BiHom-pre-alternative algebras and BiHom-alternative quadri-algebras
aa r X i v : . [ m a t h . R A ] M a r BiHom-pre-alternative algebras and BiHom-alternativequadri-algebras
Taoufik Chtioui , Sami Mabrouk , Abdenacer Makhlouf
1. University of Sfax, Faculty of Sciences Sfax, BP 1171, 3038 Sfax, Tunisia2. University of Gafsa, Faculty of Sciences Gafsa, 2112 Gafsa, Tunisia3. IRIMAS - D´epartement de Math´ematiques, 6, rue des fr`eres Lumi`ere, F-68093 Mulhouse, France
Abstract
The purpose of this paper is to introduce and study the notion of BiHom-pre-alternativealgebra which may be viewed as a BiHom-alternative algebra whose product can be decomposedinto two compatible pieces. Furthermore, we introduce the notion of BiHom-alternative quadri-algebra and show the connections between all these algebraic structures using Rota-Baxteroperators and O -operators. Key words : BiHom-alternative algebra, BiHom-pre-alternative algebra, BiHom-alternative quadri-algebra, bimodule, O -operator. Introduction
The study of periodicity phenomena in algebraic K-theory led J.-L. Loday in 1990th to introducethe notion of dendriform algebra ([17]) as the (Koszul) dual of the associative dialgebra. Thereis a remarkable fact that a Rota-Baxter operator (of weight zero), which first arose in probabilitytheory ([9]) and later became a subject in combinatorics ([22]), on an associative algebra naturallygives a dendriform algebra structure on the underlying vector space of the associative algebra ([2, 3,11, 17, 18]). Such unexpected relationships between dendriform algebras in the field of operads andalgebraic topology and Rota-Baxter operators in the field of combinatorics and probability havebeen attracting a great interest because of their connections with various fields in mathematicsand physics (see [1, 12, 13, 15] and the references therein). Later, Aguiar and Loday introducedthe notion of quadri-algebra ([4] in order to determine the algebraic structures behind a pair ofcommuting Rota-Baxter operators (on an associative algebra). This type of algebras are relatedfor example to the space of linear endomorphisms of an infinitesimal bialgebra and have deeprelationships with combinatorics and the theory of Hopf algebras ([4]). A quadri-algebra is alsoregarded as the underlying algebra structure of a dendriform algebra with a nondegenerate 2-cocycle. The connection between associative algebras and dendriform algebras were generalized toalternative algebras by X. Ni and C. Bai ([7]) who introduced the notion of pre-alternative algebra orpre-alternative-dendriform algebra, as a generalization of dendriform algebras. Analogs of quadri-algebras for alternative algebras had been obtained by S. Madariaga, using the technique of splittingof operations ([8], [19]). Twisted algebras which were motivated by q -deformations of algebras ofvector fields and also called Hom-algebras or BiHom-algebras have been intensively studied thislast decade [10, 14, 16, 20]. In ([21]), Q. Sun introduced the notion of Hom-pre-alternative algebraas a generalization of pre-alternative and Hom-dendriform algebras.1e aim in this paper to introduce the BiHom version of pre-alternative and alternative quadri-algebras which generalize the classical structures. We also introduce the notion of O -operatorsof BiHom-alternative and BiHom-pre-alternative algebras. We will prove that given a BiHom-alternative algebra and an O -operator give rise to a BiHom pre-alternative algebra. In the same waywe can construct BiHom-alternative quadri-algebras starting by a BiHom pre-alternative algebra.This paper is organized as follows. In Section 1, we summarize the definitions of BiHom-alternative, BiHom-pre-alternative algebra and their bimodules. We exploit the notion of O -operator to illustrate the relations existing between these structures. In Section 2, we defineBiHom-alternative algebras and BiHom-quadri-algebras. Therefore, we discuss their relationshipsusing O -operators.Throughout this paper K is a field of characteristic 0 and all vector spaces are over K . We referto a BiHom-algebra as quadruple ( A, µ, α, β ) where A is a vector space, µ is a multiplication and α, β are two linear maps. It is said to be regular if α, β are invertible. A BiHom-associator is atrilinear map as α,β defined for all x, y, z ∈ A by as α,β ( x, y, z ) = α ( x )( yz ) − ( xy ) β ( z ). When thereis no ambiguity, we denote for simplicity the multiplication and composition by concatenation. In this section, we recall the notion of BiHom-alternative algebras given in [10] and we introduceBiHom-pre-alternative algebras which are a generalization of Hom-type structures given in [21].Moreover we provide some key constructions.
Definition 1.1.
A left BiHom-alternative algebra ( resp. right BiHom-alternative algebra) is aquadruple ( A, µ, α, β ) consisting of a K -vector space A , a bilinear map µ : A × A −→ A and twohomomorphisms α, β : A −→ A such that αβ = βα , αµ = µα ⊗ and βµ = µβ ⊗ that satisfy theleft BiHom-alternative identity, i.e. for all x, y, z ∈ A , one has as α,β ( β ( x ) , α ( y ) , z ) + as α,β ( β ( y ) , α ( x ) , z ) = 0 , (1.1) respectively, the right BiHom-alternative identity, i.e. for all x, y, z ∈ A , one has as α,β ( x, β ( y ) , α ( z )) + as α,β ( x, β ( z ) , α ( y )) = 0 . (1.2) A BiHom-alternative algebra is one which is both a left and a right BiHom-alternative algebra.
Definition 1.2.
Let ( A, µ, α, β ) be a BiHom-alternative algebra and V be a vector space. Let L, R : A → gl ( V ) and φ, ψ ∈ gl ( V ) two commuting linear maps. Then ( V, L, R, φ, ψ ) is called abimodule, or a representation, of ( A, µ, α, β ) , if for any x, y ∈ A, v ∈ V , φL ( x ) = L ( α ( x )) φ, φR ( x ) = R ( α ( x )) φ, (1.3) ψL ( x ) = L ( β ( x )) ψ, ψR ( x ) = R ( β ( x )) ψ (1.4) L ( β ( x ) α ( x )) ψ ( v ) = L ( αβ ( x )) L ( α ( x )) v, (1.5) R ( β ( x ) α ( x )) φ ( v ) = R ( αβ ( x )) R ( β ( x )) v, (1.6) R ( β ( y )) L ( β ( x )) φ ( v ) − L ( αβ ( x )) R ( y ) φ ( v ) = R ( α ( x ) y ) φψ ( v ) − R ( β ( y )) R ( α ( x )) ψ ( v ) , (1.7) L ( α ( y )) R ( α ( x )) ψ ( v ) − R ( αβ ( x )) L ( y ) ψ ( v ) = L ( yβ ( x )) φψ ( v ) − L ( α ( y )) L ( β ( x )) φ ( v ) . (1.8) Proposition 1.1.
A tuple ( V, L, R, φ, ψ ) is a bimodule of a BiHom-alternative algebra ( A, µ, α, β ) if and only if the direct sum ( A ⊕ V, ∗ , α + φ, β + ψ ) is turned into a BiHom-alternative algebra (the emidirect product) where ( x + v ) ∗ ( x + v ) = µ ( x , x ) + L ( x ) v + R ( x ) v , (1.9)( α + φ )( x + v ) = α ( x ) + φ ( v ) , ( β + ψ )( x + v ) = β ( x ) + ψ ( v ) . (1.10) Proof.
For any x, y ∈ A and u, v ∈ V , we have as α + φ,β + ψ (cid:0) ( β + ψ )( x + u ) , ( α + φ )( x + u ) , y + v (cid:1) = (cid:16) ( β ( x ) + ψ ( u )) ∗ ( α ( x ) + φ ( u )) (cid:17) ∗ (cid:16) β ( y ) + ψ ( v ) (cid:17) − (cid:16) αβ ( x ) + φψ ( u ) (cid:17) ∗ (cid:16) ( α ( x ) + φ ( u )) ∗ ( y + v ) (cid:17) = (cid:16) β ( x ) α ( x ) + L ( β ( x )) φ ( u ) + R ( α ( x )) ψ ( u ) (cid:17) ∗ (cid:16) β ( y ) + ψ ( v ) (cid:17) − (cid:16) αβ ( x ) + φψ ( u ) (cid:17) ∗ (cid:16) α ( x ) y + L ( α ( x )) v + R ( y ) φ ( u ) (cid:17) = ( β ( x ) α ( x )) β ( y ) + L ( β ( x ) α ( x )) ψ ( v ) + R ( β ( y )) L ( β ( x )) φ ( u ) + R ( β ( y )) R ( α ( x )) ψ ( u ) − αβ ( x )( α ( x ) y ) − L ( αβ ( x )) L ( α ( x )) v − L ( αβ ( x )) R ( y ) φ ( u ) − R ( α ( x ) y ) φψ ( u ) . Hence as α + φ,β + ψ (cid:0) ( β + ψ )( x + u ) , ( α + φ )( x + u ) , y + v (cid:1) = 0 if and only if Eqs. (1.5) and (1.7) hold.Analogously, as α + φ,β + ψ (cid:0) y + v, ( β + ψ )( x + u ) , ( α + φ )( x + u ) (cid:1) = 0 if and only if (1.6) and (1.8)hold.On the other hand, ( α + φ )( β + ψ ) = ( β + ψ )( α + φ ), since φψ = ψφ and αβ = βα . Finally, themultiplicativity of α + φ and β + ψ follow from the facts that φψ = ψφ, φL ( x ) = L ( α ( x )) φ, φR ( x ) = R ( α ( x )) φ, ψL ( x ) = L ( β ( x )) ψ and ψR ( x ) = R ( β ( x )) ψ .The following result gives a construction of a bimodule of a BiHom-alternative algebra startingwith a classical one by means of the Yau twist procedure. Proposition 1.2.
Let ( V, L, R ) be a bimodule of an alternative algebra ( A, µ ) . Given four linearmaps α, β : A → A and φ, ψ : V → V such that αβ = βα, φψ = ψφ, φL ( x ) = L ( α ( x )) φ, φR ( x ) = R ( α ( x )) φ, ψL ( x ) = L ( β ( x )) ψ, and ψR ( x ) = R ( β ( x )) ψ . Then ( V, e L, e R, φ, ψ ) is a bimodule ofthe BiHom-alternative algebra ( A, µ α,β , α, β ) , where e L ( x ) = L ( α ( x )) ψ , e R ( x ) = R ( β ( x )) φ and µ α,β ( x, y ) = µ ( α ( x ) , β ( y ) .Proof. Let x, y ∈ A and v ∈ V and set µ ( x, y ) = xy , µ α,β ( x, y ) = x ∗ y . Then e L ( β ( x ) ∗ α ( x )) ψ ( v ) − e L ( αβ ( x )) e L ( α ( x )) v = L ( α β ( x ) α β ( x )) ψ ( v ) − L ( α β ( x )) L ( α β ( x )) ψ ( v ) = 0 , and e R ( β ( y )) e L ( β ( x )) φ ( v ) − e L ( αβ ( x )) e R ( y ) φ ( v ) − e R ( α ( x ) ∗ y ) φψ ( v ) + e R ( β ( y )) e R ( α ( x )) ψ ( v )= R ( β ( y )) L ( α β ( x )) φ ψ ( v ) − L ( α β ( x )) R ( β ( y )) φ ψ ( v ) − R ( α β ( x ) β ( y )) φ ψ ( v ) + R ( β ( y )) R ( α β ( x )) φ ψ ( v ) = 0 . The other identities can be shown using similar computations.3et (
V, L, R, φ, ψ ) be a bimodule of a BiHom-alternative algebra (
A, µ, α, β ) and let L ∗ , R ∗ : A → gl ( V ∗ ) , α ∗ , β ∗ : A ∗ → A ∗ , φ ∗ , ψ ∗ : V ∗ → V ∗ be the dual maps of respectively α, β, φ and ψ given by < L ∗ ( x ) u ∗ , v > = < u ∗ , L ( x ) v >, < R ∗ ( x ) u ∗ , v > = < u ∗ , R ( x ) v > (1.11) α ∗ ( x ∗ )( y ) = x ∗ ( α ( y )) , β ∗ ( x ∗ )( y ) = x ∗ ( β ( y )) (1.12) φ ∗ ( u ∗ )( v ) = u ∗ ( φ ( v )) , ψ ∗ ( u ∗ )( v ) = u ∗ ( ψ ( v )) (1.13) Proposition 1.3.
Let ( V, L, R, φ, ψ ) be a bimodule of a BiHom-alternative algebra ( A, µ, α, β ) .Then ( V ∗ , L ∗ , R ∗ , φ ∗ , ψ ∗ ) is a bimodule of ( A, µ, α, β ) provided that ψ ( L ( β ( x ) α ( x ))) u = L ( α ( x )) L ( αβ ( x )) u, (1.14) φ ( R ( β ( x ) α ( x ))) u = R ( β ( x )) R ( αβ ( x )) u, (1.15) φL ( β ( x )) R ( β ( y )) u − φR ( y ) L ( αβ ( x )) u = ψφR ( α ( x ) y ) u − ψR ( α ( x )) R ( β ( y )) u, (1.16) ψR ( α ( x )) L ( α ( y )) u − ψL ( y ) R ( αβ ( x )) u = ψφL ( yβ ( x )) u − φL ( β ( x )) L ( α ( y )) u, (1.17) for all x, y ∈ A and u ∈ V .Proof. Straightforward.
Definition 1.3. [7] A pre-alternative algebra is a triple ( A, ≺ , ≻ ) , where A is a vector space and ≺ , ≻ : A × A → A are two bilinear maps, satisfying as r ( x, y, y ) = 0 , as l ( x, x, y ) = 0 , (1.18) as m ( x, y, z ) + as r ( y, x, z ) = 0 , (1.19) as m ( x, y, z ) + as l ( x, z, y ) = 0 , (1.20) where as r ( x, y, z ) = ( x ≺ y ) ≺ z − x ≺ ( y ≺ z + y ≻ z ) (right-associator) , (1.21) as m ( x, y, z ) = ( x ≻ y ) ≺ z − x ≻ ( y ≺ z ) (middle-associator) , (1.22) as l ( x, y, z ) = ( x ≺ y + x ≻ y ) ≻ z − x ≻ ( y ≻ z ) (left-associator) . (1.23) Proposition 1.4. [7] Let ( A, ≺ , ≻ ) be a pre-alternative algebra. Then the operation x ◦ y = x ≺ y + x ≻ y, for all x, y ∈ A, defines an alternative algebra, which is called the associated alternative algebra of A and denoted by Alt ( A ) . We call ( A, ≺ , ≻ ) a compatible pre-alternative algebra structure on the alternative algebra Alt ( A ) . Now we give the BiHom version of a pre-alternative algebra.
Definition 1.4.
A BiHom-pre-alternative algebra (which may be called also BiHom-alternative-dendriform dialgebra) is a -tuple ( A, ≺ , ≻ , α, β ) consisting of a vector space A , bilinear maps , ≻ : A × A → A and two commuting linear maps α, β : A → A satisfying the following conditions(for all x, y, z ∈ A ): α ( x ≺ y ) = α ( x ) ≺ α ( y ) , α ( x ≻ y ) = α ( x ) ≻ α ( y ) , (1.24) β ( x ≺ y ) = β ( x ) ≺ β ( y ) , β ( x ≻ y ) = β ( x ) ≻ β ( y ) , (1.25) as rα,β ( x, β ( y ) , α ( z )) + as rα,β ( x, β ( z ) , α ( y )) = 0 , (1.26) as lα,β ( β ( x ) , α ( y ) , z ) + as lα,β ( β ( y ) , α ( x ) , z ) = 0 , (1.27) as mα,β ( β ( x ) , α ( y ) , z ) + as rα,β ( β ( y ) , α ( x ) , z ) = 0 , (1.28) as mα,β ( x, β ( y ) , α ( z )) + as lα,β ( x, β ( z ) , α ( y )) = 0 , (1.29) where as rα,β ( x, y, z ) = ( x ≺ y ) ≺ β ( z ) − α ( x ) ≺ ( y ◦ z ) , (1.30) as mα,β ( x, y, z ) = ( x ≻ y ) ≺ β ( z ) − α ( x ) ≻ ( y ≺ z ) , (1.31) as lα,β ( x, y, z ) = ( x ◦ y ) ≻ β ( z ) − α ( x ) ≻ ( y ≻ z ) , (1.32) named respectively, right-BiHom-associator, middle-BiHom-associator and left-BiHom-associator.We call α and β (in this order) the structure maps of A .Note that x ◦ y = x ≺ y + x ≻ y . It is obvious to see that a BiHom-dendriform algebra is BiHom-pre-alternative, exactly as anyBiHom-associative algebra is BiHom-alternative.
Remark 1.1.
Since the characteristic of K is , conditions (1.26) and (1.27) are equivalent,respectively to as rα,β ( x, β ( y ) , α ( y )) = 0 , (1.33) as lα,β ( β ( x ) , α ( x ) , y ) = 0 . (1.34)A morphism f : ( A, ≺ , ≻ , α, β ) → ( A ′ , ≺ ′ , ≻ ′ , α ′ , β ′ ) of BiHom-pre-alternative algebras is a linearmap f : A → A ′ satisfying f ( x ≺ y ) = f ( x ) ≺ ′ f ( y ), f ( x ≻ y ) = f ( x ) ≻ ′ f ( y ), for all x, y ∈ A , aswell as f ◦ α = α ′ ◦ f and f ◦ β = β ′ ◦ f . Theorem 1.1.
Let ( A, ≺ , ≻ ) be a pre-alternative algebra and α, β : A → A two commuting pre-alternative algebra morphisms. Define ≺ ( α,β ) , ≻ ( α,β ) : A × A → A by x ≺ ( α,β ) y = α ( x ) ≺ β ( y ) and x ≻ ( α,β ) y = α ( x ) ≻ β ( y ) , for all x, y ∈ A . Then A ( α,β ) := ( A, ≺ ( α,β ) , ≻ ( α,β ) , α, β ) is a BiHom-pre-alternative algebra, calledthe Yau twist of A .Moreover, assume that ( A ′ , ≺ ′ , ≻ ′ ) is another pre-alternative algebra and α ′ , β ′ : A ′ → A ′ aretwo commuting pre-alternative algebra morphisms. Let f : A → A ′ be a pre-alternative algebramorphism satisfying f ◦ α = α ′ ◦ f and f ◦ β = β ′ ◦ f . Then f : A ( α,β ) → A ′ ( α ′ ,β ′ ) is a BiHom-pre-alternative algebra morphism.Proof. Let x, y, z ∈ A . Then we have as lα,β l ( β ( x ) , α ( x ) , y ) = ( β ( x ) ◦ ( α,β ) α ( x )) ≻ ( α,β ) β ( y ) − αβ ( x ) ≻ ( α,β ) ( α ( x ) ≻ ( α,β ) y )= ( αβ ( x ) ◦ αβ ( x )) ≻ ( α,β ) β ( y ) − αβ ( x ) ≻ ( α,β ) ( α ( x ) ≻ β ( y ))= ( α β ( x ) ◦ α β ( x )) ≻ β ( y ) − α β ( x ) ≻ ( α β ( x ) ≻ β ( y ))= as l ( α β ( x ) , α β ( x ) , β ( y )) = 0 . as rα,β ( x, β ( y ) , α ( y )) = 0.On the other hand, we get as mα,β ( β ( x ) , α ( y ) , z ) + as rα,β ( β ( y ) , α ( x ) , z )= ( β ( x ) ≻ ( α,β ) α ( y )) ≺ ( α,β ) β ( z ) − αβ ( x ) ≻ ( α,β ) ( α ( y ) ≺ ( α,β ) z )+( β ( y ) ≺ ( α,β ) α ( x )) ≺ ( α,β ) β ( z ) − αβ ( y ) ≺ ( α,β ) ( α ( x ) ◦ ( α,β ) z )= ( αβ ( x ) ≻ αβ ( y )) ≺ ( α,β ) β ( z ) − αβ ( x ) ≻ ( α,β ) ( α ( y ) ≺ β ( z ))+( αβ ( y ) ≺ αβ ( x )) ≺ ( α,β ) β ( z ) − αβ ( y ) ≺ ( α,β ) ( α ( x ) ◦ β ( z ))= ( α β ( x ) ≻ α β ( y )) ≺ β ( z ) − α β ( x ) ≻ ( α β ( y ) ≺ β ( z ))+( α β ( y ) ≺ α β ( x )) ≺ β ( z ) − α β ( y ) ≺ ( α β ( x ) ◦ β ( z ))= as m ( α β ( x ) , α β ( y ) , β ( z )) + as r ( α β ( y ) , α β ( x ) , β ( z )) = 0 . The rest is left to the reader.
Remark 1.2.
More generally, let ( A, ≺ , ≻ , α, β ) be a BiHom-pre-alternative algebra and ˜ α, ˜ β : A → A two BiHom-pre-alternative algebra morphisms such that any two of the maps α, β, ˜ α, ˜ β commute.Define new multiplications on A by x ≺ ′ y = ˜ α ( x ) ≺ ˜ β ( y ) and x ≻ ′ y = ˜ α ( x ) ≻ ˜ β ( y ) , for all x, y ∈ A . Then ( A, ≺ ′ , ≻ ′ , α ˜ α, β ˜ β ) is a BiHom-pre-alternative algebra. Proposition 1.5.
Let ( A, ≺ , ≻ , α, β ) be a BiHom-pre-alternative algebra. Then ( A, ◦ , α, β ) is aBiHom-alternative algebra with the operation x ◦ y = x ≺ y + x ≻ y, for any x, y ∈ A . We say that ( A, ◦ , α, β ) is the associated BiHom-alternative algebra of ( A, ≺ , ≻ , α, β ) and ( A, ≺ , ≻ , α, β ) is called a compatible BiHom-alternative algebra structure on the BiHom-alternative algebra ( A, ◦ , α, β ) .Proof. In fact, for any x, y ∈ A , we have as α,β ( β ( x ) , α ( x ) , y ) = ( β ( x ) ◦ α ( x )) ◦ β ( y ) − αβ ( x ) ◦ ( α ( x ) ◦ y )= ( β ( x ) ◦ α ( x )) ≻ β ( y ) + ( β ( x ) ◦ α ( x )) ≺ β ( y ) − αβ ( x ) ≻ ( α ( x ) ◦ y ) − αβ ( x ) ≺ ( α ( x ) ◦ y )= ( β ( x ) ◦ α ( x )) ≻ β ( y ) + ( β ( x ) ≻ α ( x )) ≺ β ( y )+( β ( x ) ≺ α ( x )) ≺ β ( y ) − αβ ( x ) ≻ ( α ( x ) ≻ y ) − αβ ( x ) ≻ ( α ( x ) ≺ y ) − αβ ( x ) ≺ ( α ( x ) ◦ y )= as lα,β ( β ( x ) , α ( x ) , y ) + as mα,β ( β ( x ) , α ( x ) , y ) + as rα,β ( β ( x ) , α ( x ) , y ) = 0 . Similarly, we show that as α,β ( x, β ( y ) , α ( y )) = 0.In the following we show connections with BiHom-Jordan algebras and BiHom-Malcev algebras.We refer for the definitions to [10]. 6 orollary 1.1. Let ( A, ≺ , ≻ , α, β ) be a regular BiHom-pre-alternative algebra. Then ( A, ⋆, α, β ) isa BiHom-Jordan algebra with the multiplication x ⋆ y = x ≺ y + x ≻ y + α − β ( y ) ≺ αβ − ( x ) + α − β ( y ) ≻ αβ − ( x ) , for any x, y ∈ A . Corollary 1.2.
Let ( A, ≺ , ≻ , α, β ) be a regular BiHom-pre-alternative algebra. Then ( A, [ − , − ] , α, β ) is a BiHom-Malcev algebra with the multiplication [ x, y ] = x ≺ y + x ≻ y − α − β ( y ) ≺ αβ − ( x ) − α − β ( y ) ≻ αβ − ( x ) , for any x, y ∈ A . Proposition 1.6.
Let ( A, ≺ , ≻ , α, β ) be a BiHom-pre-alternative algebra. Then ( A, l ≻ , r ≺ , α, β ) isa bimodule of the associated BiHom-alternative algebra ( A, ◦ , α, β ) , where l ≻ and r ≺ are the leftand right multiplication operators corresponding respectively to the two multiplications ≺ , ≻ .Proof. Let x, y, z ∈ A . Then as rα,β ( y, β ( x ) , α ( x )) = 0, that is( y ≺ β ( x )) ≺ αβ ( x ) = α ( y ) ≺ ( β ( x ) ◦ α ( x )) , which means that r ≺ ( αβ ( x )) r ≺ ( β ( x )) y = r ≺ ( β ( x ) ◦ α ( x )) α ( y ) . Similarly, as lα,β ( β ( x ) , α ( x ) , y ) = 0 is equivalent to l ≻ ( β ( x ) ◦ α ( x )) β ( y ) = l ≻ ( αβ ( x )) l ≻ ( α ( x )) y. On the other hand, we have as mα,β ( β ( x ) , α ( y ) , z ) + as rα,β ( β ( y ) , α ( x ) , z ) = 0, that is( β ( x ) ≻ α ( y )) ≺ β ( z ) − αβ ( x ) ≻ ( α ( y ) ≺ z ) = αβ ( y ) ≺ ( α ( x ) ◦ z ) − ( β ( y ) ≺ α ( x )) ≺ β ( z ) . This means that r ≺ ( β ( z )) l ≻ ( β ( x )) α ( y ) − l ≻ ( αβ ( x )) r ≺ ( z ) α ( y ) = r ≺ ( α ( x ) ◦ z ) αβ ( y ) − r ≺ ( β ( z )) r ≺ ( α ( x )) β ( y ) . Finally, since as mα,β ( x, β ( y ) , α ( z )) + as lα,β ( x, β ( z ) , α ( y )) = 0 then( x ≻ β ( y )) ≺ αβ ( z ) − α ( x ) ≻ ( β ( y ) ≺ α ( z )) = ( x ◦ β ( z )) ≻ αβ ( y ) − α ( x ) ≻ ( β ( z ) ≻ α ( y )) . Hence r ≺ ( αβ ( z )) l ≻ ( x ) β ( y ) − l ≻ ( α ( x )) r ≺ ( α ( z )) β ( y ) = l ≻ ( x ◦ β ( z )) αβ ( y ) − l ≻ ( α ( x )) l ≻ ( β ( z )) α ( y ) . The following definition introduces the notion of bimodule of BiHom-pre-alternative algebras.7 efinition 1.5.
Let ( A ≺ , ≻ , , α, β ) be a BiHom-pre-alternative algebra. A Bimodule of A is avector space V together with two commuting linear maps φ, ψ : V → V and four linear maps L ≻ , L ≺ , R ≻ , R ≺ : A → gl ( V ) satisfying the following set of identities φL ≺ ( x ) = L ≺ ( α ( x )) φ, ψL ≺ ( x ) = L ≺ ( β ( x )) ψ, (1.35) φR ≺ ( x ) = R ≺ ( α ( x )) φ, ψR ≺ ( x ) = R ≺ ( β ( x )) ψ, φL ≻ ( x ) = L ≻ ( α ( x )) φ, (1.36) ψL ≻ ( x ) = L ≻ ( β ( x )) ψ, φR ≻ ( x ) = R ≻ ( α ( x )) φ, ψR ≻ ( x ) = R ≻ ( β ( x )) ψ, (1.37) L ≻ ( β ( x ) ◦ α ( x )) ψ = L ≻ ( αβ ( x )) L ≻ ( α ( x )) , (1.38) R ≻ ( β ( y ))( L ◦ ( β ( x )) φ + R ◦ ( α ( x )) ψ ) = L ≻ ( αβ ( x )) R ≻ ( y ) φ + R ≻ ( α ( x ) ≻ y ) φψ, (1.39) R ≺ ( αβ ( x )) R ≺ ( β ( x )) = R ≺ ( β ( x ) ◦ α ( x )) φ, (1.40) L ≺ ( α ( y ))( L ◦ ( β ( x )) φ + R ◦ ( α ( x )) ψ ) = L ≺ ( y ≺ β ( x )) φψ + R ≺ ( αβ ( x )) L ≺ ( y ) ψ, (1.41) L ≺ ( β ( x ) ≻ α ( y + β ( y ) ≺ α ( x )) ψ = L ≻ ( αβ ( x )) L ≺ ( α ( y )) + L ≺ ( αβ ( y )) L ( α ( x )) , (1.42) R ≺ ( β ( y ))( L ≻ ( β ( x )) φ + R ≺ ( α ( x )) ψ ) = L ≻ ( αβ ( x )) R ≺ ( y ) φ + R ≺ ( α ( x ) ◦ y ) φψ, (1.43) R ≺ ( β ( y ))( R ≻ ( α ( x )) ψ + L ≺ ( β ( x )) φ ) = R ≻ ( α ( x ) ≺ y ) φψ + L ≺ ( αβ ( x )) R ◦ ( y ) φ, (1.44) R ≺ ( αβ ( y )) R ≻ ( β ( x )) + R ≻ ( αβ ( x )) R ◦ ( β ( y )) = R ≻ ( β ( x ) ≺ α ( y )) φ + R ≻ ( β ( y ) ≻ α ( x )) φ, (1.45) R ≺ ( αβ ( y )) L ≻ ( x ) ψ + L ≻ ( x ◦ β ( y )) φψ = L ≻ ( α ( x ))( R ≺ ( α ( y )) ψ + L ≻ ( β ( y )) φ ) , (1.46) L ≺ ( x ≻ β ( y )) φψ + R ≻ ( αβ ( y )) L ◦ ( x ) ψ = L ≻ ( α ( x )) L ≺ ( β ( y )) φ + L ≻ ( α ( x )) R ≻ ( α ( y )) ψ, (1.47) where ◦ = ≺ + ≻ , L ◦ = L ≺ + L ≻ and R ◦ = R ≺ + R ≻ . Proposition 1.7.
A tuple ( V, L ≻ , L ≺ , R ≻ , R ≺ , φ, ψ ) is a bimodule of a BiHom-pre-alternative alge-bra ( A, ≺ , ≻ , α, β ) if and only if the direct sum ( A ⊕ V, ≪ , ≫ , α + φ, β + ψ ) is a BiHom-pre-alternativealgebra, where ( x + u ) ≪ ( y + v ) = x ≺ y + L ≺ ( x ) v + R ≺ ( y ) u, ( x + u ) ≫ ( y + v ) = x ≻ y + L ≻ ( x ) v + R ≻ ( x ) u, and ( α + φ )( x + u ) = α ( x ) + φ ( u ) , ( β + ψ )( x + u ) = β ( x ) + ψ ( u ) , for any x, y ∈ A and u, v ∈ V .Proof. Straightforward and left to the reader.
Definition 1.6.
Let ( V, L, R, φ, ψ ) be a bimodule of a BiHom-alternative algebra ( A, ◦ , α, β ) . Alinear map T : V → A is called an O -operator associated to ( V, L, R, φ, ψ ) if for all u, v ∈ VT ( u ) ◦ T ( v ) = T (cid:0) L ( T ( u )) v + R ( T ( v )) u (cid:1) , T φ = αT, T ψ = βT. (1.48) Remark 1.3.
A Rota-Baxter operator of weight on a BiHom-alternative algebra ( A, ◦ , α, β ) isjust an O -operator associated to the bimodule ( A, ℓ, r, α, β ) , where ℓ and r are the left and rightmultiplication operators corresponding to the multiplication ◦ . Proposition 1.8.
Let T : V → A be an O -operator of a BiHom-alternative algebra ( A, · , α, β ) associated to a bimodule ( V, L, R, φ, ψ ) . Then ( V, ≺ , ≻ , φ, ψ ) is a BiHom-pre-alternative algebrastructure, where u ≺ v = R ( T ( v )) u and u ≻ v = L ( T ( u )) v, for all u, v ∈ V . Therefore ( V, ◦ = ≺ + ≻ , φ, ψ ) is the associated BiHom-alternative algebra ofthis BiHom-pre-alternative algebra and T is a BiHom-alternative algebra morphism. Furthermore, ( V ) = { T ( v ); v ∈ V } ⊂ A is a BiHom-alternative subalgebra of ( A, · , α, β ) and ( T ( V ) , ≺ ′ , ≻ ′ , α, β ) is a BiHom-pre-alternative algebra given by T ( u ) ≺ ′ T ( v ) = T ( u ≺ v ) , and T ( u ) ≻ ′ T ( v ) = T ( u ≻ v ) for all u, v ∈ V .Moreover, the associated BiHom-alternative algebra ( T ( V ) , • = ≺ ′ + ≻ ′ , α, β ) is just a BiHom-alternative subalgebra structure of ( A, · , α, β ) and T is a BiHom-alternative algebra morphism.Proof. For any u, v, w ∈ V , as lφ,ψ ( ψ ( u ) , φ ( u ) , v ) = ( ψ ( u ) ◦ φ ( u )) ≻ ψ ( v ) − φψ ( u ) ≻ ( φ ( u ) ≻ v )= L ( T ( ψ ( u ) ◦ φ ( u ))) ψ ( v ) − φψ ( u ) ≻ ( L ( T ( φ ( u ))) v )= L ( T ( ψ ( u )) ◦ T ( φ ( u ))) ψ ( v ) − L ( T ( φψ ( u ))) L ( T ( φ ( u ))) v = 0 . Furthermore, as lφ,ψ ( u, ψ ( v ) , φ ( w )) = ( u ◦ ψ ( v )) ≻ φψ ( w ) − φ ( u ) ≻ ( ψ ( v ) ≻ φ ( w ))= L ( T ( u ◦ ψ ( v ))) φψ ( w ) − L ( T ( φ ( u ))) L ( T ( ψ ( v ))) φ ( w )= L ( T ( u ) ◦ β ( T ( v ))) φψ ( w ) − L ( α ( T ( u ))) L ( β ( T ( v ))) φ ( w ) , and as mφ,ψ ( u, ψ ( w ) , φ ( v )) = ( u ≻ ψ ( w )) ≺ φψ ( v ) − φ ( u ) ≻ ( ψ ( w ) ≺ φ ( v ))= R ( T ( φψ ( v ))) L ( T ( u )) ψ ( w ) − L ( T ( φ ( u ))) R ( T ( φ ( v ))) ψ ( w )= R ( αβ ( T ( v ))) L ( T ( u )) ψ ( w ) − L ( α ( T ( u ))) R ( α ( T ( v ))) ψ ( w ) . Hence as lφ,ψ ( u, ψ ( v ) , φ ( w )) + as mφ,ψ ( u, ψ ( w ) , φ ( v ))= L ( T ( u ) ◦ β ( T ( v ))) φψ ( w ) − L ( α ( T ( u ))) L ( β ( T ( v ))) φ ( w )+ R ( αβ ( T ( v ))) L ( T ( u )) ψ ( w ) − L ( α ( T ( u ))) R ( α ( T ( v ))) ψ ( w )= 0 . The other identities for ( V, ≺ , ≻ , φ, ψ ) being a BiHom-pre-alternative algebra can be verified simi-larly. Corollary 1.3.
Let ( A, µ, α, β ) be a BiHom-alternative algebra and R : A → A be a Rota-Baxteroperator of weight 0 such that Rα = αR , Rβ = βR . Define the multiplications ≺ and ≻ on A by x ≺ y = xR ( y ) and x ≻ y = R ( x ) y, for all x, y ∈ A . Then ( A, ≺ , ≻ , α, β ) is a BiHom-pre-alternative algebra.Proof. The proof is left to the reader.Now, we introduce a definition of 1-BiHom-cocycle.9 efinition 1.7.
Let ( A, · , α, β ) be a BiHom-alternative algebra and ( V, L, R, φ, ψ ) be a bimodule of A . A linear map D : A → V is said to be a -BiHom-cocycle of ( A, ., α, β ) into ( V, L, R, φ, ψ ) if,for all x, y ∈ A , D ( xy ) = L ( x ) D ( y ) + R ( y ) D ( x ) , φD = Dα, ψD = Dβ.
Proposition 1.9.
Let ( A, ., α, β ) be a BiHom-alternative algebra. Then the following statementsare equivalent(i) There is a compatible BiHom-pre-alternative algebra ( A, ≺ , ≻ , α, β ) structure on ( A, · , α, β ) .(ii) There is an invertible O -operator associated to a bimodule of ( A, · , α, β ) .(iii) There is a bijective -BiHom-cocycle of ( A, · , α, β ) into a bimodule.Proof. ( iii ) ⇒ ( ii ): If D is a bijective 1-BiHom-cocycle of ( A, · , α, β ) into a bimodule ( V, L, R, φ, ψ ),then D − is an O -operator associated to ( V, L, R, φ, ψ ). Indeed, it’s clear that D − φ = αD − and D − ψ = βD − . Take two elements u, v ∈ V , there is x, y ∈ A such that D ( x ) = u and D ( y ) = v .Since D ( xy ) = L ( x ) D ( y ) + R ( y ) D ( x ) , then D − ( u ) D − ( v ) = D − (cid:0) L ( D − ( u )) v + R ( D − ( v )) u (cid:1) . ( ii ) ⇒ ( i ): If T : V → A is an invertible O -operator associated to a bimodule ( V, L, R, φ, ψ ), then T ( V ) = A and using Proposition 1.8, there is a compatible pre-alternative algebra structure on A given by: x ≺ y = T ( R ( y ) T − ( x )) , x ≻ y = T ( L ( x ) T − ( y )) , ∀ x, y ∈ A. (1.49)( i ) ⇒ ( iii ): If ( A, ≺ , ≻ , α, β ) is a compatible BiHom-pre-alternative algebra structure on ( A, · , α, β ),then it is obvious that the identity map id is a bijective 1-BiHom-cocycle of A into the adjointbimodule ( A, l ≻ , r ≺ , α, β ). In this section, we introduce the BiHom version of alternative quadri-algebras introduced in [19].This variety of algebras is a generalization of BiHom-quadri-algebras discussed in [16].
Definition 2.1.
A BiHom-alternative quadri-algebra is a 7-tuple ( A, տ , ւ , ր , ց , α, β ) consistingof a vector space A , four bilinear maps տ , ւ , ր , ց : A × A → A and two commuting linear maps α, β : A → A which are algebra maps with respect to previous four operations and such that the ollowing axioms hold for all x, y, z ∈ A { β ( x ) , α ( y ) , z } rα,β + { β ( y ) , α ( x ) , z } mα,β = 0 , (2.50) { β ( x ) , α ( y ) , z } nα,β + { β ( y ) , α ( x ) , z } wα,β = 0 , (2.51) { β ( x ) , α ( y ) , z } neα,β + { β ( y ) , α ( x ) , z } eα,β = 0 , (2.52) { β ( x ) , α ( y ) , z } swα,β + { β ( y ) , α ( x ) , z } sα,β = 0 , (2.53) { β ( x ) , α ( y ) , z } lα,β + { β ( y ) , α ( x ) , z } lα,β = 0 , (2.54) { x, β ( y ) , α ( z ) } rα,β + { x, β ( z ) , α ( y ) } rα,β = 0 , (2.55) { x, β ( y ) , α ( z ) } nα,β + { x, β ( z ) , α ( y ) } neα,β = 0 , (2.56) { x, β ( y ) , α ( z ) } wα,β + { x, β ( z ) , α ( y ) } swα,β = 0 , (2.57) { x, β ( y ) , α ( z ) } mα,β + { x, β ( z ) , α ( y ) } lα,β = 0 , (2.58) { x, β ( y ) , α ( z ) } sα,β + { x, β ( z ) , α ( y ) } eα,β = 0; (2.59) where x ≻ y := x ր y + x ց y, x ≺ y := x տ y + x ւ y,x ∨ y := x ց y + x ւ y, x ∧ y := x ր y + x տ y,x ∗ y := x ց y + x ր y + x ւ y + x տ y = x ≻ y + x ≺ y = x ∨ y + x ∧ y. and { x, y, z } rα,β = ( x տ y ) տ β ( z ) − α ( x ) տ ( y ∗ z ) , { x, y, z } lα,β = ( x ∗ y ) ց β ( z ) − α ( x ) ց ( y ց z ) { x, y, z } neα,β = ( x ∧ y ) ր β ( z ) − α ( x ) ր ( y ≻ z ) , { x, y, z } swα,β = ( x ≺ y ) ւ β ( z ) − α ( x ) ւ ( y ∨ z ) , { x, y, z } nα,β = ( x ր y ) տ β ( z ) − α ( x ) ր ( y ≺ z ) , { x, y, z } wα,β = ( x ւ y ) տ β ( z ) − α ( x ) ւ ( y ∧ z ) , { x, y, z } sα,β = ( x ≻ y ) ւ β ( z ) − α ( x ) ց ( y ւ z ) , { x, y, z } eα,β = ( x ∨ y ) ր β ( z ) − α ( x ) ց ( y ր z ) , { x, y, z } mα,β = ( x ց y ) տ β ( z ) − α ( x ) ց ( y տ z ) . Remarks 2.1. • Note that in every BiHom-quadri-algebra, all these associators are trivial. • When α = β = id , the BiHom-alternative quadri-algebra is an alternative quadri-algebra. • Since K is a field of characteristic different from , the identities (2.54) and (2.55) can be writtenrespectively { β ( x ) , α ( x ) , y } lα,β = 0 and { x, β ( y ) , α ( y ) } rα,β = 0 . Lemma 2.1.
Let ( A, տ , ւ , ր , ց , α, β ) be a BiHom-alternative quadri-algebra. Then both ( A, ≺ , ≻ , α, β ) and ( A, ∨ , ∧ , α, β ) are BiHom-pre-alternative algebras (called, respectively, horizontal andvertical BiHom-pre-alternative structures associated to A ) and ( A, ∗ , α, β ) is a BiHom-alternativealgebra.Proof. We will prove just that ( A, ≺ , ≻ , α, β ) is a BiHom-pre-alternative algebra. For this, take x, y, z ∈ A . We will just show how to prove two identities and the other ones could be donesimilarly. We remark that( x ≺ β ( y )) ≺ αβ ( y ) − α ( x ) ≺ ( β ( y ) ∗ α ( y ))= ( x տ β ( y )) տ αβ ( y ) + ( x տ β ( y )) ւ αβ ( y ) + ( x ւ β ( y )) տ αβ ( y )+ ( x ւ β ( y )) ւ αβ ( y ) − α ( x ) տ ( β ( y ) ∗ α ( y )) − α ( x ) ւ ( β ( y ) ∨ α ( y )) − α ( x ) ւ ( β ( y ) ∧ α ( y ))= { x, β ( y ) , α ( y ) } rα,β + { x, β ( y ) , α ( y ) } wα,β + { x, β ( y ) , α ( y ) } swα,β = 0 .
11n addition, as mα,β ( β ( x ) , α ( y ) , z ) + as rα,β ( β ( y ) , α ( x ) , z )= ( β ( x ) ≻ α ( y )) ≺ β ( z ) − αβ ( x ) ≻ ( α ( y ) ≺ z ) + ( β ( y ) ≺ α ( x )) ≺ β ( z ) − αβ ( y ) ≺ ( α ( x ) ∗ z )= ( β ( x ) ց α ( y )) տ β ( z ) + ( β ( x ) ց α ( y )) ւ β ( z ) + ( β ( x ) ր α ( y )) տ β ( z )+ ( β ( x ) ր α ( y )) տ β ( z ) − αβ ( x ) ց ( α ( y ) տ z ) − αβ ( x ) ց ( α ( y ) ւ z ) − αβ ( x ) ր ( α ( y ) տ z ) − αβ ( x ) ր ( α ( y ) ւ z ) + ( β ( y ) ≺ α ( x )) ւ β ( z )+ ( β ( y ) տ α ( x )) տ β ( z ) + ( β ( y ) ւ α ( x )) տ β ( z ) − αβ ( y ) տ ( α ( x ) ∗ z ) − αβ ( y ) տ ( α ( x ) ∨ z ) − αβ ( y ) տ ( α ( x ) ∧ z )= { β ( x ) , α ( y ) , z } mα,β + { β ( x ) , α ( y ) , z } nα,β + { β ( x ) , α ( y ) , z } sα,β + { β ( y ) , α ( x ) , z } rα,β + { β ( y ) , α ( x ) , z } wα,β + { β ( y ) , α ( x ) , z } swα,β = 0 . Using a similar computation, we can get as mα,β ( x, β ( y ) , α ( z )) + as lα,β ( x, β ( z ) , α ( y ))= { x, β ( y ) , α ( z ) } mα,β + { x, β ( y ) , α ( z ) } nα,β + { x, β ( y ) , α ( z ) } sα,β + { x, β ( z ) , α ( y ) } lα,β + { x, β ( z ) , α ( y ) } neα,β + { x, β ( z ) , α ( y ) } eα,β = 0 . Similarly, we can show that ( A, ∨ , ∧ , α, β ) and ( A, ∗ , α, β ) are respectively BiHom-pre-alternativealgebra and BiHom-alternative algebra.A morphism f : ( A, տ , ւ , ր , ց , α, β ) → ( A ′ , տ ′ , ւ ′ , ր ′ , ց ′ , α ′ , β ′ ) of BiHom-alternative quadri-algebras is a linear map f : A → A ′ satisfying f ( x ր y ) = f ( x ) ր ′ f ( y ) , f ( x ց y ) = f ( x ) ց ′ f ( y ) , f ( x տ y ) = f ( x ) տ ′ f ( y ) and f ( x ւ y ) = f ( x ) ւ ′ f ( y ), for all x, y ∈ A , as well as f α = α ′ f and f β = β ′ f . Proposition 2.1.
Let ( A, տ , ւ , ր , ց ) be an alternative quadri-algebra and α, β : A → A twocommuting alternative quadri-algebra morphisms. Define ց ( α,β ) , ր ( α,β ) , ւ ( α,β ) , տ ( α,β ) : A × A → A by x ց ( α,β ) y = α ( x ) ց β ( y ) , x ր ( α,β ) y = α ( x ) ր β ( y ) ,x ւ ( α,β ) y = α ( x ) ւ β ( y ) , x տ ( α,β ) y = α ( x ) տ β ( y ) , for all x, y ∈ A . Then A ( α,β ) := ( A, տ ( α,β ) , ւ ( α,β ) , ր ( α,β ) , ց ( α,β ) , α, β ) is a BiHom-alternativequadri-algebra, called the Yau twist of A . Moreover, assume that ( A ′ , տ ′ , ւ ′ , ր ′ , ց ′ ) is anotheralternative quadri-algebra and α ′ , β ′ : A ′ → A ′ are two commuting alternative quadri-algebramorphisms and f : A → A ′ is an alternative quadri-algebra morphism satisfying f α = α ′ f and f β = β ′ f . Then f : A ( α,β ) → A ′ ( α ′ ,β ′ ) is a BiHom-alternative quadri-algebra morphism.Proof. We only prove ((2.52)) and ((2.58)) and leave the rest to the reader. We define the followingoperations x ≻ ( α,β ) y := x ր ( α,β ) y + x ց ( α,β ) y , x ≺ ( α,β ) y := x տ ( α,β ) y + x ւ ( α,β ) y , x ∨ ( α,β ) y := x ց ( α,β ) y + x ւ ( α,β ) y , x ∧ ( α,β ) y := x ր ( α,β ) y + x տ ( α,β ) y and x ∗ ( α,β ) y := x ց ( α,β ) y + x ր ( α,β ) y + x ւ ( α,β ) y + x տ ( α,β ) y , for all x, y ∈ A . It is easy to get x ≻ ( α,β ) y = α ( x ) ≻ β ( y ) , x ≺ ( α,β ) y = α ( x ) ≺ β ( y ) , x ∨ ( α,β ) y = α ( x ) ∨ β ( y ) , x ∧ ( α,β ) y = α ( x ) ∧ β ( y )and x ∗ ( α,β ) y = α ( x ) ∗ β ( y ) for all x, y ∈ A . By using the fact that α and β are two commuting12uadri-alternative algebra morphisms, one can compute, for all x, y, z ∈ A : { β ( x ) , α ( y ) , z } neα,β + { β ( y ) , α ( x ) , z } eα,β = ( β ( x ) ∧ ( α,β ) α ( y )) ր ( α,β ) β ( z ) − αβ ( x ) ր ( α,β ) ( α ( y ) ≻ ( α,β ) z )+ ( β ( y ) ∨ ( α,β ) α ( x )) ր ( α,β ) β ( z ) − αβ ( y ) ց ( α,β ) ( α ( x ) ր ( α,β ) z )= ( α β ( x ) ∧ α β ( y )) ր β ( z ) − α β ( x ) ր ( α β ( y ) ≻ β ( z ))+ ( α β ( y ) ∨ α β ( x )) ր β ( z ) − α β ( y ) ց ( α β ( x ) ր β ( z ))= { α β ( x ) , α β ( y ) , β ( z ) } ne + { α β ( y ) , α β ( x ) , β ( z ) } e = 0 , and { x, β ( y ) , α ( z ) } mα,β + { x, β ( y ) , α ( z ) } lα,β = ( x ց ( α,β ) β ( y )) տ ( α,β ) αβ ( z ) − α ( x ) ց ( α,β ) ( β ( y ) տ ( α,β ) α ( z ))+ ( x ∗ ( α,β ) β ( z )) ց ( α,β ) αβ ( y ) − α ( x ) ց ( α,β ) ( β ( z ) ց ( α,β ) α ( y ))= ( α ( x ) ց αβ ( y )) տ αβ ( z ) − α ( x ) ց ( αβ ( y ) տ αβ ( z ))+ ( α ( x ) ∗ αβ ( z )) ց αβ ( y ) − α ( x ) ց ( αβ ( z ) ց αβ ( y ))= { α ( x ) , αβ ( y ) , αβ ( z ) } m + { α ( x ) , αβ ( z ) , αβ ( y ) } l = 0 . Remark 2.1.
Let ( A, տ , ւ , ր , ց , α, β ) be a BiHom-alternative quadri-algebra and e α, e β : A → A be two BiHom-alternative quadri-algebra morphisms such that any of the maps α, β, e α, e β commute.Define new multiplications on A by: x ր ′ y = e α ( x ) ր e β ( y ) , x ց ′ y = e α ( x ) ց e β ( y ) ,x տ ′ y = e α ( x ) տ e β ( y ) , x ւ ′ y = e α ( x ) ւ e β ( y ) . Then, one can prove that ( A ′ , ր ′ , ց ′ , ւ ′ , տ ′ , α ◦ e α, β ◦ e β ) is a BiHom-alternative quadri-algebra. Definition 2.2.
Let ( A, ≺ , ≻ , α, β ) be a BiHom-pre-alternative algebra and ( V, L ≺ , R ≺ , L ≻ , R ≻ , φ, ψ ) be a bimodule. A linear map T : V → A is called an O -operator of ( A, ≺ , ≻ , α, β ) associated to ( V, L ≺ , R ≺ , L ≻ , R ≻ , φ, ψ ) if T satisfies: T φ = αT , T ψ = βT and T ( u ) ≻ T ( v ) = T ( L ≻ ( T ( u )) v + R ≻ ( T ( v )) u ) , T ( u ) ≺ T ( v ) = T ( L ≺ ( T ( u )) v + R ≺ ( T ( v )) u ) , (2.60) for all u, v ∈ V . The have the following results.
Proposition 2.2.
Let ( V, L ≺ , R ≺ , L ≻ , R ≻ , φ, ψ ) be a bimodule of a BiHom-pre-alternative algebra ( A, ≺ , ≻ , α, β ) and ( A, ◦ , α, β ) be the associated BiHom-alternative algebra. If T is an O -operatorof ( A, ≺ , ≻ , α, β ) associated to ( V, L ≺ , R ≺ , L ≻ , R ≻ , φ, ψ ) , then T is an O -operator of ( A, ◦ , α, β ) associated to ( V, L ≺ + L ≻ , R ≺ + R ≻ , φ, ψ ) . roposition 2.3. Let ( A, ≺ , ≻ , α, β ) be a BiHom-pre-alternative algebra and ( V, L ≺ , R ≺ , L ≻ , R ≻ , φ, ψ ) be a bimodule. Let T be an O -operator of ( A, ≺ , ≻ , α, β ) associated to ( V, L ≺ , R ≺ , L ≻ , R ≻ , φ, ψ ) .Then there exists a BiHom-alternative quadri-algebra structure on V given for any u, v ∈ V by u ց v = L ≻ ( T ( u )) v, u ր v = R ≻ ( T ( v )) u, u ւ v = L ≺ ( T ( u )) v, u տ v = R ≺ ( T ( v )) u. (2.61) Therefore there exists a BiHom-pre-alternative algebra structure on V given by u ≺ v = u ւ v + u տ v, u ≻ v = u ց v + u ր v, and T is a homomorphism of BiHom-pre-alternative algebras.Furthermore, T ( V ) = { T ( v ) , v ∈ V } ⊂ A is a BiHom-pre-alternative subalgebra of A and thereexists an induced BiHom-alternative quadri-algebra structure on T ( V ) given by T ( u ) ց T ( v ) = T ( u ց v ) , T ( u ) ր T ( v ) = T ( u ր v ) ,T ( u ) ւ T ( v ) = T ( u ւ v ) , T ( u ) տ T ( v ) = T ( u տ v ) . (2.62) Moreover, its corresponding associated horizontal BiHom-pre-alternative algebra structure on T ( V ) is just the BiHom-pre-alternative subalgebra structure of ( A, ≺ , ≻ , α, β ) and T a BiHom-alternativequadri-algebra homomorphism.Proof. Set L = L ≺ + L ≻ and R = R ≺ + R ≻ . For any u, v, w ∈ V , we have( ψ ( u ) ∗ φ ( u )) ց ψ ( v ) − φψ ( u ) ց ( φ ( u ) ց v )= ( L ( T ( ψ ( u ))) φ ( u ) + R ( T ( φ ( u ))) ψ ( u )) ց ψ ( v ) − φψ ( u ) ց ( L ≻ ( T ( φ ( u ))) v )= L ≻ ( T ( L ( T ( ψ ( u ))) φ ( u ) + R ( T ( φ ( u ))) ψ ( u ))) ψ ( v ) − L ≻ ( T ( φψ ( u ))) L ≻ ( T ( φ ( u ))) v = L ≻ ( T ( ψ ( u )) ◦ T ( φ ( u ))) ψ ( v ) − L ≻ ( T ( φψ ( u ))) L ≻ ( T ( φ ( u ))) v = 0 . Furthermore,( u ւ ψ ( v )) տ φψ ( w ) − φ ( u ) ւ ( ψ ( v ) տ φ ( w ) + ψ ( v ) ր φ ( w ))= R ≺ ( T ( φψ ( w ))) L ≺ ( T ( u )) ψ ( v ) − L ≺ ( T ( φ ( u )))( R ≺ ( T ( φ ( w ))) ψ ( v ) + R ≻ ( T ( φ ( w ))) ψ ( v ))= R ≺ ( T ( φψ ( w ))) L ≺ ( T ( u )) ψ ( v ) − L ≺ ( T ( φ ( u ))) R ( T ( φ ( w ))) ψ ( v ) , and( u տ ψ ( w ) + u ւ ψ ( w )) ւ φψ ( v ) − φ ( u ) ւ ( ψ ( w ) ց φ ( v ) + ψ ( w ) ւ φ ( v ))= L ≺ ( T ( R ≺ ( ψ ( w )) u + L ≺ ( u ) ψ ( w ))) φψ ( v ) − L ≺ ( T ( φ ( u )))( L ≻ ( T ( ψ ( w ))) φ ( v ) + L ≺ ( T ( ψ ( w ))) φ ( v ))= L ≺ ( T ( u ) ≺ T ( ψ ( w ))) φψ ( v ) − L ≺ ( T ( φ ( u ))) L ( T ( ψ ( w ))) φ ( v ) . This means that { u, ψ ( v ) , φ ( w ) } wφ,ψ + { u, ψ ( w ) , φ ( v ) } swφ,ψ = R ≺ ( T ( φψ ( w ))) L ≺ ( T ( u )) ψ ( v ) − L ≺ ( T ( φ ( u ))) R ( T ( φ ( w ))) ψ ( v )+ L ≺ ( T ( u ) ≺ T ( ψ ( w ))) φψ ( v ) − L ≺ ( T ( φ ( u ))) L ( T ( ψ ( w ))) φ ( v ) = 0 , since ( V, L ≺ , R ≺ , L ≻ , R ≻ , φ, ψ ) is a bimodule of ( A, ≺ , ≻ , α, β ). The rest of identities can be provedusing analogous computations, so they will be left to the reader.14 orollary 2.1. Let ( A, ≺ , ≻ , α, β ) be a BiHom-pre-alternative algebra. Then there exists a com-patible BiHom-alternative quadri-algebra structure on ( A, ≺ , ≻ , α, β ) such that ( A, ≺ , ≻ , α, β ) isthe associated horizontal BiHom-pre-alternative algebra if and only if there exists an invertible O -operator of ( A, ≺ , ≻ , α, β ) .Proof. If there exists an invertible O -operator T of ( A, ≺ , ≻ , α, β ) associated to a bimodule( V, L ≺ , R ≺ , L ≻ , R ≻ , φ, ψ ), then by Proposition 2.3, there exists a BiHom-alternative quadri-algebra structure on V . Therefore we can define a BiHom-alternative quadri-algebra structure on A by equation (2.62) such that T is a BiHom-alternative quadri-algebra isomorphism, that is, x ց y = T ( L ≻ ( x ) T − ( y )) , x ր y = T ( R ≻ ( y ) T − ( x )) ,x ւ y = T ( L ≺ ( x ) T − ( y )) , x տ y = T ( R ≺ ( y ) T − ( x )) , ∀ x, y ∈ A. Moreover it is a compatible BiHom-alternative quadri-algebra structure on ( A, ≺ , ≻ , α, β ) since forany x, y ∈ A , we have x ≻ y = T ( T − ( x ) ≻ T − ( y )) = T ( R ≻ ( y ) T − ( x ) + R ≻ ( x ) T − ( y )) = x ր y + x ց y,x ≺ y = T ( T − ( x ) ≺ T − ( y )) = T ( R ≺ ( y ) T − ( x ) + L ≺ ( x ) T − ( y )) = x տ y + x ւ y. Conversely, let ( A, ց , ր , տ , ւ ) be a BiHom-alternative quadri-algebra and ( A, ≺ , ≻ , α, β ) bethe associated horizontal BiHom-pre-alternative algebra. Then ( A, L ց , R ր , L ւ , R տ , α, β ) is abimodule of ( A, ≺ , ≻ , α, β ) and the identity map id is an invertible O -operator of ( A, ≺ , ≻ , α, β )associated to it.Now, we introduce the following concept of Rota-Baxter operator on a BiHom-alternativequadri-algebra which is a particular case of O -operator. Definition 2.3.
Let ( A, ≺ , ≻ , α, β ) be a BiHom-pre-alternative algebra. A Rota-Baxter operatorof weight on A is a linear map R : A → A such that Rα = αR , Rβ = βR and the followingconditions are satisfied, for all x, y ∈ A : R ( x ) ≻ R ( y ) = R ( x ≻ R ( y ) + R ( x ) ≻ y ) , (2.63) R ( x ) ≺ R ( y ) = R ( x ≺ R ( y ) + R ( x ) ≺ y ) . (2.64)We know that ( A, ∗ , α, β ) is a BiHom-alternative algebra. Adding equations (2.63) and (2.64),one obtains that R is also a Rota-Baxter operator of weight 0 for ( A, ∗ ): R ( x ) ∗ R ( y ) = R ( x ∗ R ( y ) + R ( x ) ∗ y ) . Analogously to what happens for BiHom-quadri-algebras [16], Rota-Baxter operators allowdifferent constructions for BiHom-alternative quadri-algebras.
Corollary 2.2.
Let ( A, ≺ , ≻ , α, β ) be a BiHom-pre-alternative algebra and R : A → A be a Rota-Baxter operator of weight 0 for A . Define new operations on A by x ց R y = R ( x ) ≻ y, x ր R y = x ≻ R ( y ) , x ւ R y = R ( x ) ≺ y and x տ R y = x ≺ R ( y ) . Then ( A, տ R , ւ R , ր R , ց R , α, β ) is a BiHom-alternative quadri-algebra. emma 2.2. Let ( A, ∗ , α, β ) be a BiHom-alternative algebra and R, P two commuting Rota-Baxteroperators on A such that Rα = αR , Rβ = βR , P α = αP and P β = βP . Then P is a Rota-Baxter operator on the BiHom-pre-alternative algebra ( A, ≺ R , ≻ R , α, β ) . where x ≺ R y = xR ( y ) and x ≻ R y = R ( x ) y .Proof. For all x, y ∈ A , we have: P ( x ) ≻ R P ( y ) = R ( P ( x )) P ( y ) = P ( R ( x )) P ( y )= P ( R ( x ) P ( y ) + P ( R ( x )) y ) = P ( R ( x ) P ( y ) + R ( P ( x )) y )= P ( x ≻ R P ( y ) + P ( x ) ≻ R y ) , and P ( x ) ≺ R P ( y ) = P ( x ) R ( P ( y )) = P ( x ) P ( R ( y ))= P ( xP ( R ( y )) + P ( x ) R ( y )) = P ( xR ( P ( y )) + P ( x ) R ( y ))= P ( x ≺ R P ( y ) + P ( x ) ≺ R y ) . Corollary 2.3.
In the setting of Lemma 2.2, there exists a BiHom-alternative quadri-algebra struc-ture on the underlying vector space ( A, ∗ , α, β ) , with operations defined by x ց y = P ( x ) ≻ R y = P ( R ( x )) y = R ( P ( x )) y,x ր y = x ≻ R P ( y ) = R ( x ) P ( y ) ,x ւ y = P ( x ) ≺ R y = P ( x ) R ( y ) ,x տ y = x ≺ R P ( y ) = xR ( P ( y )) = xP ( R ( y )) . Proof.
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