ( z ) = [( φ ( x ) − ψ ( x )) ⊣ y ] ⊢ βψ ( z ) . A study of cancellation of the two equalities allows to conclude. (cid:3)
Proposition 2.13.
Let ( A , · , α, β ) be a BiHom-associative algebra, and φ : A → A and ψ : A → A bea paire of commuting elements of cenroid. Let us definedx ⊣ y : = φ ( x ) · y and x ⊢ y : = ψ ( x ) · y . Then, ( A , ⊣ , ⊢ , α, β ) is a BiHom-associative dialgebra if and only if Im ( φ − ψ ) is contained in the setof isotropic vectors. hmed Zahari and Ibrahima Bakayoko 5 Proof.
We only prove axioms (2.3) and (2.5), the three other comes from BiHom-associativity. Sofor any x , y , z ∈ A , ( x ⊣ y ) ⊣ β ( z ) − α ( x ) ⊣ ( y ⊢ z ) = ( φ ( x ) y ) φβ ( z ) − φα ( x )( y ψ ( z )) = ( φ ( x ) y ) βφ ( z ) − αφ ( x )( y ψ ( z )) = ( φ ( x ) y ) βφ ( z ) − ( φ ( x ) y ) βψ ( z )) = ( φ ( x ) y ) β ( φ − ψ )( z ) = αφ ( x )( y ( φ − ψ )( z )) . and ( x ⊣ y ) ⊢ β ( z ) − α ( x ) ⊢ ( y ⊢ z ) = ( φ ( x ) y ) βψ ( z ) − ψα ( x )( y ψ ( z )) = ( φ ( x ) y ) βψ ( z ) − αψ ( x )( y ψ ( z )) = ( φ ( x ) y ) βψ ( z ) − ( ψ ( x ) y ) βψ ( z ) = [( φ ( x ) − ψ ( x )) y ] βψ ( z ) . A study of cancellation of the two equalities allow to conclude. (cid:3)
Remark . Proposition 2.13 may be seen as a consequence of Proposition 2.12.
Proposition 2.15.
Let ( A , · , α, β ) be a BiHom-associative algebra and ( M , ∗ L , ∗ R , α M , β M ) an A-BiHom-bimodule i.e. M is a vector space, α M : M → M and β M : M → M are two linear maps,and ∗ L : A → M and ∗ R : M → A two bilinear maps such that α ( x ) ∗ L ( y ∗ L m ) = ( x · y ) ∗ L β M ( m ) (2.10) α ( x ) ∗ L ( m ∗ R y ) = ( x ∗ L m ) ∗ R β ( y ) (2.11) α M ( m ) ∗ R ( x · y ) = ( m ∗ R x ) ∗ R β ( y ) . (2.12) Suppose that f : M → A is a morphism of A-BiHom-bimodule i.e. f is linear such that α ◦ f = f ◦ α M , β ◦ f = f ◦ β M and f ( x ∗ L m ) = x · f ( m ) (2.13) f ( m ∗ R x ) = f ( m ) · x . (2.14)(2.15) Then, ( M , ⊳, ⊲, α M , β M ) is a BiHom-associative dialgebra withm ⊳ n = f ( m ) ∗ R n and m ⊲ n = m ∗ R f ( n ) , for all m , n ∈ M.Proof.
We only prove axiom (2.6), the other being proved similarly. For any x , y , z ∈ A ,( m ⊳ n ) ⊲ β M ( p ) = ( f ( m ) ∗ L n ) ∗ R f β M ( p ) = ( f ( m ) ∗ L n ) ∗ R β f ( p ) . By (2.11), ( m ⊳ n ) ⊲ β M ( p ) = α f ( m ) ∗ L ( n ∗ R f ( p )) = f α M ( m ) ∗ L ( n ⊲ p ) = α M ( m ) ⊳ ( n ⊲ p ) . This completes the proof. (cid:3)
Remark . Any ( α , β )-element of centroid of a BiHom-associative algebra is a morphism ofBiHom-bimodule.Thanks to the above remark, we have what follows : Corollary 2.17.
Let ( A , · , α, β ) be a BiHom-associative algebra and let θ be an element of cenroidon A. Then, ( A , ⊳, ⊲, α, β ) is a BiHom-associative dialgebra withx ⊳ y = θ ( x ) · y and x ⊲ y = x · θ ( y ) , for any x , y ∈ A . Proposition 2.18.
Let ( D , ⊣ , ⊢ , α, β ) be a BiHom-associative dialgebra and R : D → D a Rota-Baxteroperator of weight on D i.e. R is linear and α ◦ R = R ◦ α , β ◦ R = R ◦ β , andR ( x ) ⊣ R ( y ) = R ( R ( x ) ⊣ y + x ⊣ R ( y )) (2.16) R ( x ) ⊢ R ( y ) = R ( R ( x ) ⊢ y + x ⊢ R ( y )) (2.17) Then, ( D , ⊳, ⊲, α, β ) is also a BiHom-associative algebra withx ⊳ y = R ( x ) ⊣ y + x ⊣ R ( y ) , (2.18) x ⊲ y = R ( x ) ⊢ y + x ⊢ R ( y ) , (2.19) for all x , y ∈ D.Proof.
We only prove axiom (2.6), the other being proved in a similar way. Thus, For any x , y , z ∈ A ,( x ⊳ y ) ⊳ β ( z ) − α ( x ) ⊳ ( y ⊲ z ) == ( x ⊣ R ( y ) + R ( x ) ⊣ y ) ⊣ R β ( z ) + R ( R ( x ) ⊣ y + x ⊣ R ( y )) ⊣ β ( z ) − α ( x ) ⊣ R ( R ( y ) ⊣ z + y ⊣ R ( z )) − R α ( x ) ⊣ ( R ( y ) ⊢ z + y ⊢ R ( z )) = ( x ⊣ R ( y )) ⊣ β R ( z ) + ( R ( x ) ⊣ y ) ⊣ β R ( z ) + ( R ( x ) ⊣ R ( y )) ⊣ β ( z ) − α ( x ) ⊣ ( R ( y ) ⊢ R ( z )) − α R ( x ) ⊣ ( y ⊢ R ( z )) α R ( x ) ⊣ ( R ( y ) ⊢ z ) . The left hand side vanishes by axiom (2.6). This ends the proof. (cid:3)
Corollary 2.19.
Let ( D , ⊣ , ⊢ , α, β ) BiHom-associative dialgebra and R : D → D a Rota-Baxter oper-ator of weight on D. Then, ( D , ∗ , α, β ) is a BiHom-associative algebra with x ∗ y = x ⊳ y + x ⊲ y. Corollary 2.20.
Let ( D , ⊣ , ⊢ , α, β ) be a BiHom-associative dialgebra and R : D → D a Rota-Baxteroperator of weight on D. Then, ( D , [ − , − ] , α, β ) is a BiHom-Lie algebra with [ x , y ] = x ∗ y − α − β ( y ) ∗ αβ − ( x ) , with x ∗ y = x ⊳ y + x ⊲ y. As in the previous proposition, it is well known that a Nijenhuis operator on an associativealgebra allows to define another associative algebra. In the next result, we try to establish an analoqof this result for BiHom-associative dialgebras.hmed Zahari and Ibrahima Bakayoko 7
Proposition 2.21.
Let ( D , ⊣ , ⊢ , α, β ) BiHom-associative dialgebra and N : D → D a Nijenhuis oper-ator on D i.e. N is linear and α ◦ N = N ◦ α , β ◦ N = N ◦ β , andN ( x ) ⊣ N ( y ) = N ( N ( x ) ⊣ y + x ⊣ N ( y ) − N ( x · y )) (2.20) N ( x ) ⊢ N ( y ) = N ( N ( x ) ⊢ y + x ⊢ N ( y ) − N ( x · y )) (2.21) Then, ( D , ⊳, ⊲, α, β ) is also a BiHom-associative algebra withx ⊳ y = N ( x ) ⊣ y + x ⊣ N ( y ) − N ( x ⊣ y ) , (2.22) x ⊲ y = N ( x ) ⊢ y + x ⊢ N ( y ) − N ( x ⊢ y ) , (2.23) for all x , y ∈ D.Proof.
We only prove axiom (2.4) for the products ⊳ and ⊲ . The others are leave to the reader.( x ⊲ y ) ⊳ β ( z ) − α ( x ) ⊲ ( y ⊳ z ) == N (cid:16) N ( x ) ⊢ y + x ⊣ y − N ( x ⊣ y ) (cid:17) ⊣ β ( z ) + (cid:16) N ( x ) ⊣ y + x ⊢ N ( y ) − N ( x ⊢ y ) (cid:17) ⊣ N β ( z ) − N (cid:16) ( N ( x ) ⊢ y + x ⊢ N ( y ) − N ( x ⊢ y )) ⊣ β ( z ) (cid:17) − N α ( x ) ⊢ (cid:16) N ( y ) ⊣ z + y ⊣ N ( z ) − N ( y ⊣ z ) (cid:17) − α ( x ) ⊢ N (cid:16) N ( y ) ⊣ z + y ⊣ N ( z ) − N ( y ⊣ z ) (cid:17) + N (cid:16) α ( x ) ⊢ ( N ( y ) ⊣ z + y ⊣ N ( z ) − N ( y ⊣ z )) (cid:17) . By (2.20) and (2.21), we have( x ⊲ y ) ⊳ β ( z ) − α ( x ) ⊲ ( y ⊳ z ) == ( N ( x ) ⊢ N ( y )) ⊣ β ( z ) + ( N ( x ) ⊣ y ) ⊣ β N ( z ) + ( x ⊢ N ( y )) ⊣ β N ( z ) − N ( x ⊢ y )) ⊣ N β ( z ) − N (cid:16) ( N ( x ) ⊢ y ) ⊣ β ( z ) (cid:17) − N (cid:16) ( x ⊢ N ( y )) ⊣ β ( z ) (cid:17) − N (cid:16) N ( x ⊢ y ) ⊣ β ( z ) (cid:17) − α N ( x ) ⊢ ( N ( y ) ⊣ z ) − α N ( x ) ⊢ ( y ⊣ N ( z )) + N α ( x ) ⊢ N ( y ⊣ z )) − α ( x ) ⊢ ( N ( y ) ⊣ N ( z )) + N (cid:16) α ( x ) ⊢ ( N ( y ) ⊣ z (cid:17) + N (cid:16) α ( x ) ⊢ ( y ⊣ N ( z )) (cid:17) − N (cid:16) α ( x ) ⊢ N ( y ⊣ z )) (cid:17) . By (2.4), we have( x ⊲ y ) ⊳ β ( z ) − α ( x ) ⊲ ( y ⊳ z ) == − N ( x ⊢ y )) ⊣ N β ( z ) − N (cid:16) ( N ( x ) ⊢ y ) ⊣ β ( z ) (cid:17) − N (cid:16) ( x ⊢ N ( y )) ⊣ β ( z ) (cid:17) − N (cid:16) N ( x ⊢ y ) ⊣ β ( z ) (cid:17) + N α ( x ) ⊢ N ( y ⊣ z )) + N (cid:16) α ( x ) ⊢ ( N ( y ) ⊣ z (cid:17) + N (cid:16) α ( x ) ⊢ ( y ⊣ N ( z )) (cid:17) − N (cid:16) α ( x ) ⊢ N ( y ⊣ z )) (cid:17) . Using again (2.20) and (2.21), it comes( x ⊲ y ) ⊳ β ( z ) − α ( x ) ⊲ ( y ⊳ z ) == − N (cid:16) N ( x ⊢ y ) ⊣ β ( z ) + ( x ⊢ y ) ⊣ β N ( z ) − N (( x ⊢ y ) ⊣ β ( z )) (cid:17) − N (cid:16) ( N ( x ) ⊢ y ) ⊣ β ( z ) (cid:17) − N (cid:16) ( x ⊢ N ( y )) ⊣ β ( z ) (cid:17) − N (cid:16) N ( x ⊢ y ) ⊣ β ( z ) (cid:17) + N (cid:16) α ( x ) ⊢ ( N ( y ) ⊣ z ) + α ( x ) ⊢ N ( y ⊣ z ) − N ( α ( x ) ⊢ ( y ⊣ z )) (cid:17) + N (cid:16) α ( x ) ⊢ ( N ( y ) ⊣ z (cid:17) + N (cid:16) α ( x ) ⊢ ( y ⊣ N ( z )) (cid:17) − N (cid:16) α ( x ) ⊢ N ( y ⊣ z )) (cid:17) . The left hand side vanishes by (2.4). (cid:3)
Corollary 2.22. If ( D , ⊣ , ⊢ , α ) is a Hom-associative dialgebra and N : D → D a Nijenhuis operatoron D, then ( D , ⊳, ⊲, α ) is also a Hom-associative algebra withx ⊳ y = N ( x ) ⊣ y + x ⊣ N ( y ) − N ( x ⊣ y ) , x ⊲ y = N ( x ) ⊢ y + x ⊢ N ( y ) − N ( x ⊢ y ) , for all x , y ∈ D. Corollary 2.23. If ( D , ⊣ , ⊢ , α, β ) is an associative dialgebra and N : D → D a Nijenhuis operator onD, then ( D , ⊳, ⊲, α, β ) is also an associative algebra withx ⊳ y = N ( x ) ⊣ y + x ⊣ N ( y ) − N ( x ⊣ y ) , x ⊲ y = N ( x ) ⊢ y + x ⊢ N ( y ) − N ( x ⊢ y ) , for all x , y ∈ D. The next proposition asserts that the twist of the products of any BiHom-associative dialgebraby an averaging operator gives rise to another BiHom-associative dialgebra.
Proposition 2.24.
Let ( D , ⊣ , ⊢ , α, β ) be a BiHom-associative dialgebra and θ : D → D an injectiveaveraging operator on D i.e. θ is an injective linear map such that α ◦ θ = θ ◦ α , β ◦ θ = θ ◦ β , and θ ( x ) ⊣ θ ( y ) = θ ( α k β l ( x ) ⊣ θ ( y )) = θ ( θ ( x ) ⊣ α k β l ( y )) , (2.24) θ ( x ) ⊢ θ ( y ) = θ ( α k β l ( x ) ⊢ θ ( y )) = θ ( θ ( x ) ⊢ α k β l ( y )) , (2.25) for any x , y ∈ D. Then, ( D , ⊳, ⊲, α, β ) is also a BiHom-associative algebra withx ⊳ y = θ ( x ) ⊣ α k β l ( y )) (2.26) x ⊲ y = α k β l ( x ) ⊢ θ ( y ) , (2.27) for all x , y ∈ D.Proof.
We only prove one identity, the others have a similar proof. For any x , y , z ∈ D , one has : θ [( x ⊳ y ) ⊲ β ( z ) − α ( x ) ⊲ ( y ⊳ z )] == θ [ θ ( θ ( x ) ⊣ α k β l ( y ))) ⊢ α k β l + ( z )) − θα ( x ) ⊢ ( θ ( y ) ⊣ α k β l ( z )))] = θ [( θ ( x ) ⊣ θ ( y ) ⊢ α k β l + ( z )] − θα ( x ) ⊢ θ ( θ ( y ) ⊣ α k β l ( z )))] = ( θ ( x ) ⊢ θ ( y )) ⊣ βθ ( z ) − αθ ( x ) ⊢ ( θ ( y ) ⊣ θ ( z )) . Which vanishes by axiom (2.4), and the conclusion holds by injectivity. (cid:3)
At this moment, we introduce ideals for BiHom-associative dialgebra in order to give anotherconstruction of BiHom-associative dialgebras.
Definition 2.25.
Let ( D , ⊣ , ⊢ , α, β ) be a BiHom-associative dialgebra and D o a subset of D . We saythat D o is a BiHom-subalgebra of D if D o is stable under α and β , and x ⊣ y , x ⊢ y ∈ D o , for any x , y ∈ D o . Example 2.26. If ϕ : D → D is a homomorphism of BiHom-associative dialgebras, the image Im ϕ is a BiHom-subalgebra of D . Definition 2.27.
A two side BiHom-ideal of a BiHom-associative dialgebra ( D , ⊣ , ⊢ , α, β ) is subspace I such that α ( I ) ⊂ I , x ∗ y , y ∗ x ∈ I for all x ∈ D , y ∈ I with ∗ = ⊣ and ⊢ . Note that I is called the leftand right BiHom-ideal if x ⊣ y , x ⊢ y and y ⊣ x , y ⊢ x are in I , respectively, for all x ∈ D . y ∈ I . Example 2.28. i) Obviously I = { } and I = D are two-sided ideals.ii) If ϕ : D → D is a homomorphism of BiHom-associative dialgebras, the kernel Ker ϕ is a twosided ideal in D .iii) If I and I are two sided ideals of D, then so is I + I . hmed Zahari and Ibrahima Bakayoko 9In the below proposition, we prove that BiHom-associative dialgebras are closed under directsummation, and give a condition for which a linear map becomes a morphism. Proposition 2.29.
Let ( A , ⊣ A , ⊢ A , α A , β A ) and ( B , ⊣ B , ⊢ B , α B , β B ) be two BiHom-associative dialge-bras. Then there exists a BiHom-associative dialgebra structure on A ⊕ B with the bilinear maps ⊳, ⊲ : ( A ⊕ B ) ⊗ → A ⊕ B given by ( a + b ) ⊣ ( a + b ) = a ⊣ A a + b ⊣ B b , ( a + b ) ⊢ ( a + b ) = a ⊢ A a + b A ⊢ B b and the linear maps α = α A + α B , β = β A + β B : A ⊕ B → A ⊕ B given by ( α A + α B )( a + b ) = α A ( a ) + α B ( b ) , ( β A + β B )( a + b ) = β A ( a ) + β B ( b ) , ∀ ( a , b ) ∈ ( A × B ) . Moreover, if ξ : A → B is a linear map. Then ξ : ( A , ⊣ A , ⊢ A , α A , β A ) to ( B , ⊣ B , ⊢ B , α B , β B ) is a morphismif and only if its graph Γ ξ = { ( x , ξ ( x )) , x ∈ A } is a BiHom-subalgebra of ( A ⊕ B , ⊳, ⊲, α, β ) .Proof. The proof of the first part of the proposition comes from a simple computation.Let us suppose that ξ : ( A , ⊣ A , ⊢ A , α A , β A ) → ( B , ⊣ B , ⊢ B , α B , β B ) is a morphism of BiHom-associativedialgebras. Then( u + ξ ( u )) ⊣ ( v + ξ ( v )) = ( u ⊣ A v + ξ ( u ) ⊣ B ξ ( v )) = ( u ⊣ A v + ξ ( u ⊣ A v )( u + ξ ( u )) ⊢ ( v + ξ ( v )) = ( u ⊢ A v + ξ ( u ) ⊢ B ξ ( v )) = ( u ⊢ A v + ξ ( u ⊢ A v ) . Thus the graph Γ ξ is closed under the operations ⊣ and ⊢ .Furthermore since ξ ◦ α A = α B ◦ ξ, and ξ ◦ β A = β B ◦ ξ, we have( α A ⊕ α B )( u , ξ ( u )) = ( α A ( u ) , α B ◦ ξ ( u )) = ( α A ( u ) , ξ ◦ α A ( u )) . and ( β A ⊕ β B )( u , ξ ( u )) = ( β A ( u ) , β B ◦ ξ ( u )) = ( β A ( u ) , ξ ◦ β A ( u )) , implies that Γ ξ is closed α A ⊕ α B and β A ⊕ β B . Thus, Γ ξ is a BiHom-subalgebra of ( A ⊗ B , ⊣ , ⊢ , α, β ) . Conversely, if the graph Γ ξ ⊂ A ⊕ B is a BiHom-subalgebra of ( A ⊕ B , ⊣ , ⊢ , α, β ) then we( u + ξ ( u )) ⊣ ( v + ξ ( v )) = ( u ⊣ A v + ξ ( u ) ⊣ B ξ ( v )) ∈ Γ ξ ( u + ξ ( u )) ⊢ ( v + ξ ( v )) = ( u ⊢ A v + ξ ( u ) ⊢ B ξ ( v )) ∈ Γ ξ . Furthermore, ( α A ⊕ α B )( Γ ξ ) ⊂ Γ ξ , ( β A ⊕ β B )( Γ ξ ) ⊂ Γ ξ , implies( α A ⊕ α B )( u , ξ ( u )) = ( α A ( u ) , α B ◦ ξ ( u )) ∈ Γ ξ , ( β A ⊕ β B )( u , ξ ( u )) = ( β A ( u ) , β B ◦ ξ ( u )) ∈ Γ ξ , which is equivalent to the condition α B ◦ ξ ( u ) = ξ ◦ α A ( u ) , i.e α B ◦ ξ = ξ ◦ α A . Similary, β B ◦ ξ = ξ ◦ β A .Therefore, ξ is a morphism BiHom-associative dialgebras. (cid:3) Proposition 2.30.
Let ( D , ⊣ , ⊢ , α, β ) be a BiHom-associative dialgebra and I be a two sided BiHom-ideal of ( D , ⊣ , ⊢ , α, β ) . Then, ( D / I , [ · , · ] , ⊣ , ⊢ , α, β ) is a BiHom-associative dialgebra wherex ⊣ y : = x ⊣ y , x ⊢ y : = x ⊢ y , α ( x ) : = α ( x ) , β ( x ) : = β ( x ) , for all x , y ∈ A / I . Proof.
We only prove left associativity, the other being proved similarly. For all x , y , z ∈ D / I , wehave ( x ⊢ y ) ⊢ β ( z ) − α ( x ) ⊢ ( y ⊢ z ) = ( x ⊢ y ) ⊢ β ( z ) − α ( x ) ⊢ ( y ⊢ z ) = . Then, ( D / I , ⊣ , ⊢ , α, β ) is BiHom-associative dialgebra. (cid:3) Now, let us recall the definition of BiHom-Lie algebra.
Definition 2.31. [7] A BiHom-Lie algebra ( L , [ · , · ] , α, β ) is a 4-tuple in where L is linear space, α, β : A → A ,are linear maps and [ · , · ] : L ⊗ L → L is a bilinear maps, such that, for all x , y , z ∈ L : α ◦ β = β ◦ α, (2.28) α ( (cid:2) x , y (cid:3) ) = (cid:2) α ( x ) , α ( y ) (cid:3) , and , β ( (cid:2) x , y (cid:3) ) = (cid:2) β ( x ) , β ( y ) (cid:3) , (2.29) (cid:2) β ( x ) , α ( y ) (cid:3) ) = − (cid:2) β ( y ) , α ( x ) (cid:3) , (BiHom-skew-symetry) , (2.30) h β ( x ) , (cid:2) β ( y ) , α ( z ) (cid:3)i + h β ( y ) , (cid:2) β ( z ) , α ( x ) (cid:3)i + h β ( z ) , (cid:2) β ( x ) , α ( y ) (cid:3)i = , (2.31)(BiHom-Jacobi identity).The maps α and β (in this order) are called the structure maps of L. Definition 2.32.
A morphism between two BiHom-Lie algebras f : ( L , [ − , − ] , α, β ) → ( L ′ , [ − , − ] ′ , α ′ , β ′ )is a linear map f : L → L ′ such that α ′ ◦ f = f ◦ α, β ′ ◦ f = f ◦ β and f ( (cid:2) x , y (cid:3) ) = (cid:2) f ( x ) , f ( y ) (cid:3) ′ , for all x , y ∈ L . The following lemma asserts that the commutator of any BiHom-associative algebra gives riseto BiHom-Lie.
Lemma 2.33. [7] Let ( A , · , α, β ) be a regular BiHom-associative algebra. ThenL ( A ) = ( A , [ − , − ] , α, β ) is a regular BiHom-Lie algebra, where [ x , y ] = x · y − α − β ( y ) · αβ − ( x ) , for any x , y ∈ A . Proposition 2.34.
Let ( L , [ − , − ] , α, β ) be a BiHom-Lie algebra and N : L → L be a Nijenhuis oper-ator on L i.e. α ◦ N = N ◦ α , β ◦ N = N ◦ β and [ N ( x ) , N ( y )] = N ([ N ( x ) , y ] + [ x , N ( y )] − N ([ x , y ])) for any x , y ∈ L. Then, ( L , [ − , − ] N , α, β ) is a BiHom-Lie algebra with [ x , y ] N = [ N ( x ) , y ] + [ x , N ( y )] − N ([ x , y ]) for all x , y ∈ L.Proof.
It follows from direct computation. (cid:3) hmed Zahari and Ibrahima Bakayoko 11
Corollary 2.35.
Let ( A , · , α, β ) be a BiHom-associative algebra and N : A → A be a Nijenhuis oper-ator on A i.e. α ◦ N = N ◦ α , β ◦ N = N ◦ β andN ( x ) · N ( y ) = N ( N ( x ) · y + x · N ( y ) − N ( x · y )) for any x , y ∈ A. Let us denote by L ( A ) the BiHom-Lie algebra associated with A as in Proposition2.33. Then, ( A , [ − , − ] N , α, β ) is a BiHom-Lie algebra. Corollary 2.36.
Let ( A , · , α, β ) be a BiHom-associative algebra and N : A → A be a Nijenhuis oper-ator on A i.e. α ◦ N = N ◦ α , β ◦ N = N ◦ β andN ( x ) · N ( y ) = N ( N ( x ) · y + x · N ( y ) − N ( x · y )) for any x , y ∈ A. Then, ( A , {− , −} , α, β ) is a BiHom-Lie algebra with { x , y } = x ∗ N y − α − β ( y ) ∗ N αβ − ( x ) and x ∗ N y = N ( x ) · y + x · N ( y ) − N ( x · y ) for all x , y ∈ A.Proof.
It is similar to the one of Proposition 2.21. And the Lemma 2.33 will end the proof. (cid:3)
Remark . The BiHom-Lie algebra generated by Corollary 2.35 and Corollary 2.36 are equal.
Proposition 2.38.
Let ( D , ⊣ , ⊢ , α, β ) be a BiHom-associative dialgebra. Then,for all x , y ∈ D, thebracket [ x , y ] = [ x , y ] L + [ x , y ] R , where [ x , y ] L = x ⊣ y − α − β ( y ) ⊣ αβ − ( x ) , [ x , y ] R = x ⊢ y − α − β ( y ) ⊢ αβ − ( x ) , is a BiHom-Lie bracket if and only if α ( x ) ⊣ ( y ⊢ z ) = ( x ⊣ y ) ⊢ β ( z ) , (2.32) α ( x ) ⊣ ( y ⊣ z ) = ( x ⊢ y ) ⊢ β ( z ) . (2.33) Proof.
It is essentialy based on Lemma 2.33. That is, an expansion of BiHom-Jacobi identity leadsto 48 terms including 8 terms which cancel pairewise by axiom (2.2), 4 terms cancel pairewise byaxiom (2.3), 12 terms cancel pairewise by axiom (2.4), 6 terms cancel pairewise by axiom (2.5) and6 terms cancel pairewise by axiom (2.6).For the of the 12 terms, 8 terms cancel pairewise by axiom (2.32) and 4 terms cancel pairewise byaxiom (2.33). (cid:3)
Definition 2.39.
A (right ) BiHom-Leibniz algebra is a 4-tuple ( L , [ · , · ] , α, β ), where L is a linearspace, [ · , · ] : L × L → L is a bilinear map and α, β : L → L are linear maps satisfying (cid:2)(cid:2) x , y (cid:3) , αβ ( z ) , (cid:3) = (cid:2)(cid:2) x , β ( z ) (cid:3) , α ( y ) (cid:3) + (cid:2) α ( x ) , (cid:2) y , α ( z ) (cid:3)(cid:3) , (2.34)for all x , y , z ∈ L .2 Example 2.40.
Let L be a two-dimensional vector space and { e , e } be a basis of L. Then, ( L , [ − , − ] , α, β ) is a BiHom-Leibniz algebra with [ e , e ] = ae , [ e , e ] = be , α ( e ) = β ( e ) = e , a , b ∈ R . Now, we introduce BiHom-Poisson dialgebras and study its connection with BiHom-associativedialgebras.
Definition 2.41.
A BiHom-Poisson dialgebra is a BiHom-associative dialgebra ( P , ⊣ , ⊢ , α, β ) and aBiHom-Leibniz algebra ( P , [ − , − ] , α, β ) such that[ x ⊣ y , αβ ( z )] = α ( x ) ⊣ [ y , α ( z )] + [ x , β ( z )] ⊣ α ( y ) , [ x ⊢ y , αβ ( z )] = α ( x ) ⊢ [ y , α ( z )] + [ x , β ( z )] ⊢ α ( y ) , { αβ ( x ) , y ⊣ z } = β ( y ) ⊢ [ α ( x ) , z ] + [ β ( x ) , y ] ⊣ β ( z ) = [ αβ ( x ) , y ⊢ z ] , are satisfied for x , y , z ∈ P . Theorem 2.42.
Let ( D , ⊣ , ⊢ , α, β ) be a BiHom-associative dialgebra. Then,P ( D ) = ( D , [ − , − ] , ⊣ , ⊢ , α, β ) is a BiHom-Poisson dialgebra, where [ x , y ] = x ⊣ y − y ⊢ x, for any x , y ∈ D.Proof.
By Theorem 2.46 P ( D ) is a BiHom-Leibniz algebra. Moreover, for any x , y , z ∈ D ,[ x ⊣ y , αβ ( z )] − α ( x ) ⊣ [ y , α ( z )] − [ x , β ( z )] ⊣ α ( y ) == ( x ⊣ y ) ⊣ αβ ( z ) − α − βαβ ( z ) ⊢ αβ − ( x ⊣ y ) − α ( x ) ⊣ ( y ⊣ α ( z ) − α − βα ( z ) ⊢ αβ − ( y )) − ( x ⊣ β ( z ) − α − ββ ( z ) ⊢ αβ − ( x )) ⊣ α ( y ) = ( x ⊣ y ) ⊣ αβ ( z ) − β ( z ) ⊢ ( αβ − ( x ) ⊣ αβ − ( y )) − α ( x ) ⊣ ( y ⊣ α ( z ) + α ( x ) ⊣ ( β ( z ) ⊢ αβ − ( y )) − ( x ⊣ β ( z )) ⊣ α ( y ) + ( α − β ( z ) ⊢ αβ − ( x )) ⊣ α ( y ) . The last three axioms are proved analagously. This completes the proof. (cid:3)
Theorem 2.43.
Let ( P , ⊣ , ⊢ , [ − , − ] , α, β ) be a BiHom-Poisson dialgebra and α ′ , β ′ : D → D two mor-phisms of BiHom-Poisson dialgebras such that the maps α, α ′ , β, β ′ commute pairewise. ThenP ( α ′ ,β ′ ) = ( D , ⊳ : = ⊣ ( α ′ ⊗ β ′ ) , ⊲ : = ⊢ ( α ′ ⊗ β ′ ) , {− , −} : = [ − , − ]( α ′ ⊗ β ′ ) , αα ′ , ββ ′ ) , is a BiHom-Poisson dialgebra.Proof. It is essentialy based on the one of Theorem 2.6. (cid:3)
Now, we introduce action of BiHom-Leibniz algebra on another one.
Definition 2.44.
Let D and L be two BiHom-Leibniz algebras. An action of D on L consists of apair of bilinear maps, D × L → L , ( x , a ) [ x , a ] and L × D → [ x , a ], such that[ α ( x ) , [ a , α ( b )]] = (cid:2) [ x , a ] , αβ ( b ) , (cid:3) − (cid:2)(cid:2) x , β ( b ) (cid:3) , α ( a ) (cid:3) (2.35)[ α ( a ) , [ x , α ( b )]] = (cid:2) [ a , x ] , αβ ( b ) , (cid:3) − (cid:2)(cid:2) a , β ( b ) (cid:3) , α ( x ) (cid:3) (2.36)[ α ( a ) , [ b , α ( x )]] = (cid:2) [ a , b ] , αβ ( x ) , (cid:3) − (cid:2)(cid:2) a , β ( x ) (cid:3) , α ( b ) (cid:3) (2.37) (cid:2) α ( a ) , (cid:2) x , α ( y ) (cid:3)(cid:3) = (cid:2) [ a , x ] , αβ ( y ) , (cid:3) − (cid:2)(cid:2) a , β ( y ) (cid:3) , α ( x ) (cid:3) (2.38) (cid:2) α ( x ) , (cid:2) a , α ( y ) (cid:3)(cid:3) = (cid:2) [ x , a ] , αβ ( y ) , (cid:3) − (cid:2)(cid:2) x , β ( y ) (cid:3) , α ( a ) (cid:3) (2.39) (cid:2) α ( x ) , (cid:2) y , α ( a ) (cid:3)(cid:3) = (cid:2)(cid:2) x , y (cid:3) , αβ ( a ) , (cid:3) − (cid:2)(cid:2) x , β ( a ) (cid:3) , α ( y ) (cid:3) (2.40)for all x , y ∈ D , a , b ∈ L .hmed Zahari and Ibrahima Bakayoko 13 Lemma 2.45.
Given a BiHom-Leibniz action of D on L, we can consider the semidirect productLeibniz algebra L Y D, which consists of vector space D ⊕ L together with the Leibniz bracket givenby [( x , a ) , ( y , b )] = ([ x , y ] + [ x , b ] + [ a , y ] , [ a , b ]) (2.41) for all ( x , a ) , ( x , b ) ∈ D × L.Proof. [ α ( x , a ) , [( y , b ) , α ( z , c )]] = [( α ( x ) , α ( a )) , ([ y , α ( z )] + [ y , α ( c )] + [ b , α ( z )] , [ b , α ( c )])] = (cid:16) [ α ( x ) , [ y , α ( z )]] + [ α ( x ) , [ y , α ( c )]] + [ α ( x ) , [ b , α ( z )]] + [ α ( x ) , [ b , α ( c )]] + [ α ( a ) , [ y , α ( z )]] + [ α ( a ) , [ y , α ( z )]] + [ α ( a ) , [ y , α ( c )]] + [ α ( a ) , [ b , α ( z )] , [ α ( a ) , [ b , α ( c )]] (cid:17) . [[( x , a ) , ( y , b )] , αβ ( z , c )] = [([ x , y ] + [ x , b ] + [ a , y ]) , [ a , b ]) , ( αβ ( z ) , αβ ( c ))] = ([[ x , y ] , αβ ( z )] + [[ x , b ] , αβ ( z )] + [[ a , y ] , αβ ( z )] + [[ x , y ] , αβ ( c )] + [[ x , b ] , αβ ( c )] + [[ a , y ] , αβ ( c )] + [[ a , b ] , αβ ( c )] , [[ a , b ] , αβ ( c )] . [[( x , a ) , β ( z , c )] , α ( y , b )] = [([ x , β ( z )] + [ x , β ( c )] + [ a , β ( z )] , [ a , β ( c )]) , ( α ( y ) , α ( b ))] = ([[ x , β ( z )] , α ( y )] + [[ x , β ( c )] , α ( y )] + [[ a , β ( z )] , α ( y )] + [[ x , β ( z )] , α ( b )] + [[ x , β ( c )] , α ( b )] + [[ a , β ( z )] , α ( b )] + [[ a , β ( c )] , α ( y )] , [[ a , β ( c )] , α ( b )]) . Using axioms in Definition 2.35, it follows that[[( x , a ) , ( y , b )] , αβ ( z , c )] = [[( x , a ) , β ( z , c )] , α ( y , b )] + [ α ( x , a ) , [( y , b ) , α ( z , c )]] . Which proves the proposition. (cid:3)
Theorem 2.46.
Let ( D , ⊣ , ⊢ , α, β ) be a regular BiHom-associative dialgebra. Then the bracket de-fined by (cid:2) x , y (cid:3) = x ⊣ y − α − β ( y ) ⊢ αβ − ( x ) , defines a structure of BiHom-Leibniz algebra on D, anddenoted Lb ( D ) .Proof. For any x , y , z ∈ D , we have[[ x , y ] , αβ ( z )] = ( x ⊣ y − α − β ( y ) ⊢ αβ − ( x )) ⊣ αβ ( z ) − α − βαβ ( z ) ⊢ ( x ⊣ y − α − β ( y ) ⊢ αβ − ) = ( x ⊣ y ) ⊣ αβ ( z ) − ( α − β ( y ) ⊢ αβ − ( x )) ⊣ αβ ( z ) − β ( z ) ⊢ ( αβ − ( x ) ⊣ αβ − ( y ) + β ( z ) ⊢ ( y ⊢ α β − ( x )) . [[ x , β ( z )] , α ( y )] = ( x ⊣ β ( z ) − α − β ( z ) ⊢ αβ − ( x ) ⊣ α ( y ) − α − βα ( z ) ⊢ ( αβ − ( x ⊣ y + α − β ( y ) ⊢ αβ − ) = ( x ⊣ β ( z )) ⊣ α ( y ) − ( α − β ( z ) ⊢ αβ − ( x )) ⊣ α ( y ) − β ( y ) ⊢ ( αβ − ( x ) ⊣ α ( z )) + β ( y ) ⊢ ( β ( z ) ⊢ α β − ( x )) . [ α ( x ) , [ y , α ( z )]] = α ( x ) ⊣ ( y ⊣ α ( z ) − α − βα ( z ) ⊢ αβ − ( y )) − α − β ( y ⊣ α ( z ) − β ( z ) ⊢ αβ − ( y )) ⊢ αβ − α ( x ) = α ( x ) ⊣ ( y ⊣ α ( z )) − α ( x ) ⊣ ( β ( z ) ⊢ αβ − ( y )) − ( α − β ( y ) ⊣ β ( z )) ⊢ α β − ( x ) − ( α − β ( z ) ⊢ y ) ⊢ α β − ( x ) . By axioms in Definition 2.1, the conclusion holds. (cid:3)
Definition 2.47.
Let D and L be dialgebras. An action of D on L consists of four linear maps, twoof them denoted by the symbol ⊣ and other two by ⊢ , ⊣ : D ⊗ L → L , ⊣ : L ⊗ D → L , ⊣ : D ⊗ L → L , ⊣ : L ⊗ D → L such that the following 30 equalities hold :(01) ( x ⊣ a ) ⊣ β ( b ) = α ( x ) ⊣ ( a ⊣ b ), (16) ( a ⊣ x ) ⊣ β ( y ) = α ( a ) ⊣ ( x ⊣ y ),(02) ( x ⊣ a ) ⊣ β ( b ) = α ( x ) ⊣ ( a ⊢ b ), (17) ( a ⊣ x ) ⊣ β ( y ) = α ( a ) ⊣ ( x ⊢ y ),(03) ( x ⊢ a ) ⊣ β ( b ) = α ( x ) ⊢ ( a ⊣ b ), (18) ( a ⊢ x ) ⊣ β ( y ) = α ( a ) ⊢ ( x ⊣ y ),(04) ( x ⊣ a ) ⊢ β ( b ) = α ( x ) ⊢ ( a ⊢ b ), (19) ( a ⊣ x ) ⊢ β ( y ) = α ( a ) ⊢ ( x ⊢ y ),(05) ( x ⊢ a ) ⊢ β ( b ) = α ( x ) ⊢ ( a ⊢ b ), (20) ( a ⊢ x ) ⊢ β ( y ) = α ( a ) ⊢ ( x ⊢ y ),(06) ( a ⊣ x ) ⊣ β ( b ) = α ( a ) ⊣ ( x ⊣ b ), (21) ( x ⊣ a ) ⊣ β ( y ) = α ( x ) ⊣ ( a ⊣ y ),(07) ( a ⊣ x ) ⊣ β ( b ) = α ( a ) ⊣ ( x ⊢ b ), (22) ( x ⊣ a ) ⊣ β ( y ) = α ( x ) ⊣ ( a ⊢ y ),(08) ( a ⊢ x ) ⊣ β ( b ) = α ( a ) ⊢ ( x ⊣ b ), (23) ( x ⊢ a ) ⊣ β ( y ) = α ( x ) ⊢ ( a ⊣ y ),(09) ( a ⊣ x ) ⊢ β ( b ) = α ( a ) ⊢ ( x ⊢ b ), (24) ( x ⊣ a ) ⊢ β ( y ) = α ( x ) ⊢ ( a ⊢ y ),(10) ( a ⊢ x ) ⊢ β ( b ) = α ( a ) ⊢ ( x ⊢ b ), (25) ( x ⊢ a ) ⊢ β ( y ) = α ( x ) ⊢ ( a ⊢ y ),(11) ( a ⊣ b ) ⊣ β ( x ) = α ( a ) ⊣ ( b ⊣ x ), (26) ( x ⊣ y ) ⊣ β ( a ) = α ( x ) ⊣ ( y ⊣ a ),(12) ( a ⊣ b ) ⊣ β ( x ) = α ( a ) ⊣ ( b ⊢ x ), (27) ( x ⊣ y ) ⊣ β ( a ) = α ( x ) ⊣ ( y ⊢ a ),(13) ( a ⊢ b ) ⊣ β ( x ) = α ( a ) ⊢ ( b ⊣ x ), (28) ( x ⊢ y ) ⊣ β ( a ) = α ( x ) ⊢ ( y ⊣ a ),(14) ( a ⊣ b ) ⊢ β ( x ) = α ( a ) ⊢ ( b ⊢ x ), (29) ( x ⊣ y ) ⊢ β ( a ) = α ( x ) ⊢ ( y ⊢ a ),(15) ( a ⊢ b ) ⊢ β ( x ) = α ( a ) ⊢ ( b ⊢ x ), (30) ( x ⊢ y ) ⊢ β ( a ) = α ( x ) ⊢ ( y ⊢ a ),for all x , y ∈ D , a , b ∈ L . The action is called trivial if these four maps are trivial. Example 2.48. i) Any BiHom-associative dialgebra may be seen as acting on itself ii)Given ahomomorphism ϕ : D → L of BiHom-associative dialgebras, then there is an action of D on L viathe maps x ⊳ a : = ϕ ( x ) ⊣ a , x ⊲ a : = ϕ ( x ) ⊲, a a ⊳ x : = a ⊢ ϕ ( x ) and a ⊲ x : = a ⊢ ϕ ( x ) .iii)If ψ : L → D is an isomorphism of BiHom-associative dialgebras, then there is an action of D onL via the maps x ⊳ a : = ψ − ( x ) ⊣ a , x ⊲ a : = ψ − ( x ) ⊲ a , a ⊳ x : = a ⊢ ψ − ( x ) and a ⊲ x : = a ⊢ ψ − ( x ) .iv) If I is an ideal of D, then the left and the right product yield an action of D on I. Lemma 2.49.
Given two regular BiHom-associative dialgebras D and L together with an action ofD on L, there is an action an action Lb ( D ) on Lb ( L ) given by [ x , a ] = x ⊣ a − α − β ( a ) ⊣ αβ − ( x ) , [ a , x ] = a ⊢ x − α − β ( x ) ⊢ αβ − ( a ) , for all x ∈ Lb ( D ) , a ∈ Lb ( L ) .Proof. For all x ∈ Lb ( D ), a ∈ Lb ( L ),[[ x , a ] , αβ ( b )] = ( x ⊣ a − α − β ( a ) ⊢ αβ − ( x )) ⊣ αβ ( b ) − α − βαβ ( b ) ⊢ αβ − ( x ⊣ a − α − β ( a ) ⊢ αβ − ( x )) = ( x ⊣ a ) ⊣ βα ( b ) − ( α − β ( a ) ⊢ αβ − ( x )) ⊣ βα ( b ) − β ( b ) ⊢ ( αβ − ( x ) ⊣ β − α ( a )) + β ( b ) ⊢ ( a ⊢ α β − ( x )) . hmed Zahari and Ibrahima Bakayoko 15On the other hand,[[ x , β ( b )] , α ( a )] + [ α ( x ) , [ a , α ( b )]] == (cid:16) x ⊣ β ( b ) − α − β ( b ) ⊢ αβ − ( x ) (cid:17) ⊣ α ( a ) − α − βα ( a ) ⊢ αβ − (cid:16) x ⊣ β ( b ) − α − β ( b ) ⊢ αβ − ( x ) (cid:17) + α ( x ) ⊣ (cid:16) a ⊣ α ( b ) − α − βα ( b ) ⊢ αβ − ( a ) (cid:17) − α − β (cid:16) a ⊣ α ( b ) − α − βα ( b ) ⊢ αβ − ( a ) (cid:17) ⊢ αβ − α ( x ) = ( x ⊣ β ( b )) ⊣ α ( a ) − ( α − β ( b ) ⊢ αβ − ( x )) ⊣ α ( a ) − β ( a ) ⊢ ( αβ − ( x ) ⊣ α ( b )) + β ( a ) ⊢ ( β ( b ) ⊢ α β − ( x )) + α ( x ) ⊣ ( a ⊣ α ( b )) − α ( x ) ⊣ ( β ( b ) ⊢ αβ − ( a )) − ( α − β ( a ) ⊣ β ( b )) ⊢ β − α ( x ) + ( α − β ( b ) ⊢ a ) ⊢ β − α ( x ) . Using axioms (2.3), (2.5), it comes[[ x , β ( b )] , α ( a )] + [ α ( x ) , [ a , α ( b )]] == − ( α − β ( b ) ⊢ αβ − ( x )) ⊣ α ( a ) − β ( a ) ⊢ ( αβ − ( x ) ⊣ α ( b )) + α ( x ) ⊣ ( a ⊣ α ( b )) + ( α − β ( b ) ⊢ a ) ⊢ α β − ( x ) . By comparing, we get the attended result. The five other axioms are proved in the same way. (cid:3)
Lemma 2.50.
Let D and L be two regular BiHom-associative dialgebras together with an actionof D on L. There is a BiHom-associative dialgebra structure on L Y D which consists with vectorspace L ⊕ D and ( a , x ) ⊳ ( b , y ) = ( a ⊣ b + a ⊣ y + x ⊣ b , x ⊣ y ) , ( a , x ) ⊲ ( b , y ) = ( a ⊢ b + a ⊢ y + x ⊢ b , x ⊢ y ) , for any ( a , x ) , ( b , y ) ∈ L × DProof.
For any a , b , c ∈ L , x , y , z ∈ D , one has (cid:16) ( a , x ) ⊳ ( b , y ) (cid:17) ⊳ β ( c , z ) − α ( a , x ) ⊳ (cid:16) ( b , y ) ⊲ ( c , z ) (cid:17) == ( a ⊣ b + a ⊣ y + x ⊣ b , x ⊣ y ) ⊳ ( β ( c ) , β ( z )) − ( α ( a ) , α ( x )) ⊳ ( b ⊢ c + b ⊢ z + y ⊢ c , y ⊢ z ) = (cid:16) ( a ⊣ b + a ⊣ y + x ⊣ b ) ⊣ β ( c ) + ( a ⊣ b + a ⊣ y + x ⊣ b ) ⊣ β ( z ) + ( x ⊣ y ) ⊣ β ( c ) , ( x ⊣ y ) ⊣ β ( z ) (cid:17) − (cid:16) α ( a ) ⊣ ( b ⊢ c + b ⊢ z + y ⊢ c ) + α ( a ) ⊣ ( y ⊢ z ) + α ( x ) ⊣ ( b ⊢ c + b ⊢ z + y ⊢ c ) , α ( x ) ⊣ ( y ⊢ z ) (cid:17) = (cid:16) ( a ⊣ b ) ⊣ β ( c ) + ( a ⊣ y ) ⊣ β ( c ) + ( x ⊣ b ) ⊣ β ( c ) + ( a ⊣ b ) ⊣ β ( z ) + ( a ⊣ y ) ⊣ β ( z ) + ( x ⊣ b ) ⊣ β ( z ) + ( x ⊣ y ) ⊣ β ( c ) − α ( a ) ⊣ ( b ⊢ c ) − α ( a ) ⊣ ( b ⊢ z ) − α ( a ) ⊣ ( y ⊢ c ) − α ( a ) ⊣ ( y ⊢ z ) − α ( x ) ⊣ ( b ⊢ c ) − α ( x ) ⊣ ( b ⊢ z ) − α ( x ) ⊣ ( y ⊢ c ) , ( x ⊣ y ) ⊣ β ( z ) − α ( x ) ⊣ ( y ⊢ z ) (cid:17) . The left hand side vanishes by axiom (2.3) and axioms (02) , (07) , (12) , (17) , (22) , (27) in Definition2.47. The other axioms are proved in the same way. (cid:3) Theorem 2.51.
Let D and L be two regular BiHom-associative dialgebras together with an actionof D on L. Then, Lb ( L Y D ) = Lb ( L ) Y Lb ( D ) .Proof. By lemma 2 . Lb ( D ) acts on Lb ( L ), so it makes sense to consider the semidirect productLeibniz algebra Lb ( L ) Y Lb ( D ). It is clear that Lb ( L Y D ) and Lb ( L ) Y Lb ( D ) are egal as vectorspace, so we only need to verify that they share the same bracket. Let ( a , x ) , ( b , y ) ∈ L × D . If we usethe bracket in Lb ( L ) Y Lb ( D ), we get :[( a , x ) , ( b , y )] = ([ a , b ] + [ x , b ] + [ a , y ] , [ x , y ]) = ( a ⊣ y − b ⊢ + x ⊣ b − b ⊢ x + a ⊣ y − y ⊢ a , x ⊣ y − y ⊢ x ) . Lb ( L Y D ) (Lemma 2.50), we get { ( a , x ) , ( b , y ) } = ( a , x ) ⊳ ( b , y ) − ( b , y ) ⊲ ( a , x ) = ( a ⊣ b + x ⊣ b + a ⊣ y , x ⊣ y ) − ( b ⊢ a + y ⊢ a + b ⊢ x , y ⊢ x ) , So the brackets are equal. (cid:3)
This section concerns the central extension of BiHom-associative dialgebras in relation with cocy-cles.
Definition 3.1.
Let ( D i , ⊣ i , ⊢ i , α i , β i ) , i = , , D is called the extension of D by D if there are homomorphisms φ : D → D and ψ : D → D such that the following sequence0 → D φ −→ D ψ → D → Definition 3.2.
An extension is called trivial if there exists a BiHom-ideal I of D complementaryto Ker ψ i.e. D = Ker ψ ⊕ I It may happen that there exist several extensions of D by D . To classify extensions the notionof equivalent extensions is defined. Definition 3.3.
Two sequences 0 → D φ −→ D ψ → D → → D φ ′ −→ D ψ ′ → D → f : D → D ′ such that f ◦ φ = φ ′ and ψ ′ ◦ f = ψ. Definition 3.4.
An extension 0 → D φ −→ D ψ → D → ψ is contained in the center Z ( D ) of D , i.e. Ker ψ ⊂ Z ( D ).Now, we introduce 2-cocycle on BiHom-associative dialgebra with values in a BiHom-module. Definition 3.5.
Let ( D , ⊣ , ⊢ , α, β ) be a BiHom-associative dialgebra and ( M , α M , β M ) a BiHom-module over the same field that D . A pair Θ = ( θ , θ ) of bilinear maps θ : D × D → V and θ : D × D → V is called a 2-cocycle on D with values in V if θ and θ satisfy θ ( x ⊣ y , β ( z )) = θ ( α ( x ) , y ⊣ z ) , (3.1) θ ( x ⊣ y , β ( z )) = θ ( α ( x ) , y ⊢ z ) , (3.2) θ ( x ⊢ y , β ( z )) = θ ( α ( x ) , y ⊢ z ) , (3.3) θ ( x ⊣ y , β ( z )) = θ ( α ( x ) , y ⊢ z ) , (3.4) θ ( x ⊢ y , β ( z )) = θ ( α ( x ) , y ⊣ z ) , (3.5)for all x , y , z ∈ D .hmed Zahari and Ibrahima Bakayoko 17The set of all 2-cocycles on D with values in M is denoted Z ( D , M ), which a vector space.In the below lemma, we give a special type of 2-cocycles which are called 2-coboundaries. Lemma 3.6.
Let ν : D → V be a linear map, and define ϕ ( x , y ) = ν ( x ⊣ y ) and ϕ ( x , y ) = ν ( x ⊢ y ) .Then, Φ = ( ϕ , ϕ ) is a -cocycle on D.Proof. We will prove one equality, the others being proved in the same way. For any x , y , z ∈ D , onehas ϕ ( α ( x ) , y ⊣ z ) = ν ( α ( x ) ⊣ ( y ⊣ z )) = ν (( x ⊣ y ) ⊣ β ( z )) = ν ( α ( x ) ⊣ ( y ⊢ z )) = ϕ ( α ( x ) , y ⊢ z ) . This finishes the proof. (cid:3)
The set of all 2-coboundaries is denoted by B ( D , M ) and it is a subgroup of Z ( D , M ). Thegroup H ( D , M ) = Z ( D , M ) / B ( D , M ) is said to be a second cohomology group of D with valuesin M . Two cocycles Θ and Θ are said to be cohomologous cocycles if Θ − Θ is a coboundary. Theorem 3.7.
Let ( D , ⊣ , ⊢ , α D , β D ) be a BiHom-associative dialgebra, ( M , α M , β M ) a BiHom-module, θ : D × D → M and θ : D × D → Mbe bilinear maps. Let us set D Θ = D ⊕ M, where
Θ = ( θ , θ ) . For any x , y ∈ D, v , w ∈ M, let us define ( x + u ) ⊳ ( y + v ) = x ⊣ y + θ ( x , y ) and ( x + u ) ⊲ ( y + v ) = x ⊢ y + θ ( x , y ) . Then, ( D Θ , ⊳, ⊲, α A ⊗ α M , β A ⊗ β M ) is a BiHom-associative dialgebra if and only if Θ is a -cocycle.Proof. For any x , y , z ∈ D , u , v , w ∈ M , we have(( x + v ) ⊳ ( y + w )) ⊳ ( β ( z ) + w ) − ( α ( x ) + v ) ⊳ (( y + w ) ⊳ ( z + w )) == (( x + v ) ⊳ ( y + w )) ⊳ ( β ( z ) + w ) − ( α ( x ) + v ) ⊳ (( y ⊣ z ) + θ ( y , z )) = (( x ⊣ y ) ⊣ β ( z )) + θ ( x ⊣ y , β ( z )) − ( α ( x ) ⊣ ( y ⊣ z )) − θ ( α ( x ) , y ⊣ z ) . The left hand vanishes by axioms (2.2) and (3.1). The other axioms are proved analagously. (cid:3)
Lemma 3.8.
Let Θ be a -cocycle and Φ a -coboundary. Then, D Θ+Φ is a BiHom-associativedialgebra with ( x + u ) E ( y + v ) = x ⊣ y + ϕ ( x , y ) + θ ( x , y ) , ( x + u ) D ( y + v ) = x ⊢ y + ϕ ( x , y ) + θ ( x , y ) . Moreover, D Θ (cid:27) D Θ+Φ .Proof.
First, we have to shown that D Θ+Φ is a BiHom-associative dialgebra. So, for any x + u , y + v , z + w ∈ D ⊕ M ,(( x + u ) E ( y + v )) E β ( z + w ) − α ( x + u ) E (( y + v )) E ( z + w )) == ( x ⊣ y + ϕ ( x , y ) + θ ( x , y )) E ( β ( z ) + β ( w )) − ( α ( x ) + α ( u )) E ( y ⊣ z + ϕ ( y , z ) + θ ( y , z )) = ( x ⊣ y ) ⊣ β ( z ) + ϕ ( x ⊣ y , β ( z )) + θ ( x ⊣ y , β ( z )) − α ( x ) ⊣ y ⊣ z − ϕ ( α ( x ) , y ⊣ z ) + θ ( α ( x ) , y ⊣ z )The left hand side vanishes by (2.2) and (3.1). The proofs of the rest of axioms are leaved to thereader.8Next, the isomorphism f : D Θ → D Θ+Φ is given by f ( x + v ) = x + ν ( x ) + v . In fact, it is clear that f isa bijective linear map and f ( α D + α M )( x + v ) = f ( α D ( x ) + α M ( v )) = α D ( x ) + να D ( x ) + α M ( v ) = α D ( x ) + α M ν ( x ) + α M ( v ) = ( α D + α M )( x + ν ( x ) + v ) = ( α D + α M ) ◦ f ( x + v ) . Thus, f commutes α D + α M , and similarly with β D + β M .Then, f (( x + v ) ⊳ ( y + w )) = f ( x ⊣ y + θ ( x , y )) = f ( x ⊣ y ) + f ( θ ( x , y )) = x ⊣ y + ν ( x ⊣ y ) + θ ( x , y ) = x ⊣ y + ϕ ( x , y ) + θ ( x , y ) . and f ( x + v ) E f ( y + w ) = ( x + ν ( x ) + v ) E ( y + ν ( y ) + w ) = ( x ⊣ y ) + ϕ ( x , y ) + θ ( x , y ) . (cid:3) Corollary 3.9.
Let Θ , Θ be two cohomologous -cocycles on a BiHom-associative dialgebra D,and D , D be the central extensions constructed with these -cocycles, respectively. The the centralextensions D and D are equivalent extensions. In particular a central extension defined by acoboundary is equivalent with a trivial central extension. The following theorem is proved Mutatis Mutandis as ([25], Theorem 4.1). So we omitted theproof.
Theorem 3.10.
There exists one to one correspondence between elements of H ( D , M ) and nonequiv-alents central extensions of associative dialgebra D by M. In this section, we give classification of BiHom-associative dialgebras in low dimension.Let ( D , ⊣ , ⊢ , α, β ) be an n -dimensional BiHom-associative dialgebra, { e i } be a basis of D . For any i , j ∈ N , ≤ i , j ≤ n , let us put e i ⊣ e j = n X k = γ ki j e k , e i ⊢ e j n X k = δ ki j e k , α ( e j ) = n X k = α k j e k , β ( e j ) = n X k = β k j e k . hmed Zahari and Ibrahima Bakayoko 19The axioms in Definition 2.1 are respectively equivalent to β k j α pk − α ji β p j = , (4.1) γ pi j β qk γ rpq − α pi γ qjk γ rpq = , (4.2) γ pi j β qk γ rpq − α pi δ qjk γ rpq = , (4.3) γ pi j β qk δ rpq − α pi δ qjk δ rpq = , (4.4) δ pi j β qk γ rpq − α pi γ qjk δ rpq = , (4.5) δ pi j β qk γ rpq − α pi δ qjk δ rpq = . (4.6) There is only one 1-dimensional BiHom-associative dialgebra ; the nul (or trivial) BiHom-associativedialgebra.
Algebras
Multiplications Morphisms α, β . A lg e ⊣ e = ae , e ⊣ e = be , e ⊢ e = ce , e ⊢ e = de , e ⊢ e = f e . α ( e ) = e ,β ( e ) = e A lg e ⊣ e = ae , e ⊣ e = ae , e ⊣ e = e , e ⊢ e = e , e ⊢ e = e , α ( e ) = e ,β ( e ) = e A lg e ⊣ e = ae , e ⊢ e = be , e ⊢ e = ce , e ⊢ e = de α ( e ) = e ,β ( e ) = e A lg e ⊣ e = e , e ⊣ e = e , e ⊣ e = ae , e ⊢ e = be , e ⊢ e = ce , e ⊢ e = de , α ( e ) = e ,β ( e ) = e Remark . In two dimensional, all of the BiHom-associative dialgebras are Hom-associative di-algebras i.e. α = β . Algebras
Multiplications Morphisms α, β . A lg e ⊣ e = e , e ⊣ e = e , e ⊣ e = ae , e ⊣ e = be , e ⊣ e = ce , e ⊢ e = e , e ⊢ e = de , e ⊢ e = f e , α ( e ) = e β ( e ) = e ,β ( e ) = be Algebras
Multiplications Morphisms α, β . A lg e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = e , α ( e ) = e β ( e ) = e ,β ( e ) = be A lg e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = e , α ( e ) = e β ( e ) = e ,β ( e ) = be A lg e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = e , α ( e ) = e β ( e ) = e ,β ( e ) = be A lg e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = e , α ( e ) = e β ( e ) = e ,β ( e ) = be Algebras
Multiplications Morphisms α, β . A lg e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊢ e = e , e ⊢ e = ce , e ⊢ e = e , e ⊢ e = de , α ( e ) = be β ( e ) = e , β ( e ) = e ,β ( e ) = e , A lg e ⊣ e = e , e ⊣ e = e , e ⊣ e = ae , e ⊣ e = be , e ⊣ e = − ce , e ⊣ e = e , e ⊢ e = e , e ⊢ e = de , e ⊢ e = f e , e ⊢ e = e , e ⊢ e = e , α ( e ) = e β ( e ) = e , β ( e ) = e ,β ( e ) = e , A lg e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = be , e ⊣ e = ce , e ⊣ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = ce , e ⊢ e = de , e ⊢ e = e , α ( e ) = e α ( e ) = e β ( e ) = e ,β ( e ) = e ,β ( e ) = e , A lg e ⊣ e = e , e ⊣ e = ae , e ⊣ e = e , e ⊣ e = e , e ⊣ e = ce , e ⊣ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = e , α ( e ) = e α ( e ) = e β ( e ) = e ,β ( e ) = e ,β ( e ) = e , hmed Zahari and Ibrahima Bakayoko 21 Algebras
Multiplications Morphisms α, β . A lg e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = e , α ( e ) = e α ( e ) = e β ( e ) = e ,β ( e ) = e ,β ( e ) = e , A lg e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = e , α ( e ) = e α ( e ) = e β ( e ) = e ,β ( e ) = e ,β ( e ) = e , A lg e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = f e , e ⊣ e = − ge , e ⊢ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = − he , e ⊢ e = ke , α ( e ) = e ,α ( e ) = e β ( e ) = e ,β ( e ) = e ,β ( e ) = e , A lg e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = e , α ( e ) = e α ( e ) = e β ( e ) = e ,β ( e ) = e ,β ( e ) = e , A lg e ⊣ e = e + e , e ⊣ e = e + e , e ⊣ e = e + e , e ⊣ e = e + e , e ⊢ e = − e + e , e ⊢ e = e , e ⊢ e = e + e , e ⊢ e = e + e , α ( e ) = e α ( e ) = e α ( e ) = e β ( e ) = e ,β ( e ) = e , A lg e ⊣ e = e , e ⊣ e = e + e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = e + e , e ⊢ e = e + e , α ( e ) = e α ( e ) = e α ( e ) = e β ( e ) = e ,β ( e ) = e , A lg e ⊣ e = f e + ge , e ⊣ e = e , e ⊣ e = e + e , e ⊣ e = e , e ⊣ e = e , e ⊢ e = e , e ⊢ e = he − ke , e ⊢ e = e + e , e ⊢ e = e + e , α ( e ) = e α ( e ) = e α ( e ) = e β ( e ) = e ,β ( e ) = e , A lg e ⊣ e = e , e ⊣ e = e , e ⊣ e = ae , e ⊣ e = e , e ⊣ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = − be , e ⊢ e = e , α ( e ) = e α ( e ) = e β ( e ) = e , β ( e ) = e + e ,β ( e ) = e + e ,β ( e ) = e + e , A lg e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = e , α ( e ) = e α ( e ) = e β ( e ) = e , β ( e ) = e + e ,β ( e ) = e + e ,β ( e ) = e + e , A lg e ⊣ e = e , e ⊣ e = − ce , e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = − ae , e ⊢ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = be , α ( e ) = e α ( e ) = e β ( e ) = e , β ( e ) = e + e ,β ( e ) = e + e ,β ( e ) = e + e , Algebras
Multiplications Morphisms α, β . A lg e ⊣ e = − e , e ⊣ e = ae , e ⊣ e = be , e ⊣ e = ce , e ⊣ e = de , e ⊣ e = e , e ⊢ e = f e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = e , e ⊢ e = ge , e ⊢ e = e , e ⊢ e = e , α ( e ) = e β ( e ) = e , β ( e ) = e + e ,β ( e ) = e + e ,β ( e ) = e + e , A lg e ⊣ e = e , e ⊣ e = e , e ⊣ e = e , e ⊣ e = ae , e ⊣ e = e , e ⊣ e = e , e ⊢ e = be , e ⊢ e = ce , e ⊢ e = de , e ⊢ e = e , e ⊢ e = e , α ( e ) = ae β ( e ) = e , β ( e ) = e + e ,β ( e ) = e + e ,β ( e ) = e + e , In this section, we introduce and study derivations of BiHom-dendrifom, BiHom-dialgebras.
Definition 5.1.
Let ( A , µ, α, β ) be a BiHom-associative algebra. A linear map D : A −→ A is calledan ( α s , β r )-derivation of ( A , µ, α, β ), if it satisfies D ◦ α = α ◦ D and D ◦ β = β ◦ DD ◦ µ ( x , y ) = µ ( D ( x ) , α s β r ( y )) + µ ( α s β r ( x ) , D ( y )) Example 5.2.
We consider the -dimensional BiHom-associative with a basis { e , e } . For µ ( e , e ) = − e , µ ( e , e ) = − e , µ ( e , e ) = , µ ( e , e ) = e and α ( e ) = e , α ( e ) = − e , β ( e ) = e , β ( e ) = e . A direct computation gives that : D ( e ) = d e , D ( e ) = d e ,α s ( e ) = α β β e + e β , α s ( e ) = α e , β r ( e ) = e β e , β r ( e ) = β e + β e . Definition 5.3.
Let ( D , ⊣ , ⊢ , α, β ) be a BiHom-associative dialgebra. A linear map D : D → D iscalled an ( α k , β l )-derivation of D if it satisfies1 . D ◦ α = α ◦ D , D ◦ β = β ◦ D ;2 . D ( x ⊣ y ) = α k β l ( x ) ⊣ D ( y ) + D ( x ) ⊣ α k β l ( y );3 . D ( x ⊢ y ) = α k β l ( x ) ⊢ D ( y ) + D ( x ) ⊢ α k β l ( y ) , for x , y ∈ D . We denote by
Der ( D ) : = M k ≥ M l ≥ Der ( α k ,β l ) ( D ), where Der ( α k ,β l ) ( D ) is the set of all ( α k , β l )-derivations of D . Proposition 5.4.
For any D ∈ Der ( α s ,β r ) ( A ) and D ′ ∈ Der ( α s ′ ,β r ′ ) ( A ) , we have [ D , D ′ ] ∈ Der ( α s + s ′ ,β r + r ′ ) ( A ) . hmed Zahari and Ibrahima Bakayoko 23 Proof.
For x , y ∈ A , we have[ D , D ′ ] ◦ µ ( x , y ) = D ◦ D ′ ◦ µ ( x , y ) − D ′ ◦ D ◦ µ ( x , y ) = D ( µ ( D ′ ( x ) , α s β r ( y )) + µ ( α s β r ( x ) , D ′ ( y ))) − D ′ ( µ ( D ( x ) , α s β r ( y )) + µ ( α s β r ( x ) , D ( y ))) = µ ( D ◦ D ′ ( x ) , α s + s ′ β r + r ′ ( y )) + µ ( α s β r ◦ D ′ ( x ) , D ◦ α s β r ( y )) + µ ( D ◦ α s β r ( x ) , α s β r ◦ D ′ ( y )) + µ ( α s + s ′ β r + r ′ ( x ) , D ◦ D ′ ( y )) − µ ( D ′ ◦ D ( x ) , α s + s ′ β r + r ′ ( y )) − µ ( α s β r ◦ D ( x ) , D ′ ◦ α s β r ( y )) − µ ( D ′ ◦ α s β r ( x ) , α s β r D ( y )) − µ ( α s + s ′ β r + r ′ ( x ) , D ′ ◦ D ( y )) . Since D and D ′ satisfy D ◦ α = α ◦ D , D ′ ◦ α = α ◦ D ′ , D ◦ β = β ◦ D , D ′ ◦ β = β ◦ D ′ .We obtain α s β r ◦ D ′ = D ′ ◦ α s β r , D ◦ α s ′ β r ′ = α s ′ β r ′ ◦ D . Therefore, we arrive at[ D , D ′ ] ◦ µ ( x , y ) = µ ( α s + s ′ β r + r ′ ( x ) , [ D , D ′ ] ( y )) + µ ([ D , D ′ ] ( x ) , α s + s ′ β r + r ′ ( y )) . Furthermore, it is straightforward to see that[ D , D ′ ] ◦ α = D ◦ D ′ ◦ α − D ′ ◦ D ◦ α = α ◦ D ◦ D ′ − α ◦ D ′ ◦ D = α ◦ [ D , D ′ ] . [ D , D ′ ] ◦ β = D ◦ D ′ ◦ β − D ′ ◦ D ◦ β = β ◦ D ◦ D ′ − β ◦ D ′ ◦ D = β ◦ [ D , D ′ ]which yields that [ D , D ′ ] ∈ Der ( α s + s ′ ,β r + r ′ ) ( A ) with µ = ⊣ = ⊢ . (cid:3) Proposition 5.5.
The space Der ( α s ,β r ) ( A ) is an invariant of the triple BiHom-associative algebra A.Proof. Let σ : ( A , ⊣ A , ⊢ A , α s , β r ) −→ ( B , ⊣ B , ⊢ B , α s , β r ) be a triple BiHom-associative algebra isomor-phism and let D be a ( α s , β r )-derivation of A. Then for any x , y , z ∈ B . We have : σ D σ − ◦ ((( x ) ⊣ B ( y )) ⊣ B ( z )) = σ D ◦ (( σ − ( x ) ⊣ A σ − ( y )) ⊣ A σ − ( z )) = σ ( D ◦ σ − ( x ) ⊢ A σ − ◦ α s β r ( y )) ⊢ A σ − ◦ α s β r ( z )) + σ ( σ − ◦ α s β r ( x ) ⊢ A D ◦ σ − ( y )) ⊢ A σ − ◦ α s β r ( z )) + σ ( σ − ◦ α s β r ( x ) ⊢ A σ − ◦ α s β r ( y ) ⊢ A D ◦ σ − ( z )) = ( D ◦ σ − ( x ) ⊣ B α s β r ( y )) ⊣ B α s β r ( z )) + ( α s β r ( x ) ⊣ B σ ◦ D ◦ σ − ( y )) ⊣ B α s β r ( z )) + ( α s β r ( x ) ⊣ B α s β r ( y )) ⊣ B D ◦ σ − ( z )) . Thus σ ◦ D ◦ σ − is a ( α s , β r )-derivation of B , hence the mapping. ψ : Der ( α s ,β r ) ( A ) −→ Der ( α s ,β r ) ( B ), D σ D σ − is an isomorphism of triple BiHom-associative algebras.In fact, it is easy to see that ψ is linear. Moreover let D , D , D be derivations of A : α s β r ◦ ψ ( D ⊣ tr D ) ⊣ tr D ) == α s β r ψ ( tr ( D )( D ⊣ D )) + α s β r ψ ( tr ( D )( D D )) + α s β r ψ ( tr ( D )( D ⊣ D )) = α s β r tr ( D ) ψ ( D ⊣ D ) + α s β r tr ( D ) ψ ( D ⊣ D ) + α s β r tr ( D ) ψ ( D ⊣ D ) = α s β r tr ( ψ ( D ))( ψ ( D ) ⊣ ψ ( D )) + α s β r tr ( ψ ( D ))( ψ ( D ) ⊣ ψ ( D )) + α s β r tr ( ψ ( D )) ψ (( ψ ( D ) ⊣ ψ ( D )) , since ψ is a morphism of the Der ( α s ,β r ) ( A ) and Der ( α s ,β r ) ( B ), and tr ( D ) = tr ( σ ◦ D ◦ σ − ) . Then α s β r ψ (( D ⊣ tr D ) ⊣ tr D )) = α s β r (( ψ ( D ) ⊣ tr ψ ( D )) ⊣ tr ψ ( D )) . (cid:3) References [1] A. Frolicher, A. Nijenhuis,
Theory of vector valued di ff erential forms , Part I. Indag Math,1956, 18: 338-360[2] A. Zahari and A. Makhlouf, Structure and Classification of Hom-Associative Algebras, Actaet commentationes universitis Tartuensis de mathematica, vol 24 (1) 2020.[3] A. Kitouni, A. Makhlouf, S. Silvestrov, On n-ary Generalization of BiHom-Lie algebras andBiHom-Associative Algebras , arXiv:1812.00094, 2018.[4] A. Majumdar, and G. Mukherjee, (2002). Deformation theory of dialgebras. K-theory,27(1):33-60.[5] A. P. Pozhidaev, (2008). Dialgebras and related triple systems. Siberian Mathematical Journal,49(4):696-708.[6] Basri, W., Rakhimov, I., Rikhsiboev, I., et al. (2015). Four-dimensional nilpotent dias- socia-tive algebras. Journal of Generalized Lie Theory and Applications, 9(1):1-7.[7] G. Graziani, A. Makhlouf, C. Menini and F. Panaite,
BiHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras , Symmetry, Integrability and Geometry: Methods andApplications SIGMA 11 (2015), 086, 34 pages.[8] H. Adimi, T. Chtioui, S. Mabrouk, S. Massoud,
Construction of BiHom-post-Lie algebras ,arXiv:math.RA / Bimodules and Rota-Baxter relations , J. Appl. Mech.Eng 4:178, doi:10.4172 / Left-Hom-symmetric and Hom-Poisson dialgebras ,Konuralp Journal of Mathematics, No.2, 42-53, 2015.[11] I.Rikhsiboev, I.Rakhimov and W. Basri, (2014). Diassociative algebras and their derivations.In Journal of Physics: Conference Series, volume 553, pages 1?9. IOP Publishing.[12] I. M.Rikhsiboev, I. Rakhimov, and W. Basri (2010). Classification of 3-dimensional complexdiassociative algebras. Malaysian Journal of Mathematical Sciences, 4(2):241-254.[13] J. Li, L. Chen, B. Sun,
Bihom-Nijienhuis operators and extensions of Bihom-Lie superalge-bras .[14] J. M. Casas, R. F. Casado, E. Khmaladze and M. Ladra, More on crossed modules of Lie,Leibniz, associative and diassociative algebras, available as arXiv:1508.01147 (05.08.2015).[15] J.-L. Loday, Dialgebras. Dialgebras and related operads, pp. 7-66, Lecture Notes in Math.,1763, Springer, Berlin, 2001.[16] J. Carinena, J. Grabowski, G. Marmo,
Quantum bi-Hamiltonian systems , Internat J. ModernPhys A, 2000, 15: 4797-4810.[17] K. Abdaoui, B. H. Abdelkader and A. Makhlouf,
BiHom-Lie colour algebras structures , arXiv1706.02188v1[math. RT] 6 Juin 2017.hmed Zahari and Ibrahima Bakayoko 25[18] L. Lin and Y. Zhang, F [x, y] as a dialgebra and a Leibniz Algebra, Comm. Algebra 38(9)(2010), 3417-3447.[19] L. Liu, A. Makhlouf, C. Menini, F. Panaite,
Rota-Baxter operators on BiHom-associativealgebras and related structures , arXiv:math.QA / BiHom-Novikov algebras and infinitesimalBiHom-bialgebras , arXiv 1903.08145v1[Math. QA] 18 Mars 2019.[21] L. Ling, A. Makhlouf, Claudia M. and Florin P.,
BiHom-pre-Lie algebras, BiHom-Leibnizalgebras and Rota-Baxter operators on BiHom-Lie algebras , arXiv 1706.00457v2[Math. QA]2 Fevrier 2020.[22] L. Liu, A. Makhlouf, C. Menini, F. Panaite, { σ, τ } -ota-Baxter operators, infinitesimal Hom-bialgebras and the associative (Bi)Hom-Yang-Baxter equation , Canad. Math. Bull., DOI:10.4153 / CMB-2018-028-8.[23] P. Leroux, Contruction of Nijenhuis operators and Dendriform trialgebras, February 2004.[24] P. Kolesnikov, and V. Y. Voronin, (2013). On special identities for dialgebras. Linear andMultilinear Algebra, 61(3):377-391.[25] S. Isamiddin and Rakhimov, On central Extensions of Associative Dialgebras, J. Physics :conf. Ser. 697 (2016).[26] Salazar-Diaz, O. Velasquez, R., and Wills-Toro, L. A. (2016). Construction of dialgebrasthrough bimodules over algebras. Linear and Multilinear Algebra, pages 1-22.[27] S. Guo, X. Zhang, S. Wang,,
The construction and deformation of BiHom-Novikov algebras ,J. Geom. Phys. 132 (2018), 460-472.[28] S. Wang, S. Guo,
BiHom-Lie superalgebra structures , arXiv:1610.02290v1 (2016).[29] X. LI,
BiHom-Poisson algebra and its application , International Journal of Algebra, Vol 13,2019, no 2, 73-81.[30] Y. Cheng, H. Qi,