Double constructions of biHom-Frobenius algebras
Mahouton Norbert Hounkonnou, Gbêvèwou Damien Houndedji, Sergei Silvestrov
aa r X i v : . [ m a t h . QA ] A ug Double constructions of biHom-Frobeniusalgebras
Mahouton Norbert Hounkonnou, Gbêvèwou Damien Houndedji, Sergei Silvestrov
Abstract
This paper addresses a Hom-associative algebra built as a direct sum ofa given Hom-associative algebra ( A , · , α ) and its dual ( A ∗ , ◦ , α ∗ ) , endowed witha non-degenerate symmetric bilinear form B , where · and ◦ are the products de-fined on A and A ∗ , respectively, and α and α ∗ stand for the corresponding algebrahomomorphisms. Such a double construction, also called Hom-Frobenius algebra,is interpreted in terms of an infinitesimal Hom-bialgebra. The same procedure isapplied to characterize the double construction of biHom-associative algebras, alsocalled biHom-Frobenius algebra. Finally, a double construction of Hom-dendriformalgebras, also called double construction of Connes cocycle or symplectic Hom-associative algebra, is performed. Besides, the concept of biHom-dendriform al-gebras is introduced and discussed. Their bimodules and matched pairs are alsoconstructed, and related relevant properties are given. Key words:
Hom-assoicative algebra, biHom-associative algebra, biHom Frobe-nius algebra, biHom-dendriform algebra
MSC2020 Classification:
Mahouton Norbert HounkonnouInternational Chair in Mathematical Physics and Applications, ICMPA-UNESCO Chair, Univer-sity of Abomey-Calavi, 072 BP 50, Cotonou, Rep. of Benin.e-mail: [email protected],[email protected]
Gbêvèwou Damien HoundedjiInternational Chair in Mathematical Physics and Applications, ICMPA-UNESCO Chair, Univer-sity of Abomey-Calavi, 072 BP 50, Cotonou, Rep. of Benin.e-mail: [email protected]
Sergei SilvestrovDivision of Applied Mathematics, School of Education, Culture and Communication, MälardalenUniversity, Box 883, 72123 Västerås, Sweden. e-mail: [email protected]
The Hom-algebraic structures originated from quasi-deformations of Lie algebras ofvector fields which gave rise to quasi-Lie algebras, defined as generalized Lie struc-tures in which the skew-symmetry and Jacobi conditions are twisted. Hom-Lie alge-bras and more general quasi-Hom-Lie algebras where introduced first by Silvestrovand his students Hartwig and Larsson in [25], where the general quasi-deformationsand discretizations of Lie algebras of vector fields using general twisted derivations, σ -derivations, and a general method for construction of deformations of Witt andVirasoro type algebras based on twisted derivations has been developed. The ini-tial motivation came from examples of q -deformed Jacobi identities discovered in q -deformed versions and other discrete modifications of differential calculi and ho-mological algebra, q -deformed Lie algebras and other algebras important in stringtheory, vertex models in conformal field theory, quantum mechanics and quantumfield theory, such as the q -deformed Heisenberg algebras, q -deformed oscillator al-gebras, q -deformed Witt, q -deformed Virasoro algebras and related q -deformationsof infinite-dimensional algebras [1, 15–21, 28, 29, 37–39].Possibility of studying, within the same framework, q -deformations of Lie al-gebras and such well-known generalizations of Lie algebras as the color and superLie algebras provided further general motivation for development of quasi-Lie alge-bras and subclasses of quasi-Hom-Lie algebras and Hom-Lie algebras. The generalabstract quasi-Lie algebras and the subclasses of quasi-Hom-Lie algebras and Hom-Lie algebras, as well as their color (graded) counterparts, color (graded) quasi-Liealgebras, color (graded) quasi-Hom-Lie algebras and color (graded) Hom-Lie al-gebras, including in particular the super quasi-Lie algebras, super quasi-Hom-Liealgebras, and super Hom-Lie algebras, have been introduced in [25, 33–35, 50, 51].In [43], Hom-associative algebras have been introduced. Hom-associative algebrasis a generalization of the associative algebras with the associativity law twisted bya linear map. In [43], Hom-Lie admissible algebras generalizing Lie-admissible al-gebras, were introduced as Hom-algebras such that the commutator product, de-fined using the multiplication in a Hom-algebra, yields a Hom-Lie algebra, andHom-associative algebras were shown to be Hom-Lie admissible. Moreover, in[43], more general G -Hom-associative algebras including Hom-associative alge-bras, Hom-Vinberg algebras (Hom-left symmetric algebras), Hom-pre-Lie algebras(Hom-right symmetric algebras), and some other Hom-algebra structures, general-izing G -associative algebras, Vinberg and pre-Lie algebras respectively, have beenintroduced and shown to be Hom-Lie admissible, meaning that for these classesof Hom-algebras, the operation of taking commutator leads to Hom-Lie algebrasas well. Also, flexible Hom-algebras have been introduced, connections to Hom-algebra generalizations of derivations and of adjoint maps have been noticed, andsome low-dimensional Hom-Lie algebras have been described. The enveloping al-gebras of Hom-Lie algebras were considered in [54] using combinatorial objects ofweighted binary trees. In [27], for Hom-associative algebras and Hom-Lie algebras,the envelopment problem, operads, and the Diamond Lemma and Hilbert seriesfor the Hom-associative operad and free algebra have been studied. Strong Hom- ouble constructions of biHom-Frobenius algebras 3 associativity yielding a confluent rewrite system and a basis for the free stronglyhom-associative algebra has been considered in [26]. An explicit constructive way,based on free Hom-associative algebras with involutive twisting, was developedin [23] to obtain the universal enveloping algebras and Poincaré-Birkhoff-Witt typetheorem for Hom-Lie algebras with involutive twisting map. Free Hom-associativecolor algebra on a Hom-module and enveloping algebra of color Hom-Lie algebraswith involutive twisting and also with more general conditions on the powers oftwisting map was constructed, and Poincaré–Birkhoff–Witt type theorem was ob-tained in [3, 4]. It is worth noticing here that, in the subclass of Hom-Lie algebras,the skew-symmetry is untwisted, whereas the Jacobi identity is twisted by a singlelinear map and contains three terms as in Lie algebras, reducing to ordinary Liealgebras when the twisting linear map is the identity map.Hom-algebra structures include their classical counterparts and open new broadpossibilities for deformations, extensions to Hom-algebra structures of representa-tions, homology, cohomology and formal deformations, Hom-modules and hom-bimodules, Hom-Lie admissible Hom-coalgebras, Hom-coalgebras, Hom-Hopf al-gebras, Hom-bialgebras, L -modules, L -comodules and Hom-Lie quasi-bialgebras, n -ary generalizations of biHom-Lie algebras and biHom-associative algebras andgeneralized derivations, Rota-Baxter operators, Hom-dendriform color algebras,Rota-Baxter bisystems and covariant bialgebras, Rota-Baxter cosystems, coquasi-triangular mixed bialgebras, coassociative Yang-Baxter pairs, coassociative Yang-Baxter equation and generalizations of Rota-Baxter systems and algebras, curved O -operator systems and their connections with tridendriform systems and pre-Liealgebras [2, 6–12, 14, 24, 30, 32, 33, 40–42, 44–49, 52–56].The notion of biHom-associative algebras was introduced in [22]. In fact, abiHom-associative algebra is a (nonassociative) algebra A endowed with two com-muting multiplicative linear maps α , β : A → A such that α ( a )( bc ) = ( ab ) β ( c ) , forall a , b , c ∈ A . This concept arose in the study of algebras in so-called group Hom-categories. In [22], the authors introduced biHom-Lie algebras (also by using thecategorical approach) and biHom-bialgebras. They discussed these new structuresby presenting some basic properties and constructions (representations, twisted ten-sor products, smash products, etc.).A Frobenius algebra is an associative algebra equipped with a non-degenerate in-variant bilinear form. This type of algebras also plays an important role in differentareas of mathematics and physics, such as statistical models over two-dimensionalgraphs [13] and topological quantum field theory [31]. In [5], Bai described asso-ciative analogs of Drinfeld’s double constructions for Frobenius algebras and forassociative algebras equipped with non-degenerate Connes cocycles. We note thattwo different types of constructions are involved:(i) the Drinfeld’s double type constructions, from a Frobenius algebra or froman associative algebra equipped with a Connes cocyle; and(ii) the Frobenius algebra obtained from anti-symmetric solution of associativeYang-Baxter equation and non-degenerate Connes cocycle obtained from asymmetric solution of a D -equation. Mahouton Norbert Hounkonnou, Gbêvèwou Damien Houndedji, Sergei Silvestrov
The aim of the present work is to establish the double constructions of biHom-Frobenius algebras and Hom-associative algebra equipped with a Connes cocyle,generalizing the double constructions of Frobenius algebras and Connes cocycledescribed in [5] by twisting the defining axioms by a certain twisting map. Whenthe twisting map happens to be the identity map, one gets the ordinary algebraicstructures. Furthermore, the bialgebras of related double constructions are built. Wedefine the antisymmetric infinitesimal biHom-bialgebras and Hom-dendriform D -bialgebras.The paper is organized as follows. In Section 2, we introduce the conceptsof matched pairs of Hom-associative algebras and establish some relevant prop-erties. In Section 3, we perform the double constructions of multiplicative Hom-Frobenius algebras and antisymmetric infinitesimal Hom-bialgebras. In Section 4,we define the bimodule of biHom-associative algebras, and achieve the double con-structions of multiplicative biHom-Frobenius algebras and antisymmetric infinites-imal biHom-bialgebras. Section 5 deals with the double constructions of involutivesymplectic Hom-associative algebras. Section 6 is devoted to the matched pairs ofbiHom-associative algebras and related important characteristics. In Section 7, weend with some concluding remarks. Definition 1 ( [43]).
A Hom-associative algebra is a triple ( A , · , α ) consisting of alinear space A over a field K , K -bilinear map · : A ⊗ A → A and a linear spacemap α : A → A satisfying the Hom-associativity property: α ( x ) · ( y · z ) = ( x · y ) · α ( z ) . (1)If, in addition, α satisfies the multiplicativity property α ( x · y ) = α ( x ) · α ( y ) , (2)then ( A , · , α ) is said to be multiplicative. Remark 1. If α = Id A , ( A , · , Id A ) , simply denoted ( A , · ) , is an associative algebra. Example 1.
Let { e , e , e } be a basis of a 3-dimensional vector space A over K .The following multiplication · and map α on A define a Hom-associative algebra: e · e = e , e · e = e · e = e , α ( e ) = a e + a e , α ( e ) = b e + b e , α ( e ) = c e + c e , (3)where a , a , b , b , c , c ∈ K . ouble constructions of biHom-Frobenius algebras 5 Definition 2.
A Hom-module is a pair ( V , β ) , where V is a K -vector space, and β : V → V is a linear map.We will use in this article a definition of bimodule of a Hom-associative alge-bras including Hom-modules maps conditions (7), (8), while we note that thereare also other definitions of Hom-modules and Hom-bimodules of Hom-associativealgebras, for example the more general notions requiring only (4), (5) and (6),[7, 8, 24, 44–46, 53, 56].In order to avoid, when necessary, the ambiguity of the general category endo-morphisms notation End ( L ) for endomorphisms of L as linear space, algebra orother structure, throughout this paper, we will use the notation gl ( L ) for the set ofall linear transformations on a linear space L , and viewing it context-dependent, as alinear space, as a associative algebra with usual associative composition product, asa Lie algebra of all linear transformations on L with the usual commutator productof the associative composition product (usual notation for Lie algebras), or as otherstructure type on the set gl ( L ) . Definition 3.
Let ( A , · , α ) be a Hom-associative algebra and let ( V , β ) be a Hom-module. Let l , r : A → gl ( V ) be two linear maps. The quadruple ( l , r , β , V ) is calleda bimodule of A if for all x , y ∈ A , v ∈ V : l ( x · y ) β ( v ) = l ( α ( x )) l ( y ) v , (4) r ( x · y ) β ( v ) = r ( α ( y )) r ( x ) v , (5) l ( α ( x )) r ( y ) v = r ( α ( y )) l ( x ) v , (6) β ( l ( x ) v ) = l ( α ( x )) β ( v ) , (7) β ( r ( x ) v ) = r ( α ( x )) β ( v ) . (8) Proposition 1.
Let ( A , · , α ) be a Hom-associative algebra and let ( V , β ) be a Hom-module. Let l , r : A → gl ( V ) be two linear maps. The quadruple ( l , r , β , V ) satisfiesa Hom-bimodule properties (4) , (5) , (6) of a Hom-associative algebra ( A , · , α ) ifand only if the direct sum of vector spaces, A ⊕ V , is turned into a Hom-associativealgebra by defining multiplication in A ⊕ V by ( x + v ) ∗ ( x + v ) = x · x + ( l ( x ) v + r ( x ) v ) , ( α ⊕ β )( x + v ) = α ( x ) + β ( v ) (9) for all x , x ∈ A , v , v ∈ V .Proof.
Let v , v , v ∈ V and x , x , x ∈ A . The left-hand side and right-hand sideof Hom-associativity of ( A ⊕ V , ∗ , α ⊕ β ) are expended as follows: (( x + v ) ∗ ( x + v )) ∗ ( α ⊕ β )( x + v )= (( x + v ) ∗ ( x + v )) ∗ ( α ( x ) + β ( v ))= ( x · x + ( l ( x ) v + r ( x ) v )) ∗ ( α ( x ) + β ( v ))= ( x · x ) · α ( x ) + ( l ( x · x ) β ( v ) + r ( α ( x ))( l ( x ) v + r ( x ) v ))) Mahouton Norbert Hounkonnou, Gbêvèwou Damien Houndedji, Sergei Silvestrov = ( x · x ) · α ( x ) + ( l ( x · x ) β ( v ) + r ( α ( x )) l ( x ) v + r ( α ( x )) r ( x ) v )( α ⊕ β )( x + v ) ∗ (( x + v ) ∗ ( x + v ))= ( α ( x ) + β ( v )) ∗ (( x + v ) ∗ ( x + v ))= ( α ( x ) + β ( v )) ∗ ( x · x + ( l ( x ) v + r ( x ) v ))= α ( x ) · ( x · x ) + l ( α ( x ))( l ( x ) v + r ( x ) v )) + r ( x · x ) β ( v )= α ( x ) · ( x · x ) + ( l ( α ( x )) l ( x ) v + l ( α ( x )) r ( x ) v + r ( x · x ) β ( v )) These elements of A ⊕ V are equal if and only if α ( x ) · ( x · x ) = ( x · x ) · α ( x ) l ( x · x ) β ( v ) + r ( α ( x )) l ( x ) v + r ( α ( x )) r ( x ) v = l ( α ( x )) l ( x ) v + l ( α ( x )) r ( x ) v + r ( x · x ) β ( v ) for all x , x , x ∈ A , v , v , v ∈ V . This holds if and only if the hom-associativityholds, and for each j = , , V terms involving v j ∈ V are equal. Ifthe terms are equal then the sums are equal. If the summs are equal then the termsshould be equal if one specifies all or two of v , v , v to zero element of V and usingthat linear transformations map zero to zero. Since,Hom-associativity ⇔ α ( x ) · ( x · x ) = ( x · x ) · α ( x ) , ∀ x , x , x ∈ A (4) ⇔ l ( x · x ) β ( v ) = l ( α ( x )) l ( x ) v , ∀ x , x , x ∈ A (6) ⇔ l ( α ( x )) r ( x ) v = r ( α ( x )) l ( x ) v , ∀ x , x , x ∈ A (5) ⇔ r ( α ( x )) r ( x ) v = r ( x · x ) β ( v ) , ∀ x , x , x ∈ A . the proof is complete. ⊓⊔ We denote such a Hom-associative algebra ( A ⊕ V , ∗ , α + β ) , or A × l , r , α , β V . Example 2.
Let ( A , · , α ) be a multiplicative Hom-associative algebra. Let L · x and R · x denote the left and right multiplication operators, respectively, i. e. L · x ( y ) = x · y , R · x ( y ) = y · x for any x , y ∈ A . Let L · : A → gl ( A ) with x L · x and R · : A → gl ( A ) with x R · x (for every x ∈ A ) be two linear maps. Then, the triples ( L · , , α ) , ( , R · , α ) and ( L · , R · , α ) are bimodules of ( A , · , α ) . Proposition 2.
Let ( l , r , β , V ) be bimodule of a multiplicative Hom-associative al-gebra ( A , · , α ) . Then, ( l ◦ α n , r ◦ α n , β , V ) is a bimodule of A for any integer n . Proof.
We have l ◦ α n ( x · y ) β ( v ) = l ( α n ( x ) · α n ( y )) β ( v ) = l ( α ( α n ( x ))) l ( α n ( y )) v = l ( α n + ( x )) l ( α n ( y )) v = l ◦ α n ( α ( x )) l ◦ α n ( y ) v . Similarly, the other relations are established. ⊓⊔ Example 3.
Let ( A , · , α ) be a multiplicative Hom-associative algebra. Then, thequadruple ( L · ◦ α n , R · ◦ α n , α , A ) is a bimodule of A for any finite integer n . ouble constructions of biHom-Frobenius algebras 7 Example 4.
Let ( A , · , α ) be a multiplicative associative algebra, and β : A → A be a morphism. Then, A β = ( A , · β = β ◦ · , α β = β ◦ α ) is a multiplicative Hom-associative algebra. Hence ( L · β ◦ α n β , R · β ◦ α n β , α β , A ) is a bimodule of A for anyfinite integer n . Theorem 1.
Let ( A , · , α ) and ( B , ◦ , β ) be two Hom-associative algebras. Supposethere are linear maps l A , r A : A → gl ( B ) and l B , r B : B → gl ( A ) such that thequadruple ( l A , r A , β , B ) is a bimodule of A , and ( l B , r B , α , A ) is a bimodule of B , satisfying, for any x , y ∈ A , a , b ∈ B , the following conditions:l A ( α ( x ))( a ◦ b ) = l A ( r B ( a ) x ) β ( b ) + ( l A ( x ) a ) ◦ β ( b ) , (10) r A ( α ( x ))( a ◦ b ) = r A ( l B ( b ) x ) β ( a ) + β ( a ) ◦ ( r A ( x ) b ) , (11) l B ( β ( a ))( x · y ) = l B ( r A ( x ) a ) α ( y ) + ( l B ( a ) x ) · α ( y ) , (12) r B ( β ( a ))( x · y ) = r B ( l A ( y ) a ) α ( x ) + α ( x ) · ( r B ( a ) y ) , (13) l A ( l B ( a ) x ) β ( b ) + ( r A ( x ) a ) ◦ β ( b ) − r A ( r B ( b ) x ) β ( a ) − β ( a ) ◦ ( l A ( x ) b ) = , (14) l B ( l A ( x ) a ) α ( y ) + ( r B ( a ) x ) · α ( y ) − r B ( r A ( y ) a ) α ( x ) − α ( x ) · ( l B ( a ) y ) = . (15) Then, there is a Hom-associative algebra structure on the direct sum A ⊕ B of theunderlying vector spaces of A and B given by ( x + a ) ∗ ( y + b ) = ( x · y + l B ( a ) y + r B ( b ) x ) + ( a ◦ b + l A ( x ) b + r A ( y ) a ) , ( α ⊕ β )( x + a ) = α ( x ) + β ( a ) (16) for all x , y ∈ A , a , b ∈ B .Proof. Let v , v , v ∈ V and x , x , x ∈ A . Set [( x + v ) ∗ ( x + v )] ∗ ( α ( x ) + β ( v )) = ( α ( x ) + β ( v )) ∗ [( x + v ) ∗ ( x + v )] , which is developed to obtain (10)-(15). Then, using the relations β ( l A ( x ) a ) = l A ( α ( x )) β ( a ) , β ( r A ( x ) a ) = r A ( α ( x )) β ( a ) , Mahouton Norbert Hounkonnou, Gbêvèwou Damien Houndedji, Sergei Silvestrov α ( l B ( a ) x ) = l B ( β ( a )) α ( x ) , α ( r B ( a ) x ) = r B ( β ( a )) α ( x ) , we show that ∗ is a Hom-associative algebra. ⊓⊔ We denote this Hom-associative algebra by ( A ⊲⊳ B , ∗ , α + β ) or A ⊲⊳ l A , r A , β l B , r B , α B . Definition 4.
Let ( A , · , α ) and ( B , ◦ , β ) be two Hom-associative algebras. Supposethat there are linear maps l A , r A : A → gl ( B ) and l B , r B : B → gl ( A ) such that ( l A , r A , β ) is a bimodule of A and ( l B , r B , α ) is a bimodule of B . If the conditions(10) - (15) are satisfied, then, ( A , B , l A , r A , β , l B , r B , α ) is called a matched pairof Hom-associative algebras . In this section, we consider the multiplicative Hom-associative algebra ( A , · , α ) such that α is involutive, i.e, α = Id A . Definition 5.
Let V , V be two vector spaces. For a linear map φ : V → V , wedenote the dual (linear) map by φ ∗ : V ∗ → V ∗ given by h v , φ ∗ ( u ∗ ) i = h φ ( v ) , u ∗ i for all v ∈ V , u ∗ ∈ V ∗ . Lemma 1.
Let ( l , r , β , V ) be a bimodule of a multiplicative Hom-associative algebra ( A , · , α ) , and let l ∗ , r ∗ : A → gl ( V ∗ ) be the linear maps given for all x ∈ A , u ∗ ∈ V ∗ , v ∈ V , by h l ∗ ( x ) u ∗ , v i : = h l ( x ) v , u ∗ i , h r ∗ ( x ) u ∗ , v i : = h r ( x ) v , u ∗ i . (17) Then (i) ( r ∗ , l ∗ , β ∗ , V ∗ ) is a bimodule of ( A , · , α ) ; (ii) ( r ∗ , , β ∗ , V ∗ ) and ( , l ∗ , β ∗ , V ∗ ) are also bimodules of A .Proof. (i) Let ( l , r , β , V ) be a bimodule of a multiplicative Hom-associative alge-bra ( A , · , α ) . We show that ( r ∗ , l ∗ , β ∗ , V ∗ ) is a bimodule of A . For x , y ∈ A , u ∗ ∈ V ∗ , v ∈ V ,(i-1) the computation h r ∗ ( x · y ) β ∗ ( u ∗ ) , v i = h β ( r ( x · y ) v ) , u ∗ i = h r ( α ( x · y )) β ( v ) , u ∗ i ouble constructions of biHom-Frobenius algebras 9 = h r ( α ( x ) · α ( y )) β ( v ) , u ∗ i = h r ( α ( y )) r ( α ( x )) v , u ∗ i = h ( r ( y ) r ( α ( x ))) ∗ u ∗ , v i = h r ∗ ( α ( x )) r ∗ ( y ) u ∗ , v i leads to r ∗ ( x · y ) β ∗ ( u ∗ ) = r ∗ ( α ( x )) r ∗ ( y ) u ∗ ;(i-2) the computation h l ∗ ( x · y ) β ∗ ( u ∗ ) , v i = h β ( l ( x · y )( v )) , u ∗ i = h l ( α ( x · y )) β ( v ) , u ∗ i = h l ( α ( x ) · α ( y )) β ( v ) , u ∗ i = h l ( α ( x )) l ( α ( y )) β ( v ) , u ∗ i = h ( l ( x ) l ( α ( y ))) ∗ u ∗ , v i = h l ∗ ( α ( y )) l ∗ ( x ) u ∗ , v i gives l ∗ ( x · y ) β ∗ ( u ∗ ) = l ∗ ( α ( y )) l ∗ ( x ) u ∗ ;(i-3) the computation h r ∗ ( α ( x )) l ∗ ( y ) u ∗ , v i = h l ( y ) r ( α ( x )) v , u ∗ i = h l ( α ( y )) r ( α ( x )) v , u ∗ i = h ( l ◦ α )( α ( y ))( r ◦ α )( x )) v , u ∗ i = h r ( α ( x )) l ( α ( y )) v , u ∗ i = h r ( x ) l ( α ( y )) v , u ∗ i = h l ∗ ( α ( y )) r ∗ ( x ) u ∗ , v i yields r ∗ ( α ( x )) l ∗ ( y ) u ∗ = l ∗ ( α ( y )) r ∗ ( x ) u ∗ . h β ∗ ( r ∗ ( x )) u ∗ , v i = h r ( x )( β ( v )) , u ∗ i = h r ( α ( x ))( β ( v )) , u ∗ i = h ( r ◦ α )( α ( x ))( β ( v )) , u ∗ i = h β ( r ( α ( x ))) v , u ∗ i = h r ∗ ( α ( x )) β ∗ ( u ∗ ) , v i . Then β ∗ ( r ∗ ( x )) u ∗ = r ∗ ( α ( x )) β ∗ ( u ∗ ) . Similarly, one can show that β ∗ ( l ∗ ( x )) u ∗ = l ∗ ( α ( x )) β ∗ ( u ∗ ) . Hence, ( r ∗ , l ∗ , β ∗ , V ∗ ) is a bimodule of A . (ii) Analogously, ( r ∗ , , β ∗ , V ∗ ) and ( , l ∗ , β ∗ , V ∗ ) are shown to be bimodules of A . ⊓⊔ Definition 6.
Let ( A , · , α ) be a Hom-associative algebra, and B : A × A → K be abilinear form on A . Then,(i) B is said to be nondegenerate if A ⊥ = { x ∈ A / B ( y , x ) = , ∀ y ∈ A } =
0; (18)(ii) B is said to be symmetric if B ( x , y ) = B ( y , x ) ; (19)(iii) B is said to be α -invariant if B ( α ( x ) · α ( y ) , α ( z )) = B ( α ( x ) , α ( y ) · α ( z )) . (20) Definition 7.
A Hom-Frobenius algebra is a Hom-associative algebra with a non-degenerate invariant bilinear form.
Definition 8.
We call ( A , α , B ) a double construction of an involutive Hom-Frobenius algebra associated to ( A , α ) and ( A ∗ , α ∗ ) if it satisfies the followingconditions:(i) A = A ⊕ A ∗ as the direct sum of vector spaces;(ii) ( A , α ) and ( A ∗ , α ∗ ) are Hom-associative subalgebras of ( A , α ) with α = α ⊕ α ∗ ;(iii) B is the natural non-degerenate ( α ⊕ α ∗ )-invariant symmetric bilinear formon A ⊕ A ∗ given, for all x , y ∈ A , a ∗ , b ∗ ∈ A ∗ , by B ( x + a ∗ , y + b ∗ ) = h x , b ∗ i + h a ∗ , y i , B (( α + α ∗ )( x + a ∗ ) , y + b ∗ ) = B ( x + a ∗ , ( α + α ∗ )( y + b ∗ )) , (21)where h , i is the natural pair between the vector space A and dual space A ∗ .Let ( A , · , α ) be an involutive Hom-associative algebra. Suppose that there is aninvolutive Hom-associative algebra structure ” ◦ ” on its dual space A ∗ . We con-struct an involutive Hom-associative algebra structure on the direct sum A ⊕ A ∗ of the underlying vector spaces of A and A ∗ such that ( A , · , α ) and ( A ∗ , ◦ , α ∗ ) are Hom-subalgebras, equipped with the non-degenerate ( α ⊕ α ∗ )-invariant sym-metric bilinear form on A ⊕ A ∗ given by the equation (21). In other words, ( A ⊕ A ∗ , α ⊕ α ∗ , B ) is an involutive symmetric Hom-associative algebra. Such aconstruction is called a double construction of an involutive Hom-Frobenius algebraassociated to ( A , · , α ) and ( A ∗ , ◦ , α ∗ ) . Theorem 2.
Let ( A , · , α ) be an involutive Hom-associative algebra. Suppose thatthere is an involutive Hom-associative algebra structure ” ◦ ” on its dual space A ∗ .Then, there is a double construction of an involutive Hom-Frobenius algebra asso-ciated to ( A , · , α ) and ( A ∗ , ◦ , α ∗ ) if and only if ( A , A ∗ , R ∗· , L ∗· , α ∗ , R ∗◦ , L ∗◦ , α ) is amatched pair of involutive Hom-associative algebras.Proof. Let us consider the following four maps: L ∗· : A → gl ( A ∗ ) , h L ∗· ( x ) u ∗ , v i = h L · ( x ) v , u ∗ i = h x · v , u ∗ i , R ∗· : A → gl ( A ∗ ) , h R ∗· ( x ) u ∗ , v i = h R · ( x ) v , u ∗ i = h v · x , u ∗ i , R ∗◦ : A ∗ → gl ( A ) , h R ∗◦ ( x ∗ ) u , v ∗ i = h R ◦ ( x ∗ ) v ∗ , u i = h v ∗ ◦ x ∗ , u i , L ∗◦ : A ∗ → gl ( A ) , h L ∗◦ ( x ∗ ) u , v ∗ i = h L ◦ ( x ∗ ) v ∗ , u i = h x ∗ ◦ v ∗ , u i , for all x , v , u ∈ A , x ∗ , v ∗ , u ∗ ∈ A ∗ . If ( A , A ∗ , R ∗· , L ∗· , α ∗ , R ∗◦ , L ∗◦ , α ) is a matchedpair of multiplicative Hom-associative algebras, then ( A ⊲⊳ A ∗ , ∗ , α + α ∗ ) is amultiplicative Hom-associative algebra with the product ∗ given by the equa-tion (16), and the bilinear form B ( · , · ) defined by the equation (21) is ( α ⊕ α ∗ ) -invariant, that is B [( α ( x ) + α ∗ ( a ∗ )) ∗ ( α ( y ) + α ∗ ( b ∗ )) , ( α ( z ) + α ∗ ( c ∗ ))] = B [ α ( x ) + α ∗ ( a ∗ ) , ( α ( y ) + α ∗ ( b ∗ )) ∗ ( α ( z ) + α ∗ ( c ∗ ))] for all x , y ∈ A ∗ , a ∗ , b ∗ ∈ A ∗ , and ( x + a ∗ ) ∗ ( y + b ∗ ) = ( x · y + l B ( a ) y + r B ( b ) x ) + ( a ◦ b + l A ( x ) b + r A ( y ) a ) , with l A = R ∗· , r A = L ∗· , l B = R ∗◦ , r B = L ∗◦ . Indeed, we have ouble constructions of biHom-Frobenius algebras 11 B [( α ( x ) + α ∗ ( a ∗ )) ∗ ( α ( y ) + α ∗ ( b ∗ )) , ( α ( z ) + α ∗ ( c ∗ ))]= B [( α ( x ) · α ( y ) + l A ∗ ( α ∗ ( a ∗ )) α ( y ) + r A ∗ ( α ∗ ( b ∗ )) α ( x )) + ( α ∗ ( a ∗ ) ◦ α ∗ ( b ∗ )+ l A ( α ( x )) α ∗ ( b ∗ ) + r A ( α ( y )) α ∗ ( a ∗ )) , α ( z ) + α ∗ ( c ∗ )]= h α ( x ) · α ( y ) , α ∗ ( c ∗ ) i + h α ∗ ( c ∗ ) ◦ α ∗ ( a ∗ ) , α ( y ) i + h α ∗ ( b ∗ ) ◦ α ∗ ( c ∗ ) , α ( x ) i + h α ∗ ( a ∗ ) ◦ α ∗ ( b ∗ ) , α ( z ) i + h α ( z ) · α ( x ) , α ∗ ( b ∗ ) i + h α ( y ) · α ( z ) , α ∗ ( a ∗ ) i and B [ α ( x ) + α ∗ ( a ∗ ) , ( α ( y ) + α ∗ ( b ∗ )) ∗ ( α ( z ) + α ∗ ( c ∗ ))]= B [ α ( x ) + α ∗ ( a ∗ ) , ( α ( y ) · α ( z ) + l A ∗ ( α ∗ ( b ∗ )) α ( z ) + r A ∗ ( α ∗ ( c ∗ )) α ( y ))+( α ∗ ( b ∗ ) ◦ α ∗ ( c ∗ ) + l A ( α ( y )) α ∗ ( c ∗ ) + r A ( α ( z )) α ∗ ( b ∗ ))]= h α ( x ) , α ∗ ( b ∗ ) ◦ α ∗ ( c ∗ ) i + h α ∗ ( c ∗ ) , α ( x ) · α ( y ) i + h α ∗ ( b ∗ ) , α ( z ) · α ( x ) i + h α ( y ) · α ( z ) , α ∗ ( a ∗ ) i + h α ∗ ( a ∗ ) ◦ α ∗ ( b ∗ ) , α ( z ) i + h α ( c ∗ ) ◦ α ∗ ( a ∗ ) , α ( y ) i . Thus, B is well ( α ⊕ α ∗ ) -invariant. Conversely, set x ∗ a ∗ = l A ( x ) a ∗ + r A ∗ ( a ∗ ) x , a ∗ ∗ x = l A ∗ ( a ∗ ) x + r A ( x ) a ⋆ , for x ∈ A , a ∗ ∈ A ∗ . Then, ( A , A ∗ , R ∗· , L ∗· , α ∗ , R ∗◦ , L ∗◦ , α ) is a matched pair of mul-tiplicative Hom-associative algebras, since the double construction of the involutiveHom-Frobenius algebra associated to ( A , · , α ) and ( A ∗ , ◦ , α ∗ ) produces the equa-tions (10) - (15). ⊓⊔ Theorem 3.
Let ( A , · , α ) be an involutive Hom-associative algebra. Suppose thatthere is an involutive Hom-associative algebra structure ” ◦ ” on its dual space ( A ∗ , α ∗ ) . Then, ( A , A ∗ , R ∗· , L ∗· , α ∗ , R ∗◦ , L ∗◦ , α ) is a matched pair of involutive Hom-associative algebras if and only if, for any x ∈ A and a ∗ , b ∗ ∈ A ∗ , R ∗· ( α ( x ))( a ∗ ◦ b ∗ ) = R ∗· ( L ∗◦ ( a ∗ ) x ) α ∗ ( b ∗ ) + ( R ∗· ( x ) a ∗ ) ◦ α ∗ ( b ∗ ) , (22) R ∗· ( R ∗◦ ( a ∗ ) x ) α ∗ ( b ∗ ) + L ∗· ( x ) a ∗ ◦ α ∗ ( b ∗ ) = L ∗· ( L ∗◦ ( b ∗ ) x ) α ∗ ( a ∗ ) + α ∗ ( a ∗ ) ◦ ( R ∗· ( x ) b ∗ ) . (23) Proof.
Obviously, (22) gives (10), and (23) reduces to (14) when l A = R ∗· , r A = L ∗· , l B = l A ∗ = R ∗◦ , r B = r A ∗ = L ∗◦ . Now, show that(10) ⇐⇒ (11) ⇐⇒ (12) ⇐⇒ (13)and (14) ⇐⇒ (15) . Suppose (22) and (23) are satisfied and show that one has L ∗· ( α ( x ))( a ∗ ◦ b ∗ ) = L ∗· ( R ∗◦ ( b ∗ ) x ) α ∗ ( a ∗ ) + α ∗ ( a ∗ ) ◦ ( L ∗· ( x ) b ∗ ) R ∗◦ ( α ∗ ( a ∗ ))( x · y ) = R ∗◦ ( L ∗· ( x ) a ∗ ) α ( y ) + ( R ∗◦ ( a ) x ) · α ( y ) L ∗◦ ( α ∗ ( a ∗ ))( x · y ) = L ∗◦ ( R ∗· ( y ) a ∗ ) α ( x ) + α ( x ) · ( L ∗◦ ( a ∗ ) y ) R ∗◦ ( R ∗· ( x ) a ∗ ) α ( y ) + ( L ∗◦ ( a ∗ ) x ) · α ( y ) − L ∗◦ ( L · ( y ) a ∗ ) α ( x ) − α ( x ) · ( R ∗◦ ( a ) y ) = . We have h R ∗· ( x ) a ∗ , y i = h L ∗· ( y ) a ∗ , x i = h y · x , a ∗ i , h R ∗◦ ( b ∗ ) x , a ∗ i = h L ∗◦ ( a ∗ ) x , b ∗ i = h a ∗ ◦ b ∗ , x i , α ∗ ( R ∗· ( x ) a ∗ ) = R ∗· ( α ( x )) α ∗ ( a ∗ ) , α ∗ ( L ∗· ( x ) a ∗ ) = L ∗· ( α ( x )) α ∗ ( a ∗ ) , α ( R ∗◦ ( a ∗ ) x ) = R ∗◦ ( α ∗ ( a ∗ )) α ( x ) , α ( L ∗◦ ( a ∗ ) x ) = L ∗◦ ( α ∗ ( a ∗ )) α ( x ) , for all x , y ∈ A , a ∗ , b ∗ ∈ A ∗ . Set α ( x ) = z , α ( y ) = t , α ∗ ( a ∗ ) = c ∗ and α ∗ ( b ∗ ) = d ∗ . Then(i) the statement (10) ⇐⇒ (11) follows from h R ∗· ( α ( x ))( a ∗ ◦ b ∗ ) , y i = h y · α ( x ) , a ∗ ◦ b ∗ i = h ( L · ( y ) ◦ α ) x , a ∗ ◦ b ∗ i = h x , α ∗ ( L ∗· ( y )( a ∗ ◦ b ∗ )) i = h L ∗· ( α ( y )) α ∗ ( a ∗ ◦ b ∗ ) , x i = h L ∗· ( α ( y ))( α ∗ ( a ∗ ) ◦ α ∗ ( b ∗ )) , x i = h L ∗· ( α ( y ))( c ∗ ◦ d ∗ ) , x i ; h R ∗· ( L ∗◦ ( a ∗ ) x ) α ( b ∗ ) , y i = h y · L ∗◦ ( a ∗ ) x , α ∗ ( b ∗ ) i = h L ∗· ( y )( α ∗ ( b ∗ )) , L ∗◦ ( a ∗ ) x i = h L ∗◦ ( a ∗ ) x , L ∗· ( y )( α ∗ ( b ∗ )) i = h a ∗ ◦ ( L ∗· ( y )( α ∗ ( b ∗ ))) , x i = h α ∗ ( c ∗ ) ◦ ( L ∗· ( y )( d ∗ )) , x i ; h ( R ∗· ( x ) a ∗ ) ◦ α ∗ ( b ∗ ) , y i = h R ∗◦ ( α ∗ ( b ∗ )) y , R ∗· ( x ) a ∗ i = h a ∗ , ( R ∗◦ ( α ∗ ( b ∗ )) y ) · x i = h L ∗· [ R ∗◦ ( α ∗ ( b ∗ )) y ] a ∗ , x i = h L ∗· ( R ∗◦ ( d ∗ ) y ) α ∗ ( c ∗ ) , x i ;(ii) the statement (11) ⇐⇒ (12) follows from h L ∗ ( α ( x ))( a ∗ ◦ b ∗ ) , y i = h a ∗ ◦ b ∗ , α ( x ) · y i = h R ∗◦ ( b ∗ )( α ( x ) · y ) , a ∗ i = h R ∗◦ ( α ∗ ( d ∗ ))( z · y ) , a ∗ i ; h α ∗ ( a ∗ ) ◦ ( L ∗· ( x ) b ∗ ) , y i = h α ∗ ( a ∗ ) , R ∗◦ ( L ∗· ( x ) b ∗ ) y i = h a ∗ , α [ R ∗◦ ( L ∗· ( x ) b ∗ ) y ] i = h a ∗ , R ∗◦ [ α ∗ ( L ∗· ( x ) b ∗ )] α ( y ) i = h a ∗ , R ∗◦ [ L ∗· ( α ( x )) α ∗ ( b ∗ )] α ( y ) i = h a ∗ , R ∗◦ ( L ∗· ( z ) d ∗ ) α ( y ) i ; h L ∗· ( R ∗◦ ( b ∗ ) x ) α ∗ ( a ∗ ) , y i = h ( R ∗◦ ( b ∗ ) x ) ◦ y , α ∗ ( a ∗ ) i = h α [( R ∗◦ ( b ∗ ) x ) ◦ y ] , a ∗ i = h ( R ∗◦ ( α ∗ ( b ∗ )) α ( x )) ◦ α ( y ) , a ∗ i = h R ∗◦ ( d ∗ ) z · α ( y ) , a ∗ i ;(iii) the statement (10) ⇐⇒ (13) follows from h R ∗ ( α ( x ))( a ∗ ◦ b ∗ ) , y i = h a ∗ ◦ b ∗ , y · α ( x ) i = h L · ( a ∗ ) b ∗ , y · z i = h L ∗◦ ( a ∗ )( y · z ) i = h L ∗◦ ( α ∗ ( c ∗ ))( y · z ) i ; h ( R ∗· ( x ) a ∗ ) ◦ α ∗ ( b ∗ ) , y i = h α ∗ ( b ∗ ) , L ∗· ( R ∗· ( x ) a ∗ ) y i = h b ∗ , α ∗ [ L ∗· ( R ∗· ( x ) a ∗ ) y ] i = h b ∗ , L ∗· ( R ∗· ( α ( x )) α ∗ ( a ∗ )) α ( y ) i = h b ∗ , L ∗· ( R ∗· ( z ) c ∗ ) α ( y ) i ; h R ∗· ( L ∗◦ ( a ∗ ) x ) α ∗ ( b ∗ ) , y i = h y · L ∗◦ ( a ∗ ) x , α ∗ ( b ∗ ) i = h α ( y ) · α ( L ∗◦ ( a ∗ ) x ) , b ∗ i = h α ( y ) · L ∗◦ ( α ∗ ( a ∗ )) α ( x ) , b ∗ i = h α ( y ) · L ∗◦ ( c ∗ ) z , b ∗ i ;(iv) the statement (14) ⇐⇒ (15) follows from h L ∗· ( L ∗◦ ( b ∗ ) x ) α ∗ ( a ∗ ) , y i = h ( L ∗◦ ( b ∗ ) x ) · y , α ∗ ( a ∗ ) i = h a ∗ , α ( L ∗◦ ( b ∗ ) x ) · α ( y ) i = h a ∗ , L ∗◦ ( α ∗ ( b ∗ )) α ( x ) · α ( y ) i = h a ∗ , L ∗◦ ( d ∗ ) z · α ( y ) i ; h α ∗ ( a ∗ ) ◦ ( R ∗· ( x ) b ∗ ) , y i = h R ∗◦ ( R ∗◦ ( x ) b ∗ ) y , α ∗ ( a ∗ ) i = h α ∗ ( a ∗ ) ◦ ( R ∗· ( x ) b ∗ ) , y i = h α [ R ∗◦ ( R ∗◦ ( x ) b ∗ ) y ] , a ∗ i = h R ∗◦ [ R ∗◦ ( α ( x )) α ∗ ( b ∗ )] α ( y ) , a ∗ i ouble constructions of biHom-Frobenius algebras 13 = h R ∗◦ ( R ∗· ( z ) d ∗ ) α ( y ) , a ∗ i ; h ( L ∗· ( x ) a ∗ ) ◦ α ∗ ( b ∗ ) , y i = h R ∗◦ ( α ∗ ( b ∗ )) y , L ∗· ( x ) a ∗ i = h x · ( R ∗◦ ( d ∗ ) y ) , a ∗ i = h α ( z ) · ( R ∗◦ ( d ∗ ) y ) , a ∗ i ; h R ∗· ( R ∗◦ ( a ∗ ) x ) α ∗ ( b ∗ ) , y i = h y · R ∗◦ ( a ∗ ) x , α ∗ ( b ∗ ) i = h α ∗ ( b ∗ ) , L · ( y )( R ∗◦ ( a ∗ ) x ) i = h ( L ∗· ( y )( d ∗ ) , R ∗◦ ( a ∗ ) x i = h L ∗· ( y ) d ∗ ◦ a ∗ , x i = h L ∗◦ ( L ∗· ( y ) d ∗ ) x , a ∗ i = h L ∗◦ ( L ∗· ( y ) d ∗ ) α ( z ) , a ∗ i which completes the proof. ⊓⊔ Let A be a multiplicative Hom-associative algebra. Let σ : A ⊗ A → A ⊗ A bethe exchange operator defined as σ ( x ⊗ y ) = y ⊗ x , for all x , y ∈ A . Proposition 3.
Let ( A , · , α ) be a multiplicative Hom-associative algebra. Then, ( α ⊗ L · , R · ⊗ α , α ⊗ α , A ⊗ A ) given, for any x , a , b ∈ A : by ( α ⊗ L )( x )( a ⊗ b ) = ( α ⊗ L ( x ))( a ⊗ b ) = α ( a ) ⊗ x · b , ( R · ⊗ α )( x )( a ⊗ b ) = ( R ( x ) ⊗ α )( a ⊗ b ) = a · x ⊗ α ( b ) , is a bimodule of A . Proof.
Let x , y , v , v ∈ A .(i) By formulas for the maps and hom-associativity, ( α ⊗ L · )( x · y )( α ⊗ α )( v ⊗ v ) = [ α ⊗ L · ( x · y )]( α ( v ) ⊗ α ( v ))= v ⊗ ( x · y ) · α ( v ) ; ( α ⊗ L · ( α ( x )))( α ⊗ L · ( y ))( v ⊗ v ) = ( α ⊗ L · ( α ( x )))( α ( v )) ⊗ ( y · v )= v ⊗ α ( x ) · ( y · v ) give ( α ⊗ L · )( x · y )( α ⊗ α )( v ⊗ v ) = ( α ⊗ L ( α ( x )))( α ⊗ L ( y ))( v ⊗ v ) . (ii) By formulas for the maps and hom-associativity, ( R · ⊗ α )( x · y )( α ⊗ α )( v ⊗ v ) = ( R · ( x · y ) ⊗ α )( α ( v ) ⊗ α ( v ))= α ( v ) · ( x · y ) ⊗ v ( R · ( α ( y )) ⊗ α )( R · ( x ) ⊗ α )( v ⊗ v ) = ( R · ( α ( y ))(( v · x ) ⊗ α ( v ))= ( v · x ) · α ( y ) ⊗ v yield ( R · ⊗ α )( x · y )( α ⊗ α )( v ⊗ v ) = ( R · ( α ( y )) ⊗ α )( R · ( x ) ⊗ α )( v ⊗ v ) . (iii) By formulas for the maps, ( α ⊗ L · ( α ( x )))( R · ( y ) ⊗ α )( v ⊗ v ) = ( α ⊗ L · ( α ( x )))( v · y ⊗ α ( v ))= α ( v ) · α ( y ) ⊗ α ( x ) · α ( v ) ; ( R · ( α ( y )) ⊗ α )( α ⊗ L · ( x ))( v ⊗ v ) = ( R · ( α ( y )) ⊗ α )( α ( v ) ⊗ x · v )= α ( v ) · α ( y ) ⊗ α ( x ) · α ( v ) give ( α ⊗ L · ( α ( x )))( R · ( y ) ⊗ α )( v ⊗ v ) = ( R · ( α ( y )) ⊗ α )( α ⊗ L · ( x ))( v ⊗ v ) . (iv) By formulas for the maps, ( α ⊗ α )( α ⊗ L · )( v ⊗ v ) = ( α ⊗ α )( α ( v ) ⊗ x · v ) = v ⊗ α ( x ) · α ( v ) ; ( α ⊗ L · ( α ( x )))( v ⊗ v ) = ( α ⊗ L · α ( x ))( α ( v ) ⊗ α ( v )) = v ⊗ α ( x ) · α ( v ) imply ( α ⊗ α )( α ⊗ L · )( v ⊗ v ) = α ⊗ L · ( α ( x )))( v ⊗ v ) . (v) By formulas for the maps, ( α ⊗ α )( R · α ( x ))( v ⊗ v ) = ( α ⊗ α )( v · x ⊗ α ( v ))= α ( v ) · α ( x ) ⊗ v ; ( R · ( α ( x )) ⊗ α )( α ⊗ α )( v ⊗ v ) = ( R · ( α ( x )) ⊗ α )( α ( v ) ⊗ α ( v ))= α ( v ) · α ( x ) ⊗ v yield ( α ⊗ α )( R · α ( x ))( v ⊗ v ) = ( R · ( α ( x )) ⊗ α )( α ⊗ α )( v ⊗ v ) . Hence, the proof is achieved. ⊓⊔ Remark 2.
The quadruple ( L · ⊗ α , α ⊗ R · , α ⊗ α , A ⊗ A ) is also a bimodule of A . Theorem 4.
Let ( A , · , α ) be an involutive Hom-associative algebra. Suppose thereis an involutive Hom-associative algebra structure ” ◦ ” on its dual space A ∗ givenby a linear map ∆ ∗ : A ∗ ⊗ A ∗ → A ∗ . Then, ( A , A ∗ , R ∗· , L ∗· , α ∗ , R ∗◦ , L ∗◦ , α ) is amatched pair of involutive Hom-associative algebras if and only if ∆ : A → A ⊗ A satisfies the following conditions: ∆ ◦ α ( x · y ) = ( α ⊗ L · ( x )) △ ( y ) + ( R · ( y ) ⊗ α ) △ ( x ) , (24) ( L · ( y ) ⊗ α − α ⊗ R · ( y )) ∆ ( x ) + σ [( L · ( x ) ⊗ α − α ⊗ R · ( x )) ∆ ( y )] = for all x , y ∈ A .Proof. Let { e , ..., e n } be a basis of A , and { e ∗ , ..., e ∗ n } be its dual basis. Set e i · e j = ∑ nk = c ki j e k and e ∗ i ◦ e ∗ j = ∑ nk = f ki j e ∗ k . Therefore, we have ∆ ( e k ) = ∑ ni , j = f ki j e i ⊗ e j , and R ∗· ( e i ) e ∗ j = n ∑ k = c jki e ∗ k , L ∗· ( e i ) e ∗ j = n ∑ k = c jik e ∗ k , α ( e i ) = n ∑ q = b iq e q , R ∗◦ ( e ∗ i ) e j = n ∑ k = f jki e k , L ∗◦ ( e ∗ i ) e j = n ∑ k = f jik e k , α ∗ ( e ∗ i ) = n ∑ q = b ∗ iq e ∗ q . ouble constructions of biHom-Frobenius algebras 15 We have h α ∗ ( e ∗ i ) , e j i = b ∗ ji = h e ∗ i , α ( e j ) i = b ij which implies b ∗ ji = b ij . Also, fromthe identity α = Id and α ( e i ) = ∑ nk = b ik e k , we get ∑ nk = ∑ nl = b ik b kl e l = ∑ nl = δ il e l = e i , with b ik b kl = δ il . Hence, collecting the coefficient of e u ⊗ e v (for any i, j, k, m)yields ∆ ( α ( e m ) · α ( e i )) = ( α ⊗ L · ( e m )) ∆ ( e i ) + ( R · ( e i ) ⊗ α ) ∆ ( e m )= ( α ⊗ L ( e m ))( n ∑ u , v = f iuv e u ⊗ e v ) + ( R ( e i ) ⊗ α )( n ∑ u , v = f muv e u ⊗ e v )= n ∑ u , v = f iuv α ( e u ) ⊗ e m · e v + n ∑ u , v = f muv e u · e i ⊗ α ( e v )= n ∑ u , v = f iuv ( n ∑ j = b uj e j ) ⊗ ( n ∑ k = c kmv e k ) + n ∑ u , v = f muv ( n ∑ j = c jui e u ) ⊗ ( n ∑ k = b vk e k )= n ∑ j , k , u , v = ( f iuv b uj c kmv + f muv c jui b vk ) e j ⊗ e k ; ∆ ◦ α ( e m · e i ) = ∆ ◦ α ( n ∑ l = c lmi e l ) = n ∑ l = c lmi ∆ ( α ( e l ))= n ∑ l = c lmi ∆ ( n ∑ q = b lq e q ) = n ∑ l , q , j , k = c lmi b lq f qjk e j ⊗ e k , since ∆ ◦ α ( e m · e i ) = ( α ⊗ α ) ◦ ∆ ( e m · e i ) . Then, n ∑ l , u , v , j , k = c lmi f luv b uj b vk e j ⊗ e k = n ∑ l , q , j , k = c lmi b lq f qjk e j ⊗ e k . We obtain the relation n ∑ q = c lmi b lq f qjk = n ∑ u , v = ( f iuv b uj c kmv + f muv c jui b vk ) m n ∑ u , v = c lmi f luv b uj b vk = n ∑ u , v = ( f iuv b uj c kmv + f muv c jui b vk ) , and the identity given by the coefficient of e m in R ∗· ( α ( e i ))( e ∗ j ◦ e ∗ k ) = R ∗· ( L ∗◦ ( e ∗ j ) e i ) α ∗ ( e ∗ k ) + ( R ∗· ( e i ) e ∗ j ) ◦ α ∗ ( e ∗ k ) . = R ∗· ( n ∑ u = f iju e u ) α ∗ ( e ∗ v ) + ( n ∑ u = c jui e ∗ u ) ◦ α ∗ ( e ∗ k )= n ∑ u = f iju R ∗· ( e u )( n ∑ v = b vk e ∗ v ) + u ∑ u = c jui ( e ∗ u ◦ ( n ∑ v = b vk e ∗ v ))= n ∑ u , v = f iju b vk R ∗· ( e u ) e ∗ v + n ∑ u , v = c jui b vk ( e ∗ u ◦ e ∗ v ) = n ∑ u , v = f iju b vk ( n ∑ m = c vmu e ∗ m ) + n ∑ u , v , m = c jui b vk f muvp e ∗ m = n ∑ u , v , m = ( f iju b vk c vmu + c jui b vk f muv ) e ∗ m ; R ∗· ( α ( e i ))( e ∗ j ◦ e ∗ k ) = R ∗· ( α ( e i ))( n ∑ l = f ljk e ∗ l ) = n ∑ l = f ljk R ∗· ( n ∑ q = b iq e q )( e ∗ l ) = n ∑ l , q = f ljv b uq R ∗· ( e q ) e ∗ l = n ∑ l , q = f ljk b iq ( n ∑ m = c lmq e ∗ m ) = n ∑ l , q , m = f ljk b iq c lmq e ∗ m . Then, we arrive at n ∑ q = f ljk b iq c lmq = n ∑ u , v = ( f iju b vk c vmu + c jui b vk f muv ) m n ∑ u , v = f luv b uj b vk c lmq = n ∑ u , v = ( f iju b vk c vmu + c jui b vk f muv ) . Thus, taking c lmi = b iq c lmq , b lq f qjk = f ljk , f iuv b uj c kmv = f iju c vmu b vk , we obtain that (24)corresponds to (22). Similarly, we have ( L · ( e i ) ⊗ α − α ⊗ R · ( e i )) ∆ ( e m ) + σ [( L · ( e m ) ⊗ α − α ⊗ R · ( e m )) ∆ ( e i )] = ⇔ ( L · ( e i ) ⊗ α − α ⊗ R · ( e i ))( n ∑ k , l = f mlk e l ⊗ e k )+ σ [( L · ( e m ) ⊗ α − α ⊗ R · ( e m ))( n ∑ k , l = f ilk e l ⊗ e k )] = ⇔ n ∑ k , l = f mlk ( e i · e l ⊗ α ( e k ) − α ( e l ) ⊗ e k · e i )+ σ [ n ∑ k , l = f ikl ( e m · e l ⊗ α ( e k ) − α ( e l ) ⊗ e k · e m )] = ⇔ n ∑ k , l = f mlk (( n ∑ j = c jil e j ) ⊗ ( n ∑ p = b kp e p ) − ( n ∑ p = b lp e p ) ⊗ ( n ∑ j = c jki e j ))+ σ [ n ∑ j , l = f ikl (( n ∑ j = c jml e j ) ⊗ ( n ∑ p = b kp e p ) − ( n ∑ p = b lp e p ) ⊗ ( n ∑ j = c jkm e j ))] = ⇔ n ∑ k , l , j , p = ( f mlk c jil d kp e j ⊗ e p − f mlk c jki b lp e p ⊗ e j )+ σ [ n ∑ k , l , j , p = ( f ikl c jml b kp e j ⊗ e p − f ikl c jkm b lp e p ⊗ e j )] = R ∗· ( R ∗◦ ( e ∗ j ) e i ) α ∗ ( e ∗ k ) + ( L ∗· ( e i ) e ∗ j ) ◦ α ∗ ( e ∗ k )= L ∗· ( L ∗◦ ( e ∗ k ) e i ) α ∗ ( e ∗ j ) + α ∗ ( e ∗ j ) ◦ ( R ∗· ( e i ) e ∗ k ) ⇔ ouble constructions of biHom-Frobenius algebras 17 R ∗· ( n ∑ l = f il j e l )( n ∑ p = d kp e ∗ p ) + ( n ∑ l = c jil e ∗ l ) ◦ ( n ∑ p = d kp e ∗ p )= L ∗· ( n ∑ l = f ikl e l )( n ∑ q = d jq e ∗ q ) + ( n ∑ q = d jq e ∗ q ) ◦ ( n ∑ l = c kli e ∗ l ) ⇔ n ∑ l , p = f il j d kp R ∗· ( e l ) e ∗ p + n ∑ l , p = c jil d kp e ∗ l ◦ e ∗ p = n ∑ l , q = f ikl d jq L ∗· ( e l ) e ∗ q + n ∑ q , l = d jq c kli e ∗ q ◦ e ∗ l ⇔ n ∑ l , p = f il j d kp ( n ∑ m = c pml e ∗ m ) + n ∑ l , p , m = c jil d kp f mlp e ∗ m = n ∑ l , q , m = f ikl d jq c qlm e ∗ m + n ∑ q , l , m = d jq c kli f mql e ∗ m ⇔ n ∑ l , m , p = ( f il j d kp c pml + f mlp d kp c jil ) e ∗ m = n ∑ l , m , q = ( f ikl d jq c qlm + f mql d jq c kli ) e ∗ m . Thus, we conclude that (25) corresponds to (23). ⊓⊔ Definition 9.
Let ( A , · , α ) be an involutive Hom-associative algebra. An antisym-metric infinitesimal Hom-bialgebra structure on A is a linear map ∆ : A → A ⊗ A such that(i) ∆ ∗ : A ∗ ⊗ A ∗ → A ∗ defines an involutive Hom-associative algebra structureon A ∗ ;(ii) ∆ satisfies (24) and (25).We denote such an antisymmetric infinitesimal Hom-bialgebra by ( A , △ , α ) or ( A , A ∗ , α , α ∗ ) . Corollary 1.
Let ( A , · , α ) and ( A ∗ , ◦ , α ∗ ) be two involutive associative algebras.Then, the following conditions are equivalent: (i) There is a double construction of an involutive Hom-Frobenius algebra as-sociated to ( A , · , α ) and ( A ∗ , ◦ , α ∗ ) ; (ii) ( A , A ∗ , R ∗· , L ∗· , α ∗ R ∗◦ , L ∗◦ , α ) is a matched pair of involutive associative al-gebras; (iii) ( A , A ∗ , α , α ∗ ) is an antisymmetric infinitesimal Hom-bialgebra.Proof. From Theorems 2 and 4, we have the equivalences. ⊓⊔ Definition 10 ( [22]).
A biHom-associative algebra is a quadruple ( A , · , α , β ) con-sisting of a linear space A , K -bilinear map · : A ⊗ A → A , linear maps α , β : A → A satisfying, for all x , y , z ∈ A , the following conditions: α ◦ β = β ◦ α , (commutativity); α ( x · y ) = α ( x ) · α ( y ) , β ( x · y ) = β ( x ) · β ( y ) , (multiplicativity); α ( x ) · ( y · z ) = ( x · y ) · β ( z ) , (biHom-associativity) . Remark 3. If α = β , ( A , · , α , α ) is a Hom-associative algebra. Definition 11.
A biHom-module is a triple ( M , α , β ) , where M is a K -vector space,and α , β : M → M are two linear maps. Definition 12 ( [22]).
Let ( A , µ A , α A , β A ) be a biHom-associative algebra. A left A -module is a triple ( M , α M , β M ) , where M is a linear space, α M , β M : M → M arelinear maps, with, in addition, another linear map: A ⊗ M → M , a ⊗ m a · m , suchthat, for all a , a ′ ∈ A , m ∈ M : α M ◦ β M = β M ◦ α M , α M ( a · m ) = α A ( a ) · α M ( m ) , β M ( a · m ) = β A ( a ) · β M ( m ) , α A ( a ) · ( a ′ · m ) = ( aa ′ ) · β M ( m ) . Let us give now the definition of bimodule of a biHom-associative algebra.
Definition 13.
Let ( A , · , α , α ) be a biHom-associative algebra, and let ( V , β , β ) be a biHom-module. Let l , r : A → gl ( V ) be two linear maps. The quintuple ( l , r , β , β , V ) is called a bimodule of A if l ( x · y ) β ( v ) = l ( α ( x )) l ( y ) v , r ( x · y ) β ( v ) = r ( α ( y )) r ( x ) v , (26) l ( α ( x )) r ( y ) v = r ( α ( y )) l ( x ) v , (27) β ( l ( x ) v ) = l ( α ( x )) β ( v ) , β ( r ( x ) v ) = r ( α ( x )) β ( v ) , (28) β ( l ( x ) v ) = l ( α ( x )) β ( v ) , β ( r ( x ) v ) = r ( α ( x )) β ( v ) (29)for all x , y ∈ A , v ∈ V . Proposition 4.
Let ( l , r , β , β , V ) be a bimodule of a biHom-associative algebra ( A , · , α , α ) . Then, the direct sum A ⊕ V of vector spaces is turned into a biHom-associative algebra by defining multiplication in A ⊕ V by ( x + v ) ∗ ( x + v ) = x · x + ( l ( x ) v + r ( x ) v ) , ( α ⊕ β )( x + v ) = α ( x ) + β ( v ) , ( α ⊕ β )( x + v ) = α ( x ) + β ( v ) , for all x , x ∈ A , v , v ∈ V .Proof.
Let v , v , v ∈ V and x , x , x ∈ A . Setting and computing [( x + v ) ∗ ( x + v )] ∗ ( α ( x ) + β ( v )) =( α ( x ) + β ( v )) ∗ [( x + v ) ∗ ( x + v )] , and similarly for the other relations, give the required conditions. ⊓⊔ We denote such a biHom-associative algebra by ( A ⊕ V , ∗ , α + β , α + β ) , or A × l , r , α , α , β , β V . ouble constructions of biHom-Frobenius algebras 19 Example 5.
Let ( A , · , α , β ) be a multiplicative biHom-associative algebra. Then, ( L · , , α , β ) , ( , R · , α , β ) and ( L · , R · , α , β ) are bimodules of ( A , · , α , β ) . Proposition 5.
Let ( l , r , β , β , V ) be bimodule of a multiplicative biHom-associativealgebra ( A , · , α , α ) . Then, ( l ◦ α n , r ◦ α n , β , β , V ) is a bimodule of A for any fi-nite integer n . Proof.
We have ( l ◦ α n )( x · y ) β ( v ) = l ( α n ( x ) · α n ( y )) β ( v ) = l ( α ( α n ( x ))) l ( α n ( y )) v = l ( α n ( α ( x ))) l ( α n ( y )) v = ( l ◦ α n )( α ( x ))( l ◦ α n )( y ) v . Similarly, the other relations are established. ⊓⊔ Example 6.
Let ( A , · , α , α ) be a multiplicative biHom-associative algebra. Then, ( L · ◦ α n , R · ◦ α n , α , α , A ) is a bimodule of A for any finite integer n . Theorem 5.
Let ( A , · , α , α ) and ( B , ◦ , β , β ) be two biHom-associative alge-bras. Suppose there exist linear maps l A , r A : A → gl ( B ) , and l B , r B : B → gl ( A ) such that ( l A , r A , β , β , B ) is a bimodule of A , and ( l B , r B , α , α , A ) is a bimod-ule of B , satisfying, for any x , y ∈ A , a , b ∈ B the following conditions:l A ( α ( x ))( a ◦ b ) = l A ( r B ( a ) x ) β ( b ) + ( l A ( x ) a ) ◦ β ( b ) , (30) r A ( α ( x ))( a ◦ b ) = r A ( l B ( b ) x ) β ( a ) + β ( a ) ◦ ( r A ( x ) b ) , (31) l B ( β ( a ))( x · y ) = l B ( r A ( x ) a ) α ( y ) + ( l B ( a ) x ) · α ( y ) , (32) r B ( β ( a ))( x · y ) = r B ( l A ( y ) a ) α ( x ) + α ( x ) · ( r B ( a ) y ) , (33) l A ( l B ( a ) x ) β ( b ) + ( r A ( x ) a ) ◦ β ( b ) − r A ( r B ( b ) x ) β ( a ) − β ( a ) ◦ ( l A ( x ) b ) = , (34) l B ( l A ( x ) a ) α ( y ) + ( r B ( a ) x ) · α ( y ) − r B ( r A ( y ) a ) α ( x ) − α ( x ) · ( l B ( a ) y ) = . (35) Then, there is a biHom-associative algebra structure on the direct sum A ⊕ B ofthe underlying vector spaces of A and B given by ( x + a ) ∗ ( y + b ) = ( x · y + l B ( a ) y + r B ( b ) x ) + ( a ◦ b + l A ( x ) b + r A ( y ) a )( α ⊕ β )( x + a ) = α ( x ) + β ( a ) , ( α ⊕ β )( x + a ) = α ( x ) + β ( a ) (36) for all x , y ∈ A , a , b ∈ B .Proof. Let v , v , v ∈ V and x , x , x ∈ A . Setting and computing [( x + v ) ∗ ( x + v )] ∗ ( α ( x ) + β ( v )) =( α ( x ) + β ( v )) ∗ [( x + v ) ∗ ( x + v )] , we obtain (30)-(35). Then, using the following relations: β ( l A ( x ) a ) = l A ( α ( x )) β ( a ) , β ( r A ( x ) a ) = r A ( α ( x )) β ( a ) , β ( l A ( x ) a ) = l A ( α ( x )) β ( a ) , β ( r A ( x ) a ) = r A ( α ( x )) β ( a ) , α ( l B ( a ) x ) = l B ( β ( a )) α ( x ) , α ( r B ( a ) x ) = r B ( β ( a )) α ( x ) , α ( l B ( a ) x ) = l B ( β ( a )) α ( x ) , α ( r B ( a ) x ) = r B ( β ( a )) α ( x ) , we show that ∗ is a biHom-associative algebra structure. ⊓⊔ We denote this biHom-associative algebra by ( A ⊲⊳ B , ∗ , α + β , α + β ) or A ⊲⊳ l A , r A , β , β l B , r B , α , α B . Definition 14.
Let ( A , · , α , α ) and ( B , ◦ , β , β ) be two biHom-associative alge-bras. Suppose there exist linear maps l A , r A : A → gl ( B ) , and l B , r B : B → gl ( A ) such that ( l A , r A , β , β ) is a bimodule of A , and ( l B , r B , α , α ) is a bimodule of B . Then, ( A , B , l A , r A , β , β , l B , r B , α , α ) is called a matched pair of biHom-associative algebras , if the conditions (30) - (35) are satisfied. Now, we consider the multiplicative biHom-associative algebra ( A , · , α , α ) suchthat α ◦ α = α ◦ α = Id A , i.e, α − = α , and α = α = Id A . Lemma 2.
Let ( l , r , β , β , V ) be a bimodule of ( A , · , α , α ) . Then (i) ( r ∗ , l ∗ , β ∗ , β ∗ , V ∗ ) is a bimodule of ( A , · , α , α ) ;(ii) ( r ∗ , , β ∗ , β ∗ , V ∗ ) and ( , l ∗ , β ∗ , β ∗ , V ∗ ) are also bimodules of A .Proof. (i) Let ( l , r , β , β , V ) be a bimodule of an involutive biHom-associative al-gebra ( A , · , α , α ) . Show that ( r ∗ , l ∗ , β ∗ , β ∗ , V ∗ ) is a bimodule of A . Let x , y ∈ A , u ∗ ∈ V ∗ , v ∈ V . Then(i-1) the following computation h r ∗ ( x · y ) β ∗ ( u ∗ ) , v i = h β ( r ( x · y ) v ) , u ∗ i = h r ( α ( x · y )) β ( v ) , u ∗ i = h r ( α ( x ) · α ( y )) β ( v ) , u ∗ i = h r [ α ( α ( y ))] r ( α ( x )) v , u ∗ i = h ( r ( y ) r ( α ( x ))) ∗ u ∗ , v i = h r ∗ ( α ( x )) r ∗ ( y ) u ∗ , v i ;leads to r ∗ ( x · y ) β ∗ ( u ∗ ) = r ∗ ( α ( x )) r ∗ ( y ) u ∗ ;(i-2) the following computation h l ∗ ( x · y ) β ∗ ( u ∗ ) , v i = h β ( l ( x · y )( v )) , u ∗ i = h l ( α ( x · y )) β ( v ) , u ∗ i = h l ( α ( x ) · α ( y )) β ( v ) , u ∗ i = h l [ α ( α ( x ))] l ( α ( y )) β ( v ) , u ∗ i = h ( l ( x ) l ( α ( y ))) ∗ u ∗ , v i = h l ∗ ( α ( y )) l ∗ ( x ) u ∗ , v i gives l ∗ ( x · y ) β ∗ ( u ∗ ) = l ∗ ( α ( y )) l ∗ ( x ) u ∗ ; ouble constructions of biHom-Frobenius algebras 21 (i-3) the following computation h r ∗ ( α ( x )) l ∗ ( y ) u ∗ , v i = h l ( y ) r ( α ( x )) v , u ∗ i = h ( l ◦ α )( α ( y ))( r ◦ α )( x )) v , u ∗ i = h ( r ◦ α )( α ( x ))( l ◦ α )( y ) v , u ∗ i = h r ( x ) l ( α ( y )) v , u ∗ i = h l ∗ ( α ( y )) r ∗ ( x ) u ∗ , v i yields r ∗ ( α ( x )) l ∗ ( y ) u ∗ = l ∗ ( α ( y )) r ∗ ( x ) u ∗ .Furthermore, h β ∗ ( r ∗ ( x )) u ∗ , v i = h r ( x )( β ( v )) , u ∗ i = h ( r ◦ α )( α ( x ))( β ( v )) , u ∗ i = h β ( r ( α ( x ))) v , u ∗ i = h r ∗ ( α ( x )) β ∗ ( u ∗ ) , v i . Hence, β ∗ ( r ∗ ( x )) u ∗ = r ∗ ( α ( x )) β ∗ ( u ∗ ) . By analogy, we establish the otherconditions. Hence, ( r ∗ , l ∗ , β ∗ , β ∗ , V ∗ ) is a bimodule of A .(ii) Similarly, one can show that ( r ∗ , , β ∗ , β ∗ , V ∗ ) and ( , l ∗ , β ∗ , β ∗ , V ∗ ) are bimod-ules of A as well. ⊓⊔ Definition 15.
Let ( A , · , α , β ) be a biHom-associative algebra, and B : A × A → K be a bilinear form on A . B is said αβ -invariant if B ( β ( x ) · α ( y ) , α ( z )) = B ( α ( x ) , β ( y ) · α ( z )) . (37) Definition 16.
A biHom-Frobenius algebra is a biHom-associative algebra with anon-degenerate invariant bilinear form.
Definition 17.
We call ( A , α , β , B ) a double construction of an involutive biHom-Frobenius algebra associated to ( A , α ) and ( A ∗ , α ∗ ) if it satisfies the conditions:1) A = A ⊕ A ∗ as the direct sum of vector spaces;2) ( A , α , α ) and ( A ∗ , α ∗ , α ∗ ) are biHom-associative subalgebras of ( A , α ) with α = α ⊕ α ∗ and β = α ⊕ α ∗ ;3) B is the natural non-degerenate ( α ⊕ α ∗ )( α ⊕ α ∗ ) -invariant symmetric bi-linear form on A ⊕ A ∗ given by B ( x + a ∗ , y + b ∗ ) = h x , b ∗ i + h a ∗ , y i , B (( α + α ∗ )( x + a ∗ ) , y + b ∗ ) = B ( x + a ∗ , ( α + α ∗ )( y + b ∗ )) , B (( α + α ∗ )( x + a ∗ ) , y + b ∗ ) = B ( x + a ∗ , ( α + α ∗ )( y + b ∗ )) (38)for all x , y ∈ A , a ∗ , b ∗ ∈ A ∗ , where h , i is the natural pair between the vectorspace A and its dual space A ∗ .Let ( A , · , α , α ) be an involutive biHom-associative algebra. Suppose there alsoexists an involutive biHom-associative algebra structure ” ◦ ” on its dual space A ∗ . We construct an involutive biHom-associative algebra structure on the direct sum A ⊕ A ∗ of the underlying vector spaces of A and A ∗ such that ( A , · , α , α ) and ( A ∗ , ◦ , α ∗ , α ∗ ) are biHom-subalgebras, and the non-degenerate invariant symmet-ric bilinear form on A ⊕ A ∗ is given by (38). Hence, ( A ⊕ A ∗ , α ⊕ α ∗ , α ⊕ α ∗ , B ) is a symmetric multiplicative biHom-associative algebra. Such a construction iscalled a double construction of an involutive biHom-Frobenius algebra associatedto ( A , · , α , α ) and ( A ∗ , ◦ , α ∗ , α ∗ ) . Theorem 6.
Let ( A , · , α , α ) be an involutive biHom-associative algebra. Sup-pose there is an involutive biHom-associative algebra structure ” ◦ ” on its dualspace A ∗ . Then, there is a double construction of an involutive symmetric biHom-associative algebra associated to ( A , · , α , α ) and ( A ∗ , ◦ , α ∗ , α ∗ ) if and onlyif ( A , A ∗ , R ∗· , L ∗· , α ∗ , α ∗ , R ∗◦ , L ∗◦ , α , α ) is a matched pair of involutive biHom-associative algebras.Proof. By a similar proof as for Theorem 2, we obtain the results. Let us show that B is well ( α ⊕ α ∗ )( α ⊕ α ∗ ) -invariant. Let x , y , z ∈ A and a ∗ , b ∗ , c ∗ ∈ A ∗ . We have B [( α ( x ) + α ∗ ( a ∗ )) ∗ ( α ( y ) + α ∗ ( b ∗ )) , ( α ( z ) + α ∗ ( c ∗ ))]= h α ( x ) · α ( y ) , α ∗ ( c ∗ ) i + h α ∗ ( c ∗ ) ◦ α ∗ ( a ∗ ) , α ( y ) i + h α ∗ ( b ∗ ) ◦ α ∗ ( c ∗ ) , α ( x ) i + h α ∗ ( a ∗ ) ◦ α ∗ ( b ∗ ) , α ( z ) i + h α ( z ) · α ( x ) , α ∗ ( b ∗ ) i + h α ( y ) · α ( z ) , α ∗ ( a ∗ ) i ; B [ α ( x ) + α ∗ ( a ∗ ) , ( α ( y ) + α ∗ ( b ∗ )) ∗ ( α ( z ) + α ∗ ( c ∗ ))]= h α ( x ) , α ∗ ( b ∗ ) ◦ α ∗ ( c ∗ ) i + h α ∗ ( c ∗ ) , α ( x ) · α ( y ) i + h α ∗ ( b ∗ ) , α ( z ) · α ( x ) i + h α ( y ) · α ( z ) , α ∗ ( a ∗ ) i + h α ∗ ( a ∗ ) ◦ α ∗ ( b ∗ ) , α ( z ) i + h α ( c ∗ ) ◦ α ∗ ( a ∗ ) , α ( y ) i . Using α = α = Id A , α ∗ = α ∗ = Id A ∗ , α = α − and α ∗ = α ∗− , we obtain B [( α ( x ) + α ∗ ( a ∗ )) ∗ ( α ( y ) + α ∗ ( b ∗ )) , ( α ( z ) + α ∗ ( c ∗ ))]= B [ α ( x ) + α ∗ ( a ∗ ) , ( α ( y ) + α ∗ ( b ∗ )) ∗ ( α ( z ) + α ∗ ( c ∗ ))]= h x , b ∗ ◦ c ∗ i + h c ∗ , x · y i + h b ∗ , z · x i + h y · z , a ∗ i + h a ∗ ◦ b ∗ , z i + h c ∗ ◦ a ∗ , y i . This completes the proof. ⊓⊔ Theorem 7.
Let ( A , · , α , α ) be an involutive biHom-associative algebra. Supposethere exists an involutive biHom-associative algebra structure ” ◦ ” on its dual space ( A ∗ , α ∗ , α ∗ ) . Then, ( A , A ∗ , R ∗· , L ∗· , α ∗ , α ∗ , R ∗◦ , L ∗◦ , α , α ) is a matched pair of in-volutive biHom-associative algebras if and only if, for any x ∈ A and a ∗ , b ∗ ∈ A ∗ ,R ∗· ( α ( x ))( a ∗ ◦ b ∗ ) = R ∗· ( L ∗◦ ( a ∗ ) x ) α ∗ ( b ∗ ) + ( R ∗· ( x ) a ∗ ) ◦ α ∗ ( b ∗ ) , (39) R ∗· ( R ∗◦ ( a ∗ ) x ) α ∗ ( b ∗ ) + L ∗· ( x ) a ∗ ◦ α ∗ ( b ∗ ) = L ∗· ( L ∗◦ ( b ∗ ) x ) α ∗ ( a ∗ ) + α ∗ ( a ∗ ) ◦ ( R ∗· ( x ) b ∗ ) . (40) Proof.
By a similar proof as for Theorem 3, and using the following valid relations α ∗ ( R ∗· ( x ) a ∗ ) = R ∗· ( α ( x )) α ∗ ( a ∗ ) , α ∗ ( L ∗· ( x ) a ∗ ) = L ∗· ( α ( x )) α ∗ ( a ∗ ) α ∗ ( R ∗· ( x ) a ∗ ) = R ∗· ( α ( x )) α ∗ ( a ∗ ) , α ∗ ( L ∗· ( x ) a ∗ ) = L ∗· ( α ( x )) α ∗ ( a ∗ ) α ( R ∗◦ ( a ∗ ) x ) = R ∗◦ ( α ∗ ( a ∗ )) α ( x ) , α ( L ∗◦ ( a ∗ ) x ) = L ∗◦ ( α ∗ ( a ∗ )) α ( x ) α ( R ∗◦ ( a ∗ ) x ) = R ∗◦ ( α ∗ ( a ∗ )) α ( x ) , α ( L ∗◦ ( a ∗ ) x ) = L ∗◦ ( α ∗ ( a ∗ )) α ( x ) , the equivalences ouble constructions of biHom-Frobenius algebras 23 (30) ⇐⇒ (31) ⇐⇒ (32) ⇐⇒ (33) and (34) ⇐⇒ (35) . are obtained. ⊓⊔ Theorem 8.
Let ( A , · , α , α ) be an involutive biHom-associative algebra. Supposethere is an involutive biHom-associative algebra structure ” ◦ ” on its dual space A ∗ given by a linear map ∆ ∗ : A ∗ ⊗ A ∗ → A ∗ . Then, ( A , A ∗ , R ∗· , L ∗· , α ∗ , α ∗ , R ∗◦ , L ∗◦ , α , α ) is a matched pair of involutive biHom-associative algebras if and only if ∆ : A → A ⊗ A satisfies the following two conditions: ∆ ◦ α ( x · y ) = ( α ⊗ L · ( x )) △ ( y ) + ( R · ( y ) ⊗ α ) △ ( x ) , (41) ( L · ( y ) ⊗ α − α ⊗ R · ( y )) ∆ ( x ) + σ [( L · ( x ) ⊗ α − α ⊗ R · ( x )) ∆ ( y )] = for all x , y ∈ A .Proof. This proof is simillar to that of Theorem 4. ⊓⊔ Definition 18.
Let ( A , · , α , α ) be an involutive biHom-associative algebra. An antisymmetric infinitesimal biHom-bialgebra structure on A is a linear map ∆ : A → A ⊗ A such that(a) ∆ ∗ : A ∗ ⊗ A ∗ → A ∗ defines an involutive biHom-associative algebra struc-ture on A ∗ ;(b) ∆ satisfies (41) and (42).We denote it by ( A , △ , α , α ) or ( A , A ∗ , α , α , α ∗ , α , α ∗ ) . Corollary 2.
Let ( A , · , α , α ) and ( A ∗ , ◦ , α ∗ , α ∗ ) be two biHom-associative alge-bras. Then, the following conditions are equivalent: There is a double construction of an involutive biHom-Frobenius algebraassociated to ( A , · , α , α ) and ( A ∗ , ◦ , α ∗ , α ∗ ) ; ( A , A ∗ , R ∗· , L ∗· , α ∗ , α ∗ , R ∗◦ , L ∗◦ , α , α ) is a matched pair of multiplicativebiHom-associative algebras; ( A , A ∗ , α , α , α ∗ , α ∗ ) is an antisymmetric infinitesimal biHom-bialgebra.Proof. From Theorems 6 and 8, we have the equivalences. ⊓⊔ Definition 19.
A Hom-dendriform algebra is a quadruple ( A , ≺ , ≻ , α ) consistingof a vector space A on which the operations ≺ , ≻ : A ⊗ A → A , and α : A → A are linear maps satisfying ( x ≺ y ) ≺ α ( z ) = α ( x ) ≺ ( y ∗ z ) , ( x ≻ y ) ≺ α ( z ) = α ( x ) ≻ ( y ≺ z ) , α ( x ) ≻ ( y ≻ z ) = ( x ∗ y ) ≻ α ( z ) , where x ∗ y = x ≺ y + x ≻ y . (43) Definition 20.
Let ( A , ≺ , ≻ , α ) and ( A ′ , ≺ ′ , ≻ ′ , α ′ ) be two Hom-dendriform alge-bras. A linear map f : A → A ′ is a Hom-dendriform algebra morphism if ≺ ′ ◦ ( f ⊗ f ) = f ◦ ≺ , ≻ ′ ◦ ( f ⊗ f ) = f ◦ ≻ and f ◦ α = α ′ ◦ f . Proposition 6.
Let ( A , ≺ , ≻ , α ) be a Hom-dendriform algebra. Then, ( A , ∗ , α ) isa Hom-associative algebra.Proof. For all x , y , z ∈ A , ( x ∗ y ) ∗ α ( z ) = ( x ≺ y ) ≺ α ( z ) + ( x ≺ y ) ≻ α ( z ) + ( x ≻ y ) ≺ α ( z ) + ( x ≻ y ) ≻ α ( z )= ( x ≺ y ) ≺ α ( z ) + ( x ≻ y ) ≺ α ( z ) + ( x ∗ y ) ≻ α ( z )= α ( x ) ≺ ( y ∗ z ) + α ( x ) ≻ ( y ≺ z ) + α ( x ) ≻ ( y ≻ z )= α ( x ) ≺ ( y ∗ z ) + α ( x ) ≻ ( y ∗ z ) = α ( x ) ∗ ( y ∗ z ) , which completes the proof. ⊓⊔ We call ( A , ∗ , α ) the associated Hom-associative algebra of ( A , ≺ , ≻ , α ) , and ( A , ≻ , ≺ , α ) is called a compatible Hom-dendriform algebra structure on the Hom-associative algebra ( A , ∗ , α ) .Let ( A , ≺ , ≻ , α ) be a Hom-dendriform algebra. For any x ∈ A , let L ≻ ( x ) , R ≻ ( x ) and L ≺ ( x ) , R ≺ ( x ) denote the left and right multiplication operators of ( A , ≺ ) and ( A , ≻ ) , respectively, i. e., L ≻ ( x ) y = x ≻ y , R ≻ ( x ) y = y ≻ x , L ≺ ( x ) y = x ≺ y , L ≺ ( x ) y = y ≺ x , for all x , y ∈ A . Moreover, let L ≻ , R ≻ , L ≺ , R ≺ : A → gl ( A ) be four linear mapswith x L ≻ ( x ) , x R ≻ ( x ) , x L ≺ ( x ) , and x R ≺ ( x ) , respectively. Proposition 7.
The quadruple ( L ≻ , R ≺ , α , A ) is a bimodule of the associated Hom-associative algebra ( A , ∗ , α ) . ouble constructions of biHom-Frobenius algebras 25 Proof.
For all x , y , v ∈ A , L ≻ ( x ∗ y ) α ( v ) = ( x ∗ y ) ≻ α ( v ) = α ( x ) ≻ ( y ≻ v ) = L ≻ ( α ( x )) L ≻ ( y ) v , R ≺ ( x ∗ y ) α ( v ) = α ( v ) ≺ ( x ∗ y ) = ( v ≺ x ) ≺ α ( y ) = R ≺ ( α ( y )) R ≺ ( x ) v , L ≻ ( α ( x )) R ≺ ( y ) v = α ( x ) ≻ ( v ≺ y ) = ( x ≻ v ) ≺ α ( y ) = R ≺ ( α ( y )) L ≻ ( x ) v , α ( L ≻ ( x ) v ) = α ( x ≻ v ) = α ( x ) ≻ α ( v ) = L ≻ ( α ( x )) α ( v ) , α ( R ≺ ( x ) v ) = α ( v ≺ x ) = α ( v ) ≺ α ( x ) = R ≺ ( α ( x )) α ( v ) , which completes the proof. ⊓⊔ O -operators and Hom-dendriform algebras Definition 21.
Let ( A , · , α ) be a Hom-associative algebra, and ( l , r , β , V ) be a bi-module. Then, a linear map T : V → A is called an O -operator associated to ( l , r , β , V ) , if T satisfies α T = T β and T ( u ) · T ( v ) = T ( l ( T ( u )) v + r ( T ( v )) u ) for all u , v ∈ V . Example 7.
Let ( A , · , α ) be a multiplicative Hom-associative algebra. Then, theidentity map Id is an O -operator associated to the bimodule ( L , , α ) or ( , R , α ) . Example 8.
Let ( A , · , α ) be a multiplicative Hom-associative algebra. A linear map f : A → A is called a Rota-Baxter operator on A of weight zero if f satisfies f ( x ) · f ( y ) = f ( f ( x ) · y + x · f ( y )) for all x , y ∈ A . In fact, a Rota-Baxter operator on A is just an O -operator associated to the bimod-ule ( L , R , α ) . Theorem 9.
Let ( A , · , α ) be a Hom-associative algebra, and ( l , r , β , V ) be a bimod-ule. Let T : V → A be an O -operator associated to ( l , r , β , V ) . Then, there exists aHom-dendriform algebra structure on V given byu ≻ v = l ( T ( u )) v , u ≺ v = r ( T ( v )) ufor all u , v ∈ V . So, there is an associated Hom-associative algebra structure on Vgiven by the equation (43) , and T is a homomorphism of Hom-associative algebras.Moreover, T ( V ) = { T ( v ) \ v ∈ V } ⊂ A is a Hom-associative subalgebra of A , andthere is an induced Hom-dendriform algebra structure on T ( V ) given byT ( u ) ≻ T ( v ) = T ( u ≻ v ) , T ( u ) ≺ T ( v ) = T ( u ≺ v ) (44) for all u , v ∈ V . Its corresponding associated Hom-associative algebra structure onT ( V ) given by the equation (43) is just the Hom-associative subalgebra structure of A , and T is a homomorphism of Hom-dendriform algebras. Proof.
For any x , y , z ∈ V , we have ( x ≻ y ) ≺ β ( z ) − β ( x ) ≻ ( y ≺ z )= l ( T ( x ) y ) ≺ β ( z ) − β ( x ) ≻ r ( T ( z ) y )= r ( T β ( z )) l ( T ( x )) y − l ( T β ( x ) y ) r ( T ( z ) y )= r ( α ( T ( z ))) l ( T ( x )) y − l ( α ( T ( x ))) r ( T ( z )) y = . The two other axioms are similarly checked. ⊓⊔ Corollary 3.
Let ( A , ∗ , α ) be a multiplicative Hom-associative algebra. Then,there is a compatible multiplicative Hom-dendriform algebra structure on A if andonly if there exists an invertible O -operator of ( A , ∗ , α ) .Proof. If T is an invertible O − operator associated to a bimodule ( l , r , β , V ) , then,the compatible multiplicative Hom-dendriform algebra structure on A is given by x ≻ y = T ( l ( x ) T − ( y )) , x ≺ y = T ( r ( y ) T − ( x )) for all x , y ∈ A . Conversely, let ( A , ≻ , ≺ , α ) be a Hom-dendriform algebra, and ( A , ∗ , α ) be theassociated multiplicative Hom-associative algebra. Then, the identity map Id is an O − operator associated to the bimodule ( L ≻ , R ≺ , α ) of ( A , ∗ , α ) . ⊓⊔ Definition 22.
Let ( A , ≻ , ≺ , α ) be a Hom-dendriform algebra, and V be a vectorspace. Let l ≻ , r ≻ , l ≺ , r ≺ : A → gl ( V ) and β : V → V be linear maps. Then, thesextuple ( l ≻ , r ≻ , l ≺ , r ≺ , β , V ) is called a bimodule of A if the following equationshold, for any x , y ∈ A and v ∈ V : l ≺ ( x ≺ y ) β ( v ) = l ≺ ( α ( x )) l ∗ ( y ) v , r ≺ ( α ( x )) l ≺ ( y ) v = l ≺ ( α ( y )) r ∗ ( x ) v , r ≺ ( α ( y )) r ≺ ( y ) v = r ≺ ( x ∗ y ) β ( v ) , l ≺ ( x ≻ y ) β ( v ) = l ≻ ( α ( x )) l ≺ ( y ) v , r ≺ ( α ( x )) l ≻ ( y ) v = l ≻ ( α ( y )) r ≺ ( x ) v , r ≺ ( α ( x )) r ≻ ( y ) v = r ≻ ( y ≺ x ) β ( v ) , l ≻ ( x ∗ y ) β ( v ) = l ≻ ( α ( x )) l ≻ ( y ) v , r ≻ ( α ( x )) l ∗ ( y ) v = l ≻ ( α ( y )) r ≻ ( x ) v , r ≻ ( α ( x )) r ∗ ( y ) v = r ≻ ( y ≻ x ) β ( v ) , β ( l ≻ ( x ) v ) = l ≻ ( α ( x )) β ( v ) , β ( l ≺ ( x ) v ) = l ≺ ( α ( x )) β ( v ) , β ( r ≻ ( x ) v ) = r ≻ ( α ( x )) β ( v ) , β ( r ≺ ( x ) v ) = r ≺ ( α ( x )) β ( v ) , where x ∗ y = x ≻ y + x ≺ y , l ∗ = l ≻ + l ≺ , r ∗ = r ≻ + r ≺ . Proposition 8.
Let ( l ≻ , r ≻ , l ≺ , r ≺ , β , V ) be a bimodule of a Hom-dendriform algebra ( A , ≻ , ≺ , α ) . Then, there exists a Hom-dendriform algebra structure on the directsum A ⊕ V of the underlying vector spaces of A and V given by ( x + u ) ≻ ( y + v ) = x ≻ y + l ≻ ( x ) v + r ≻ ( y ) u , ( x + u ) ≺ ( y + v ) = x ≺ y + l ≺ ( x ) v + r ≺ ( y ) u , ouble constructions of biHom-Frobenius algebras 27 ( α ⊕ β )( x + u ) = α ( x ) + β ( u ) for all x , y ∈ A , u , v ∈ V .Proof.
By a straightforward calculation, we obtain the result. ⊓⊔ We denote this algebra by A × l ≻ , r ≻ , l ≺ , r ≺ , α , β V . Proposition 9.
Let (l ≻ , r ≻ , l ≺ , r ≺ , β , V ) be a bimodule of a Hom-dendriform algebra ( A , ≻ , ≺ , α ) . Let ( A , ∗ , α ) be the associated Hom-associative algebra. Then thefollowing statements hold. ( l ≻ , r ≺ , β , V ) and ( l ≻ + l ≺ , r ≻ + r ≺ , β , V ) are bimodules of ( A , ∗ , β ) . For any bimodule ( l , r , β , V ) of ( A , ∗ , α ) , ( l , , , r , β , V ) is a bimoduleof ( A , ≻ , ≺ , α ) . ( l ≻ + l ≺ , , , r ≻ + r ≺ , β , V ) and ( l ≻ , , , r ≺ , β , V ) are bimodulesof ( A , ≻ , ≺ , α ) . The dendriform algebras A × l ≻ , r ≻ , l ≺ , r ≺ , α , β V and A × l ≻ + l ≺ , , , r ≻ + r ≺ , α , β Vhave the same associated Hom-associative algebra A × l ≻ + l ≺ , r ≻ + r ≺ , α , β V . Proof.
It results from a direct computation. ⊓⊔ Theorem 10.
Let ( A , ≻ A , ≺ A , α ) and ( B , ≻ B , ≺ B , β ) be two Hom-dendriformalgebras. Suppose there are linear maps l ≻ A , r ≻ A , l ≺ A , r ≺ A : A → gl ( B ) , andl ≻ B , r ≻ B , l ≺ B , r ≺ B : B → gl ( A ) such that (l ≻ A , r ≻ A , l ≺ A , r ≺ A , β , B ) is a bimoduleof A , and (l ≻ B , r ≻ B , l ≺ B , r ≺ B , α , A ) ) is a bimodule of B , satisfying the followingrelations: r ≺ A ( α ( x ))( a ≺ B b ) = β ( a ) ≺ B ( r A ( x ) b ) + r ≺ A ( l B ( x ) β ( a )) , (45) l ≺ A ( l ≺ B ( x )) β ( b ) +( r ≺ A ( x ) a ) ≺ B β ( b ) = β ( a ) ≺ B ( l ≺ A ( x ) b ) + r ≺ A ( r ≺ B ( b ) x ) β ( a ) , (46) l ≺ A ( α ( x ))( a ∗ B b ) = ( l ≺ A ( x ) a ) ∗ B β ( b ) + l ≺ A ( r ≺ A ( a ) x ) β ( b ) , (47) r ≺ A ( α ( x ))( a ≻ B b ) = r ≻ A ( l ≺ B ( b ) x ) β ( a ) + β ( a ) ≻ B ( r ≺ A ( x ) b ) , (48) l ≺ A ( l ≻ B ( a ) x ) β ( b ) +( r ≻ A ( x ) a ) ≺ B β ( b ) = β ( a ) ≻ B ( l ≺ A ( x ) b ) + r ≻ A ( r ≺ B ( b ) x ) β ( a ) (49) l ≻ A ( α ( x ))( a ≺ B b ) = ( l ≻ A ( x ) a ) ≺ B β ( b ) + l ≺ A ( r ≻ B ( a ) x ) β ( b ) , (50) r ≻ A ( α ( x ))( a ∗ B b ) = β ( a ) ≻ B ( r ≻ A ( x ) b ) + r ≻ A ( l ≻ B ( b ) x ) β ( a ) , (51) β ( a ) ≻ B ( l ≻ A ( x ) b ) + r ≻ A ( r ≻ B ( b ) x ) β ( a ) = l ≻ A ( l B ( a ) x ) β ( b ) + ( r A ( x ) a ) ≻ B β ( b ) , (52) l ≻ A ( α ( x ))( a ≻ B b ) = ( l A ( x ) a ) ≻ B β ( b ) + l ≻ A ( r B ( a ) x ) β ( b ) , (53) r ≺ B ( β ( a ))( x ≺ A y ) = α ( x ) ≺ A ( r B ( a ) y ) + r ≺ B ( l A ( y ) a ) α ( x ) , (54) l ≺ B ( l ≺ A ( x ) a ) α ( y ) +( r ≺ B ( a ) x ) ≺ A α ( y ) = α ( x ) ≺ A ( l B ( a ) y ) + r ≺ B ( r A ( y ) a ) α ( x ) , (55) l ≺ B ( β ( a ))( x ∗ A y ) = ( l ≺ B ( a ) x ) ≺ A α ( y ) + l ≺ B ( r ≺ A ( x ) a ) α ( y ) , (56) r ≺ B ( β ( a ))( x ≻ A y ) = r ≻ B ( l ≺ B ( y ) a ) α ( x ) + α ( x ) ≻ A ( r ≺ B ( a ) y ) , (57) l ≺ B ( l ≻ A ( x ) a ) α ( y ) +( r ≻ B ( a ) x ) ≺ A α ( y ) = α ( x ) ≻ A ( l ≺ B ( a ) y ) + r ≻ B ( r ≺ A ( y ) a ) α ( x ) , (58) l ≻ B ( β ( a ))( x ≺ A y ) = ( l ≻ B ( a ) x ) ≺ A α ( y ) + l ≺ B ( r ≻ A ( x ) a ) α ( y ) , (59) r ≻ B ( β ( a ))( x ∗ A y ) = α ( x ) ≻ A ( r ≻ B ( a ) y ) + r ≻ B ( l ≻ A ( y ) a ) α ( x ) , (60) α ( x ) ≻ A ( l ≻ B ( a ) y ) + r ≻ B ( r ≻ A ( y ) a ) α ( x ) = l ≻ B ( l A ( x ) a ) α ( y ) + ( r B ( a ) x ) ≻ A α ( y ) , (61) l ≻ B ( β ( a ))( x ≻ A y ) = ( l B ( a ) x ) ≻ A α ( y ) + l ≻ B ( r A ( x ) a ) α ( y ) (62) for any x , y ∈ A , a , b ∈ B and l A = l ≻ A + l ≺ A , r A = r ≻ A + r ≺ A , l B = l ≻ B + l ≺ B , r B = r ≻ B + r ≺ B . Then, there is a Hom-dendriform algebra structure on thedirect sum A ⊕ B of the underlying vector spaces of A and B given by ( x + a ) ≻ ( y + b ) = ( x ≻ A y + r ≻ B ( b ) x + l ≻ B ( a ) y )+( l ≻ A ( x ) b + r ≻ A ( y ) a + a ≻ B b ) , ( x + a ) ≺ ( y + b ) = ( x ≺ A y + r ≺ B ( b ) x + l ≺ B ( a ) y )+( l ≺ A ( x ) b + r ≺ A ( y ) a + a ≺ B b ) , ( α ⊕ β )( x + a ) = α ( x ) + β ( a ) for any x , y ∈ A , a , b ∈ B .Proof. It is obtained in a similar way as for Theorem 1. ⊓⊔ Let A ⊲⊳ l ≻ A , r ≻ A , l ≺ A , r ≺ A , β l ≻ B , r ≻ B , l ≺ B , r ≺ B , α B or simply A ⊲⊳ B denote this Hom-dendriform alge-bra. Definition 23.
Let ( A , ≻ A , ≺ A , α ) and ( B , ≻ B , ≺ B , β ) be two Hom-dendriformalgebras. Suppose there are linear maps l ≻ A , r ≻ A , l ≺ A , r ≺ A : A → gl ( B ) and l ≻ B , r ≻ B , l ≺ B , r ≺ B : B → gl ( A ) such that ( l ≻ A , r ≻ A , l ≺ A , r ≺ A , β ) is a bimoduleof A , and ( l ≻ B , r ≻ B , l ≺ B , r ≺ B , α ) is a bimodule of B . If (45) - (62) are satisfied,then ( A , B , l ≻ A , r ≻ A , l ≺ A , r ≺ A , β , l ≻ B , r ≻ B , l ≺ B , r ≺ B , α ) is called a matched pairof Hom-dendriform algebras . Corollary 4. If ( A , B , l ≻ A , r ≻ A , l ≺ A , r ≺ A , β , l ≻ B , r ≻ B , l ≺ B , r ≺ B , α ) is a matchedpair of Hom-dendriform algebras, then ( A , B , l ≻ A + l ≺ A , r ≻ A + r ≺ A , l ≻ B + l ≺ B , r ≻ B + r ≺ B , α + β ) is a matched pair of the associated Hom-associative algebras ( A , ∗ A , α ) and ( B , ∗ B , β ) .Proof. In fact, the associated Hom-associative algebra ( A ⊲⊳ B , ∗ , α + β ) is exactlythe Hom-associative algebra obtained from the matched pair of Hom-associativealgebras ( A , B , l A , r A , β , l B , r B , α ) with ( x + a ) ∗ ( y + b ) = x ∗ A y + l B ( a ) y + r B ( b ) x + a ∗ B b + l A ( x ) b + r A ( y ) a , ouble constructions of biHom-Frobenius algebras 29 ( α ⊕ β )( x + a ) = α ( x ) + β ( a ) for all x , y ∈ A , a , b ∈ B , where l A = l ≻ A + l ≺ A , r A = r ≻ A + r ≺ A , l B = l ≻ B + l ≺ B , r B = r ≻ B + r ≺ B . ⊓⊔ In this sequel, we suppose that α is involutive. Proposition 10.
Let (l ≻ , r ≻ , l ≺ , r ≺ , β , V ) be a bimodule of a Hom-dendriform alge-bra ( A , ≻ , ≺ , α ) , and let ( A , ∗ , α ) be the associated involutive Hom-associativealgebra. Let l ∗≻ , r ∗≻ , l ∗≺ , r ∗≺ : A → gl ( V ∗ ) be the linear maps given by h l ∗≻ ( x ) a ∗ , y i = h l ≻ ( x ) y , a ∗ i , h r ∗≻ ( x ) a ∗ , y i = h r ≻ ( x ) y , a ∗ i , h l ∗≺ ( x ) a ∗ , y i = h l ≺ ( x ) y , a ∗ i , h r ∗≺ ( x ) a ∗ , y i = h r ≺ ( x ) y , a ∗ i . Then, ( r ∗≻ + r ∗≺ , − l ∗≺ , − r ∗≻ , l ∗≻ + l ∗≺ , β ∗ , V ∗ ) is a bimodule of ( A , ≻ , ≺ , α ) ;2) ( r ∗≻ + r ∗≺ , , , l ∗≻ + l ∗≺ , β ∗ , V ∗ ) and ( r ∗≺ , , , l ∗≻ , β ∗ , V ∗ ) are bimodulesof ( A , ∗ , α ) ;3) ( r ∗≻ + r ∗≺ , l ∗≻ + l ∗≺ , β ∗ , V ∗ ) and ( r ∗≺ , l ∗≻ , β ∗ , V ∗ ) are bimodules of ( A , ∗ , α ) ;4) The Hom-dendriform algebras A × r ∗≻ + r ∗≺ , − l ∗≺ , − r ∗≻ , l ∗≻ + l ∗≺ , α , β ∗ V ∗ and A × r ∗≺ , , , l ∗≻ , α , β ∗ V ∗ have the same Hom-associative algebra A × r ∗≺ , l ∗≻ , α , β ∗ V ∗ .Example 9. Let ( A , ≺ , ≻ , α ) be an involutive Hom-dendriform algebra. Then, ( L ≻ , R ≻ , L ≺ , R ≺ , α , A ) , ( L ≻ , , , R ≺ , α , A ) , ( L ≻ + L ≺ , , , R ≻ + R ≺ , α , A ) are bimodules of ( A , ≺ , ≻ , α ) . On the other hand, ( R ∗≻ + R ∗≺ , − L ∗≺ , − R ∗≻ , L ∗≻ + L ∗≺ , α ∗ , A ∗ ) , ( R ∗≺ , , , L ∗≻ , α ∗ , A ∗ ) , ( R ∗≻ + R ∗≺ , , , L ∗≻ + L ∗≺ , α ∗ , A ∗ ) are bimodules of ( A , ≻ , ≺ , α ) too. There are two compatible Hom-dendriform al-gebra structures, A × R ∗≻ + R ∗≺ , − L ∗≺ , − R ∗≻ , L ∗≻ + L ∗≺ , α , α ∗ A ∗ and A × R ∗≻ + R ∗≺ , , , L ∗≻ + L ∗≺ , α , α ∗ A ∗ , on the same Hom-associative algebra A × R ∗≺ , L ∗≻ , α , α ∗ A ∗ . Definition 24.
Let ( A , α ) be a Hom-associative algebra. We say that ( A , α , ω ) is a symplectic Hom-associative algebra if ω is a non-degenerate skew-symmetric bilinear form on A such that the following identity (invariance condition) is satisfiedfor all x , y , z ∈ A : ω ( α ( x ) α ( y ) , α ( z )) + ω ( α ( y ) α ( z ) , α ( x )) + ω ( α ( z ) α ( x ) , α ( y )) = . (63) Theorem 11.
Let ( A , ∗ , α ) be an involutive Hom-associative algebra, and let ω be an α -invariant non-degenerate skew-symmetric bilinear form on A . Then, thereexists a compatible Hom-dendriform algebra structure ≻ , ≺ on ( A , α ) given by ω ( x ≻ y , z ) = ω ( y , z ∗ x ) , ω ( x ≻ y , z ) = ω ( x , y ∗ z ) for all x , y ∈ A . (64) Proof.
Define a linear map T : A → A ∗ by h T ( x ) , y i = ω ( x , y ) for all x , y ∈ A .Then, T is invertible and T − is an O − operator of the involutive Hom-associativealgebra ( A , ∗ , α ) associated to the bimodule ( R ∗∗ , L ∗∗ , α ∗ ) . By Corollary 3, there is acompatible Hom-dendriform algebra structure ≻ , ≺ on ( A , ∗ ) given by x ≻ y = T − R ∗∗ ( x ) T ( y ) , x ≺ y = T − L ∗∗ ( y ) T ( x ) for all x , y ∈ A , which gives exactly the equation (64). ⊓⊔ Definition 25.
We call ( A , α , B ) a double construction of involutive symplec-tic Hom-associative algebra associated to ( A , α ) and ( A ∗ , α ∗ ) if it satisfies thefollowing conditions:1) A = A ⊕ A ∗ as the direct sum of vector spaces;2) ( A , α ) and ( A ∗ , α ∗ ) are Hom-associative subalgebras of ( A , α ) with α = α ⊕ α ∗ ;3) ω is the natural non-degenerate antisymmetric ( α ⊕ α ∗ )-invariant bilinearform on A ⊕ A ∗ given by ω ( x + a ∗ , y + b ∗ ) = −h x , b ∗ i + h a ∗ , y i , ω (( α + α ∗ )( x + a ∗ ) , y + b ∗ ) = ω ( x + a ∗ , ( α + α ∗ )( y + b ∗ )) (65)for all x , y ∈ A , a ∗ , b ∗ ∈ A ∗ , where h , i is the natural pair between the vectorspace A and its dual space A ∗ .Let ( A , ∗ A , α ) be an involutive Hom-associative algebra, and suppose that thereis an involutive Hom-associative algebra structure ∗ A ∗ on its dual space A ∗ . Weconstruct an involutive symplectic Hom-associative algebra structure on the directsum A ⊕ A ∗ of the underlying vector spaces of A and A ∗ such that both A and A ∗ are Hom-subalgebras, equipped with the natural non-degenerate antisymmetric( α ⊕ α ∗ )-invariant bilinear form on A ⊕ A ∗ given by the equation (65). Such aconstruction is called double construction of involutive symplectic Hom-associativealgebras associated to ( A , ∗ A , α ) and ( A ∗ , ∗ A ∗ , α ∗ ) , denoted by ( T ( A ) = A ⊲⊳ α ∗ α A ∗ , ω ) . Corollary 5.
Let ( T ( A ) = A ⊲⊳ α ∗ α A ∗ , ω ) be a double construction of involutivesymplectic Hom-associative algebras. Then, there exists a compatible involutive ouble constructions of biHom-Frobenius algebras 31 Hom-dendriform algebra structure ≻ , ≺ on ( T ( A ) , α ⊕ α ∗ ) defined by the equa-tion (65) . Moreover, A and A ∗ , endowed with this product, are Hom-dendriformsubalgebras.Proof. The first part follows from Theorem 11. Let x , y ∈ A . Set x ≻ y = a + b ∗ , where a ∈ A , b ∗ ∈ A ∗ . Since A is a Hom-associative subalgebra of ( T ( A ) , α ⊕ α ∗ ) and ω ( A , A ) = ω ( A ∗ , A ∗ ) =
0, we have ω ( b ∗ , A ∗ ) = ω ( b ∗ , A ) = ω ( x ≻ y , A ) = ω ( y , A ∗ x ) = . Therefore, b ∗ = ω . Hence, x ≻ y = a ∈ A . Similarly, x ≺ y ∈ A . Thus, A is a Hom-dendriform subalgebra of T ( A ) with the product ≺ , ≻ . By symmetry of A , A ∗ is also a Hom-dendriform subalgebra. ⊓⊔ Theorem 12.
Let ( A , ≻ A , ≺ A , α ) be an involutive Hom-dendriform algebra, and ( A , ∗ A , α ) be the associated involutive Hom-associative algebra. Suppose thereis an involutive Hom-dendriform algebra structure ” ≻ A ∗ , ≺ A ∗ , α ∗ ” on its dualspace A ∗ , and ( A ∗ , ∗ A ∗ , α ∗ ) is the associated involutive Hom-associative algebra.Then, there exists a double construction of involutive symplectic Hom-associativealgebras associated to ( A , ∗ A , α ) and ( A , ∗ A ∗ , α ∗ ) if and only if the octuple ( A , A ∗ , R ∗≺ A , L ∗≻ A , α ∗ , R ∗≺ A ∗ , L ∗≻ A ∗ , α ) is a matched pair of Hom-associative al-gebras.Proof. The conclusion can be obtained by a similar proof as in Theorem 2. Then, if ( A , A ∗ , R ∗≺ A , L ∗≻ A , α ∗ , R ∗≺ A ∗ , L ∗≻ A ∗ , α ) is a matched pair of the involutive Hom-associative algebras ( A , ∗ A , α ) and ( A , ∗ A ∗ , α ∗ ) , it is straightforward to showthat the bilinear form (65) is an ( α ⊕ α ∗ )-invariant on the Hom-associative alge-bra A ⊲⊳ R ∗≺ A , L ∗≻ A , α ∗ R ∗≺ A ∗ , L ∗≻ A ∗ , α A ∗ given by ( x + a ∗ ) ∗ A ⊕ A ∗ ( y + b ∗ ) = ( x ∗ A y + R ∗≺ A ∗ ( a ∗ ) y + L ∗≻ A ∗ ( b ∗ ) x )+( a ∗ ∗ A ∗ b ∗ + R ∗≺ A ( x ) b ∗ + L ∗≻ A ( y ) a ∗ ) . In fact, we have ω [( α ( x ) + α ∗ ( a ∗ )) ∗ A ⊕ A ∗ ( α ( y ) + α ∗ ( b ∗ )) , α ( z ) + α ∗ ( c ∗ )]+ ω [( α ( y ) + α ∗ ( b ∗ )) ∗ A ⊕ A ∗ ( α ( z ) + α ∗ ( c ∗ )) , α ( x ) + α ∗ ( a ∗ )]+ ω [( α ( z ) + α ∗ ( c ∗ )) ∗ A ⊕ A ∗ ( α ( x ) + α ∗ ( a ∗ )) , α ( y ) + α ∗ ( b ∗ )]= −h α ( x ) ∗ A α ( y ) + R ∗≺ A ∗ ( α ∗ ( a ∗ )) α ( y ) + L ∗≻ A ∗ ( α ∗ ( b ∗ )) α ( x ) , α ∗ ( c ∗ ) i + h α ∗ ( a ∗ ) ∗ A ∗ α ∗ ( b ∗ ) + R ∗≺ A ( α ( x )) α ∗ ( b ∗ ) + L ∗≻ A ( α ( y )) α ∗ ( a ∗ ) , α ( z ) i−h α ( y ) ∗ A α ( z ) + R ∗≺ A ∗ ( α ∗ ( b ∗ )) α ( z ) + L ∗≻ A ∗ ( α ∗ ( c ∗ )) α ( y ) , α ∗ ( a ∗ ) i + h α ∗ ( b ∗ ) ∗ A ∗ α ∗ ( c ∗ ) + R ∗≺ A ( α ∗ ( y )) α ∗ ( c ∗ ) + L ∗≻ A ( α ( z )) α ∗ ( b ∗ ) , α ( x ) i−h α ( z ) ∗ A α ( x ) + R ∗≺ A ∗ ( α ∗ ( c ∗ )) α ( x ) + L ∗≻ A ∗ ( α ∗ ( a ∗ )) α ( z ) , α ∗ ( b ∗ ) i + h α ∗ ( c ∗ ) ∗ A ∗ α ( a ∗ ) + R ∗≺ A ( α ( z )) α ∗ ( a ∗ ) + L ∗≺ A ( α ( x )) α ∗ ( c ∗ ) , α ( y ) i = −h α ( x ) ≺ A α ( y ) , α ∗ ( c ∗ ) i − h α ( x ) ≻ A α ( y ) , α ∗ ( c ∗ ) i−h α ∗ ( c ∗ ) ≺ A ∗ α ∗ ( a ∗ ) , α ( y ) i−h α ∗ ( b ∗ ) ≻ A ∗ α ∗ ( c ∗ ) , α ( x ) i + h α ∗ ( a ∗ ) ≺ A ∗ α ∗ ( b ∗ ) , α ( z ) i + h α ∗ ( a ∗ ) ≻ A ∗ α ∗ ( b ∗ ) , α ( z ) i + h α ( z ) ≺ A α ( x ) , α ∗ ( b ∗ ) i + h α ( y ) ≻ A α ( z ) , α ∗ ( a ∗ ) i−h α ( y ) ≻ A α ( z ) , α ∗ ( a ∗ ) i − h α ( y ) ≺ A α ( z ) , α ∗ ( a ∗ ) i−h α ∗ ( a ∗ ) ≺ A ∗ α ∗ ( b ∗ ) , α ( z ) i − h α ∗ ( a ∗ ) ≻ A ∗ α ∗ ( c ∗ ) , α ( y ) i + h α ∗ ( b ∗ ) ≺ A ∗ α ∗ ( c ∗ ) , α ( x ) i + h α ∗ ( b ∗ ) ≻ A ∗ α ∗ ( c ∗ ) , α ( x ) i + h α ( x ) ≺ A α ( y ) , α ( c ∗ ) i + h α ( z ) ≻ A α ( x ) , α ∗ ( b ∗ ) i−h α ( z ) ≺ A α ( x ) , α ∗ ( b ∗ ) i − h α ( z ) ≻ A α ( x ) , α ∗ ( b ∗ ) i−h α ∗ ( b ∗ ) ≺ A ∗ α ∗ ( c ∗ ) , α ( x ) i − h α ∗ ( a ∗ ) ≻ A ∗ α ∗ ( b ∗ ) , α ( z ) i + h α ∗ ( c ∗ ) ≺ A ∗ α ∗ ( a ∗ ) , α ( y ) i + h α ∗ ( a ∗ ) ≻ A ∗ α ∗ ( c ∗ ) , α ( y ) i + h α ( y ) ≺ A α ( z ) , α ∗ ( a ∗ i + h α ( x ) ≻ A α ( y ) , α ∗ ( c ∗ ) i = . Conversely, if there exists a double construction of involutive symplectic Hom-associative algebras associated to ( A , ∗ A , α ) and ( A , ∗ A ∗ , α ∗ ) , then, the octuple ( A , A ∗ , R ∗≺ A , L ∗≻ A , α ∗ , R ∗≺ A ∗ , L ∗≻ A ∗ , α ) is a matched pair of the involutive Hom-associative algebras given by the following equations: R ∗≺ A ( α ( x ))( a ∗ ∗ A ∗ b ∗ ) = R ∗≺ A ( L ≺ A ∗ ( a ∗ ) x ) α ∗ ( b ∗ ) + ( R ∗≺ A ( x ) a ∗ ) ∗ A ∗ α ∗ ( b ∗ ) , L ∗≻ A ( α ( x ))( a ∗ ∗ A ∗ b ∗ ) = L ∗≻ A ( R ≺ A ( b ∗ ) x ) α ∗ ( a ∗ ) + α ∗ ( a ∗ ) ∗ A ∗ ( L ∗≻ A ( x ) b ∗ ) , R ∗≺ A ∗ ( α ∗ ( a ∗ ))( x ∗ A y ) = R ∗≺ A ∗ ( L ≻ A ( x ) a ∗ ) α ( y ) + ( R ∗≺ A ∗ ( a ∗ ) x ) ∗ A α ( y ) , L ∗≻ A ∗ ( α ∗ ( a ∗ ))( x ∗ A y ) = L ∗≻ A ∗ ( R ≺ A ( y ) a ∗ ) α ( x ) + α ( x ) ∗ A ( L ∗≻ A ∗ ( a ∗ ) y ) , R ∗≺ A ( R ∗≺ A ∗ ( a ∗ ) x ) α ∗ ( b ∗ ) +( L ∗≺ A ( x ) a ∗ ) ∗ A ∗ α ∗ ( b ∗ ) − L ∗≻ A ( L ∗≻ A ∗ ( b ∗ ) x ) α ∗ ( a ∗ ) − α ∗ ( a ∗ ) ∗ A ∗ ( R ∗≺ A ( x ) b ∗ ) = , R ∗≺ A ( R ∗≺ A ∗ ( x ) a ∗ ) α ( y ) +( L ∗≻ A ∗ ( a ∗ ) x ) ∗ A α ( y ) − L ∗≻ A ∗ ( L ∗≻ A ( y ) a ∗ ) α ( x ) − α ( x ) ∗ A ( R ∗≺ A ∗ ( a ∗ ) y ) = , since the operation ∗ A ⊕ A ∗ is Hom-associative. ⊓⊔ Corollary 6.
Let ( A , ≻ , ≺ , α ) be an involutive Hom-dendriform algebra, and thetriple ( R ∗≺ , L ∗≻ , α ∗ ) be the bimodule of the associated involutive Hom-associativealgebra ( A , ∗ , α ) . Then, ( T ( A ) = A × R ∗≺ , L ∗≻ , α , α ∗ A ∗ , ω ) is a double constructionof the involutive symplectic Hom-associative algebras. Theorem 13.
Let ( A , ≻ A , ≺ A , α ) be an involutive Hom-dendriform algebra, and ( A , ∗ A , α ) be the associated involutive Hom-associative algebra. Suppose thatthere is an involutive Hom-dendriform algebra structure ≻ A ∗ , ≺ A ∗ , α ∗ on its dualspace A ∗ , and ( A ∗ , ∗ A ∗ , α ∗ ) is its associated involutive Hom-associative algebra.Then, ( A , A ∗ , R ∗≺ A , L ∗≻ A , α ∗ , R ∗≺ A ∗ , L ∗≻ A ∗ , α ) is a matched pair of involutive Hom-associative algebras if and only if ( A , A ∗ , R ∗≻ A + R ∗≺ A , − L ∗≺ A , − R ∗≻ A , L ∗≻ A + L ∗≺ A , α ∗ , R ∗≻ A ∗ + R ∗≺ A ∗ , − L ∗≺ A ∗ , − R ∗≻ A ∗ , L ∗≻ A ∗ + L ∗≺ A ∗ , α ) is a matched pair of involutive Hom-dendriform algebras. ouble constructions of biHom-Frobenius algebras 33 Proof.
The necessary condition follows from Corollary 4. We need to prove the suf-ficient condition only. If ( A , A ∗ , R ∗≺ A , L ∗≻ A , α ∗ , R ∗≺ A ∗ , L ∗≻ A ∗ , α ) is a matched pairof involutive Hom-associative algebras, then ( A ⊲⊳ R ∗≺ A , L ∗≻ A , α ∗ R ∗≺ A ∗ , L ∗≻ A ∗ , α A ∗ , ω ) is a doubleconstruction of involutive symplectic Hom-associative algebras. Hence, there existsa compatible involutive Hom-dendriform algebra structure on A ⊲⊳ R ∗≺ A , L ∗≻ A , α ∗ R ∗≺ A ∗ , L ∗≻ A ∗ , α A ∗ given by (64). By a simple and direct computation, we show that A and A ∗ are itssubalgebras, and the other products are given for any x ∈ A , a ∗ ∈ A ∗ by x ≻ a ∗ = ( R ∗≻ A + R ∗≺ A )( x ) a ∗ − L ∗≺ A ∗ x , x ≺ a ∗ = − R ∗≻ A ( x ) a ∗ + ( L ∗≻ A ∗ + L ∗≺ A ∗ )( a ∗ ) x , a ∗ ≻ x = ( R ∗≻ A ∗ + R ∗≺ A ∗ )( a ∗ ) x − L ∗≺ A ( x ) a ∗ , a ∗ ≺ x = − R ∗≻ A ∗ ( a ∗ ) x + ( L ∗≻ A + L ∗≺ A )( x ) a ∗ . Hence, ( A , A ∗ , R ∗≻ A + R ∗≺ A , − L ∗≺ A , − R ∗≻ A , L ∗≻ A + L ∗≺ A , α ∗ , R ∗≻ A ∗ + R ∗≺ A ∗ , − L ∗≺ A ∗ , − R ∗≻ A ∗ , L ∗≻ A ∗ + L ∗≺ A ∗ , α ) is a matched pair of involutive Hom-dendriform algebras. ⊓⊔ Theorem 14.
Let ( A , ≻ A , ≺ A , α ) be an involutive Hom-dendriform algebra whoseproducts are given by two linear maps β ∗≻ , β ∗≺ : A ⊗ A → A . Further, supposethat there is an involutive Hom-dendriform algebra structure ≻ A ∗ , ≺ A ∗ , α ∗ onits dual space A ∗ given by two linear maps ∆ ∗≻ , ∆ ∗≺ : A ∗ ⊗ A ∗ → A ∗ . Then, ( A , A ∗ , R ∗≺ A , L ∗≻ A , α ∗ , R ∗≺ A ∗ , L ∗≻ A ∗ , α ) is a matched pair of involutive Hom-associ-ative algebras if and only if ∆ ≺ ◦ α ( x ∗ A y ) = ( α ⊗ L ≺ A ( x )) ∆ ≺ ( y ) + ( R A ( y ) ⊗ α ) ∆ ≺ ( y ) , (66) ∆ ≻ ◦ α ( x ∗ A y ) = ( α ⊗ L ≺ A ( x )) ∆ ≻ ( y ) + ( R ≺ A ( y ) ⊗ α ) ∆ ≻ ( y ) , (67) β ≺ ◦ α ∗ ( a ∗ ∗ A ∗ b ∗ ) = ( α ∗ ⊗ L ≺ A ∗ ( a ∗ )) β ≺ ( b ∗ ) + ( R A ∗ ( b ∗ ) ⊗ α ∗ ) β ≺ ( a ∗ ) (68) β ≻ ◦ α ∗ ( a ∗ ∗ A ∗ b ∗ ) = ( α ∗ ⊗ L A ∗ ( a ∗ )) β ≻ ( b ∗ ) + ( R ≺ A ∗ ( b ∗ ) ⊗ α ∗ ) β ≻ ( a ∗ ) , (69) ( L A ( x ) ⊗ α − α ⊗ R ≺ A ( x )) ∆ ≺ ( y )+ σ [( L ≻ A ( y ) ⊗ ( − α ) ⊗ R A ( y )) ∆ ≺ ( y )] = , (70) ( L A ∗ ( a ∗ ) ⊗ α ∗ − α ∗ ⊗ R ≺ A ∗ ( a ∗ )) β ≺ ( b ∗ )+ σ [( L ≻ A ∗ ( b ∗ ) ⊗ ( − α ∗ ) ⊗ R A ∗ ( b ∗ )) β ≻ ( a ∗ )] = hold for any x , y ∈ A and a ∗ , b ∗ ∈ A ∗ , whereL A = L ≻ A + L ≺ A , R A = R ≻ A + R ≺ A , L A ∗ = L ≻ A ∗ + L ≺ A ∗ , R A ∗ = R ≻ A ∗ + R ≺ A ∗ . Proof.
Let { e , . . . , e n } be a basis of A , and { e ∗ , . . . , e ∗ n } be its dual basis. Set e i ≻ A e j = n ∑ k = a ki j e k , e i ≺ A e j = n ∑ k = b ki j e k , α ( e i ) = n ∑ q = f iq e q , e ∗ i ≻ A ∗ e ∗ j = n ∑ k = c ki j e ∗ k , e ∗ i ≺ A ∗ e ∗ j = n ∑ k = d ki j e ∗ k , α ∗ ( e ∗ i ) = n ∑ q = f ∗ iq e ∗ q . We have h α ∗ ( e ∗ i ) , e j i = f ∗ ji = h e ∗ , α ( e j ) i = f ij ⇒ f ∗ ji = f ij , β ≻ ( e ∗ k ) = n ∑ i , j = a ki j e ∗ i ⊗ e ∗ j , β ≺ ( e ∗ k ) = n ∑ i , j = b ki j e ∗ i ⊗ e ∗ j , ∆ ≻ ( e k ) = n ∑ i , j = c ki j e i ⊗ e j , ∆ ≺ ( e k ) = n ∑ i , j d ki j e i ⊗ e j , R ∗≻ A ( e i ) e ∗ j = n ∑ k = a jki e ∗ k , R ∗≺ A ( e i ) e ∗ j = n ∑ k = b jki e ∗ k , R ∗≻ A ∗ ( e ∗ i ) e j = n ∑ k = c jki e k , R ∗≺ A ∗ ( e ∗ i ) e j = n ∑ k = d jki e k , L ∗≻ A ( e i ) e ∗ j = n ∑ k = a jik e ∗ k , L ∗≺ A ( e i ) e ∗ j = n ∑ k = b jik e ∗ k , L ∗≻ A ∗ ( e ∗ i ) e j = n ∑ k = c jik e k , L ∗≺ A ∗ ( e ∗ i ) e j = n ∑ k = d jik e k . Therefore, the coefficient of e ∗ l in R ∗≺ A ( α ( e i ))( e ∗ j ∗ A ∗ e ∗ k ) = R ∗≺ A ( L ≺ A ∗ ( e ∗ j ) e i ) α ∗ ( e ∗ k ) + ( R ∗≺ A ( e i ) e ∗ j ) ∗ A ∗ α ∗ ( e ∗ k ) gives the following relation for any i , j , l , k , q : n ∑ m = f iq b mlq ( c mjk + d mjk ) = n ∑ m = f kq [ c ijm b qlm + b jmi ( c lmq + d lmq )] . (72)In fact, we have R ∗≺ A ( α ( e i ))( e ∗ j ∗ A ∗ e ∗ k ) = R ∗≺ A ( α ( e i ))( e ∗ j ≻ A ∗ e ∗ k + e ∗ j ≺ A ∗ e ∗ k )= R ∗≺ A ( n ∑ q = f iq e q ) n ∑ m = ( c mjk + d mjk ) e ∗ m = n ∑ m , q = f iq ( c mjk + d mjk ) R ∗≺ A ( e q ) e ∗ m = n ∑ m , q = f iq ( c mjk + d mjk )( n ∑ l = b mlq ) e ∗ l = n ∑ l = [ n ∑ m , q = f iq b mlq ( c mjk + d mjk )] e ∗ l , ouble constructions of biHom-Frobenius algebras 35 R ∗≺ A ( L ≺ A ∗ ( e ∗ j ) e i ) α ∗ ( e ∗ k ) = R ∗≺ A ( n ∑ m = c ijm e m )( n ∑ q = f kq e ∗ q )= n ∑ m , q = c ijm f kq R ∗≺ A ( e m ) e ∗ q = n ∑ m , q = c ijm f kq ( n ∑ l = b qlm e ∗ l ) = n ∑ l = ( n ∑ m , q = c ijm f kq b qlm ) e ∗ l , ( R ∗≺ A ( e i ) e ∗ j ) ∗ A ∗ α ( e ∗ k ) = ( n ∑ m = b jmi e ∗ m ) ∗ A ∗ ( n ∑ q = f kq e ∗ q )= n ∑ m , q = f kq b jmi ( e ∗ m ≻ A ∗ e ∗ q + e ∗ m ≺ A ∗ e ∗ q ) = n ∑ m , q = f kq b jmi [ n ∑ l = ( c lmq + d lmq )] e ∗ l = n ∑ l = [ n ∑ m , q = f kq b jmi ( c lmq + d lmq )] e ∗ l , giving the equation (72). Also, the coefficient of e ∗ l ⊗ e ∗ i in β ≺ ◦ α ∗ ( e ∗ j ∗ A ∗ e ∗ k ) = ( α ∗ ⊗ L ≺ A ∗ ( e ∗ j )) β ≺ ( e ∗ k ) + ( R A ∗ ( e ∗ k ) ⊗ α ∗ ) β ≺ ( e ∗ j ) , β ≺ ◦ α ∗ ( e ∗ j ∗ A ∗ e ∗ k ) = β ≺ ◦ α ∗ ( e ∗ j ≻ A ∗ e ∗ k + e ∗ j ≺ A ∗ e ∗ k )= β ≺ ◦ α ∗ [ n ∑ m = ( c mjk + d mjk ) e ∗ m ] = n ∑ m = ( c mjk + d mjk ) β ≺ ◦ α ∗ ( e ∗ m )= n ∑ m = ( c mjk + d mjk )( n ∑ l , i , q = f mq b qli e ∗ l ⊗ e ∗ i ) = n ∑ l , i , q = [ n ∑ m = f mq b qli ( c mjk + d mjk )] e ∗ l ⊗ e ∗ i , ( α ∗ ⊗ L ≺ A ∗ ( e ∗ j )) β ≺ ( e ∗ k ) = ( α ∗ ⊗ L ≺ A ∗ ( e ∗ j ))( n ∑ l , m = b klm e ∗ l ⊗ e ∗ m )= n ∑ l , m = b klm α ∗ ( e ∗ l ) ⊗ ( e ∗ j ≺ A ∗ e ∗ m ) = n ∑ l , m , q = b klm f lq e ∗ q ⊗ ( n ∑ i = c ijm e ∗ i )= n ∑ l , i , q = ( n ∑ m = f lq b klm c ijm ) e ∗ q ⊗ e ∗ i , ( R A ∗ ( e ∗ k ) ⊗ α ∗ ) β ≺ ( e ∗ j ) = ( R A ∗ ( e ∗ k ) ⊗ α ∗ )( n ∑ m , i = b jmi e ∗ m ⊗ e ∗ i )= n ∑ m , i , q = f iq b jmi ( e ∗ m ∗ A ∗ e ∗ k ) ⊗ e ∗ q = n ∑ m , i , q = f iq b jmi [( e ∗ m ≻ A ∗ e ∗ k + e ∗ m ≺ A ∗ e ∗ k ) ⊗ e ∗ q ]= n ∑ m , i , q = f iq b jmi [( e ∗ m ≻ A ∗ e ∗ k ) ⊗ e ∗ q + ( e ∗ m ≺ A ∗ e ∗ k ) ⊗ e ∗ q ]= n ∑ m , i , q = f iq b jmi [ n ∑ l = c lmk e ∗ l ⊗ e ∗ q + n ∑ l d lmk e ∗ l ⊗ e ∗ q ]= n ∑ l , i , q = [ n ∑ m = f iq b jmi ( c lmk + d lmk )] e ∗ l ⊗ e ∗ q gives the relation n ∑ m = [ f mq b qli ( c mjk + d mjk ) + f lq b klm c ijm ] = n ∑ m = [ f iq b jmi ( c lmk + d lmk )] . (73) Thus, (72) corresponds to (73). Therefore, (66) ⇔ (68).So, in the case l A = R ∗≺ A , r A = L ∗≻ A , l B = l A ∗ = R ∗≺ A ∗ , r B = r A ∗ = L ∗≻ A ∗ , wehave (10) ⇔ (66) ⇔ (68). Similarly, in this situation,(11) ⇔ (66) ⇔ (69) , (12) ⇔ (66) ⇔ (66) , (13) ⇔ (66) ⇔ (67) , (14) ⇔ (66) ⇔ (71) , (15) ⇔ (66) ⇔ (70) . Therefore, the conclusion holds due to Theorem 1. ⊓⊔ Definition 26.
Let A be a vector space. A Hom-dendriform D-bialgebra struc-ture on A is a set of linear maps ( ∆ ≺ , ∆ ≻ , α , β ≺ , β ≻ , α ∗ ) , ∆ ≺ , ∆ ≻ : A → A ⊗ A , β ≺ , β ≻ : A ∗ → A ∗ ⊗ A ∗ , α : A → A , α ∗ : A ∗ → A ∗ , such thata) ( ∆ ∗≺ , ∆ ∗≻ , α ∗ ) : A ∗ ⊗ A ∗ → A ∗ defines a Hom-dendriform algebra structure ( ≻ A ∗ , ≺ A ∗ , α ∗ ) on A ∗ ;b) ( β ∗≺ , β ∗≻ , α ) : A ⊗ A → A defines a Hom-dendriform algebra structure ( ≻ A , ≺ A , α ) on A ;c) the equations (66) - (71) are satisfied.We denote it by ( A , A ∗ , ∆ ≻ , ∆ ≺ , α , β ≻ , β ≺ , α ∗ ) or simply ( A , A ∗ , α , α ∗ ) . Theorem 15.
Let ( A , ≺ A , ≻ A , α ) and ( A ∗ , ≺ A ∗ , ≻ A ∗ , α ∗ ) be involutive Hom-dendriform algebras. Let ( A , ∗ A , α ) and ( A ∗ , ∗ A ∗ , α ∗ ) be the corresponding as-sociated involutive Hom-associative algebras. Then, the following conditions areequivalent: (i) There is a double construction of involutive symplectic Hom-associative al-gebras associated to ( A , ∗ A , α ) and ( A ∗ , ∗ A ∗ , α ∗ ) ; (ii) ( A , A ∗ , R ∗≺ A , L ∗≻ A , α ∗ , R ∗≺ A ∗ , L ∗≻ A ∗ , α ) is a matched pair of involutive Hom-associative algebras; (iii) ( A , A ∗ , R ∗≻ A + R ∗≺ A , − L ∗≺ A , − R ∗≻ A , L ∗≻ A + L ∗≺ A , α ∗ , R ∗≻ A ∗ + R ∗≺ A ∗ , − L ∗≺ A ∗ , − R ∗≻ A ∗ , L ∗≻ A ∗ + L ∗≺ A ∗ , α ) is a matched pair of involutive Hom-dendriformalgebras; (iv) ( A , A ∗ , α , α ∗ ) is an involutive Hom-dendriform D − bialgebra. Definition 27.
A biHom-dendriform algebra is a quintuple ( A , ≺ , ≻ , α , β ) con-sisting of a vector space A on which the operations ≺ , ≻ : A ⊗ A → A and α , β : A → A are linear maps satisfying ouble constructions of biHom-Frobenius algebras 37 α ◦ β = β ◦ α , α ( x ≺ y ) = α ( x ) ≺ α ( y ) , α ( x ≻ y ) = α ( x ) ≻ α ( y ) , β ( x ≺ y ) = β ( x ) ≺ β ( y ) , β ( x ≻ y ) = β ( x ) ≻ β ( y ) , ( x ≺ y ) ≺ β ( z ) = α ( x ) ≺ ( y ∗ z ) , ( x ≻ y ) ≺ β ( z ) = α ( x ) ≻ ( y ≺ z ) , α ( x ) ≻ ( y ≻ z ) = ( x ∗ y ) ≻ β ( z ) , where x ∗ y = x ≺ y + x ≻ y . Definition 28.
Let ( A , ≺ , ≻ , α , β ) and ( A ′ , ≺ ′ , ≻ ′ , α ′ , β ′ ) be biHom-dendriformalgebras. A linear map f : A → A ′ is a biHom-dendriform algebra morphism if ≺ ′ ◦ ( f ⊗ f ) = f ◦ ≺ , ≻ ′ ◦ ( f ⊗ f ) = f ◦ ≻ , f ◦ α = α ′ ◦ f and f ◦ β = β ′ ◦ f . Proposition 11.
Let ( A , ≺ , ≻ , α , β ) be a biHom-dendriform algebra.Then, ( A , ∗ , α , β ) is a biHom-associative algebra.Proof. We have, for all x , y , z ∈ A , ( x ∗ y ) ∗ β ( z ) = ( x ≺ y ) ≺ β ( z ) + ( x ≺ y ) ≻ β ( z ) + ( x ≻ y ) ≺ β ( z ) + ( x ≻ y ) ≻ β ( z )= ( x ≺ y ) ≺ β ( z ) + ( x ≻ y ) ≺ β ( z ) + ( x ≺ y ) ≻ β ( z ) + ( x ≻ y ) ≻ β ( z )= ( x ≺ y ) ≺ β ( z ) + ( x ≻ y ) ≺ β ( z ) + ( x ∗ y ) ≻ β ( z )= α ( x ) ≺ ( y ∗ z ) + α ( x ) ≻ ( y ≺ z ) + α ( x ) ≻ ( y ≻ z )= α ( x ) ≺ ( y ∗ z ) + α ( x ) ≻ ( y ∗ z ) = α ( x ) ∗ ( y ∗ z ) , α ( x ∗ y ) = α ( x ≻ y ) + α ( x ≺ y )= α ( x ) ≻ α ( y ) + α ( x ) ≺ α ( y )= α ( x ) ∗ α ( y ) , which completes the proof. ⊓⊔ We call ( A , ∗ , α , β ) the biHom-associative algebra of ( A , ≺ , ≻ , α , β ) , and thequintuple ( A , ≻ , ≺ , α , β ) is called a compatible biHom-dendriform algebra struc-ture on the biHom-associative algebra ( A , ∗ , α , β ) . Proposition 12.
Let ( A , ≺ , ≻ , α , β ) be a biHom-dendriform algebra. Suppose that ( A , ∗ , β , α ) is a biHom-associative algebra. Then, ( L ≻ , R ≺ , β , α , A ) is a bimoduleof ( A , ∗ , β , α ) .Proof. For x , y , v ∈ A , we have L ≻ ( x ∗ y ) β ( v ) = ( x ∗ y ) ≻ β ( v ) = α ( x ) ≻ ( y ≻ v ) = L ≻ ( α ( x )) L ≻ ( y ) v , R ≺ ( x ∗ y ) α ( v ) = α ( v ) ≺ ( x ∗ y ) = ( v ≺ x ) ≺ β ( y ) = R ≺ ( β ( y )) R ≺ ( x ) v , L ≻ ( α ( x )) R ≺ ( y ) v = α ( x ) ≻ ( v ≺ y ) = ( x ≻ v ) ≺ β ( y ) = R ≺ ( β ( y )) L ≻ ( x ) v , which completes the proof. ⊓⊔ Remark 4. If ( A , ≺ , ≻ , α , β ) is a biHom-dendriform algebra, ( L ≻ , R ≺ , α , β , A ) isnot a bimodule of associated biHom-associative algebra ( A , ∗ , α , β ) . Proposition 13.
Let ( A , ≺ , ≻ , α , β ) be a biHom-dendriform algebra, and suppose α = β = α ◦ β = β ◦ α = Id . Then, ( A , ≺ , ≻ , α , β ) ∼ = ( A , ≺ , ≻ , β , α ) . Proof.
Let x , y , z ∈ A . We have α ( x )( yz ) = ( xy ) β ( z ) ⇔ α (( α ◦ β )( x ))( yz ) = ( xy ) β (( β ◦ α )( z )) ⇔ α ( β ( x ))( yz ) = ( xy ) β ( α ( z )) ⇔ β ( x )( yz ) = ( xy ) α ( z ) . Then ( A , ≺ , ≻ , α , β ) ∼ = ( A , ≺ , ≻ , β , α ) . ⊓⊔ O -operators and biHom-dendriform algebras Definition 29.
Let ( A , · , α , α ) be a biHom-associative algebra, and ( l , r , β , β , V ) be a bimodule. A linear map T : V → A is called an O -operator associated to ( l , r , β , β , V ) , if T satisfies α T = T β , α T = T β and T ( u ) · T ( v ) = T ( l ( T ( u )) v + r ( T ( v )) u ) for all u , v ∈ V . Example 10.
Let ( A , · , α , α ) be a multiplicative biHom-associative algebra. Then,the identity map Id is an O -operator associated to the bimodule ( L , , α , α ) or ( , R , α , α ) . Example 11.
Let ( A , · , α , β ) be a multiplicative biHom-associative algebra. A lin-ear map f : A → A is called a Rota-Baxter operator on A of weight zero if f satisfies f ◦ α = α ◦ f , f ◦ β = β ◦ f and f ( x ) · f ( y ) = f ( f ( x ) · y + x · f ( y )) for all x , y ∈ A . A Rota-Baxter operator on A is just an O -operator associated to the bimodule ( L , R , α , β ) . Theorem 16.
Let ( A , · , α , α ) be a biHom-associative algebra, and ( l , r , β , β , V ) be a bimodule. Let T : V → A be an O -operator associated to ( l , r , β , β , V ) . Then,there exists a biHom-dendriform algebra structure on V given byu ≻ v = l ( T ( u )) v , u ≺ v = r ( T ( v )) ufor all u , v ∈ V . So, there is an associated biHom-associative algebra structure onV given by the equation (43) , and T is a homomorphism of biHom-associative alge-bras. Moreover, T ( V ) = { T ( v ) \ v ∈ V } ⊂ A is a biHom-associative subalgebra of A , and there is an induced biHom-dendriform algebra structure on T ( V ) given byT ( u ) ≻ T ( v ) = T ( u ≻ v ) , T ( u ) ≺ T ( v ) = T ( u ≺ v ) (74) ouble constructions of biHom-Frobenius algebras 39 for all u , v ∈ V . Its corresponding associated biHom-associative algebra structureon T ( V ) given by the equation (43) is just the biHom-associative subalgebra struc-ture of A , and T is a homomorphism of biHom-dendriform algebras.Proof. For any x , y , z ∈ V , we have ( x ≻ y ) ≺ β ( z ) − β ( x ) ≻ ( y ≺ z ) = l ( T ( x ) y ) ≺ β ( z ) − β ( x ) ≻ r ( T ( z ) y )= r ( T β ( z )) l ( T ( x )) y − l ( T β ( x ) y ) r ( T ( z ) y )= r ( α ( T ( z ))) l ( T ( x )) y − l ( α ( T ( x ))) r ( T ( z )) y = . The two other axioms are checked in a similar way. ⊓⊔ Corollary 7.
Let ( A , ∗ , α , β ) be a multiplicative biHom-associative algebra. Thereis a compatible multiplicative biHom-dendriform algebra structure on A if and onlyif there exists an invertible O -operator of ( A , ∗ , α , β ) .Proof. In fact, if the homomorphism T is an invertible O − operator associated to abimodule ( l , r , α , β , V ) , then the compatible multiplicative biHom-dendriform alge-bra structure on A is given by x ≻ y = T ( l ( x ) T − ( y )) , x ≺ y = T ( r ( y ) T − ( x )) for all x , y ∈ A . Conversely, let ( A , ≻ , ≺ , α , β ) be a multiplicative biHom-dendriform algebra, and ( A , ∗ , α , β ) be its associated biHom-associative algebra. Then, the identity map Idis an O − operator associated to the bimodule ( L ≻ , R ≺ , α , β ) of ( A , ∗ , α , β ) . ⊓⊔ Definition 30.
Let ( A , ≻ , ≺ , α , α ) be a biHom-dendriform algebra, and V be avector space. Let l ≻ , r ≻ , l ≺ , r ≺ : A → gl ( V ) , and β , β : V → V be six linear maps.Then, ( l ≻ , r ≻ , l ≺ , r ≺ , β , β , V ) is called a bimodule of A if the following equationshold for any x , y ∈ A and v ∈ V : l ≺ ( x ≺ y ) β ( v ) = l ≺ ( α ( x )) l ∗ ( y ) v , r ≺ ( α ( x )) l ≺ ( y ) v = l ≺ ( α ( y )) r ∗ ( x ) v , r ≺ ( α ( y )) r ≺ ( y ) v = r ≺ ( x ∗ y ) β ( v ) , l ≺ ( x ≻ y ) β ( v ) = l ≻ ( α ( x )) l ≺ ( y ) v , r ≺ ( α ( x )) l ≻ ( y ) v = l ≻ ( α ( y )) r ≺ ( x ) v , r ≺ ( α ( x )) r ≻ ( y ) v = r ≻ ( y ≺ x ) β ( v ) , l ≻ ( x ∗ y ) β ( v ) = l ≻ ( α ( x )) l ≻ ( y ) v , r ≻ ( α ( x )) l ∗ ( y ) v = l ≻ ( α ( y )) r ≻ ( x ) v , r ≻ ( α ( x )) r ∗ ( y ) v = r ≻ ( y ≻ x ) β ( v ) , where x ∗ y = x ≻ y + x ≺ y , l ∗ = l ≻ + l ≺ , r ∗ = r ≻ + r ≺ . Proposition 14.
Let ( l ≻ , r ≻ , l ≺ , r ≺ , β , β , V ) be a bimodule of a biHom-dendriformalgebra ( A , ≻ , ≺ , α , α ) . Then, there exists a biHom-dendriform algebra structureon the direct sum A ⊕ V of the underlying vector spaces of A and V given by ( x + u ) ≻ ( y + v ) = x ≻ y + l ≻ ( x ) v + r ≻ ( y ) u , ( x + u ) ≺ ( y + v ) = x ≺ y + l ≺ ( x ) v + r ≺ ( y ) ufor all x , y ∈ A , u , v ∈ V . We denote it by A × l ≻ , r ≻ , l ≺ , r ≺ , α , α , β , β V .Proof.
Let v , v , v ∈ V and x , x , x ∈ A . Setting and computing [( x + v ) ≺ ( x + v )] ≺ ( α ( x ) + β ( v )) =( α ( x ) + β ( v )) ≺ [( x + v ) ∗ ( x + v )] , [( x + v ) ≻ ( x + v )] ≺ ( α ( x ) + β ( v )) =( α ( x ) + β ( v )) ≻ [( x + v ) ≺ ( x + v )] , [ α ( x ) + β ( v )] ≻ [( x ) + v ) ≻ ( x + v )] =[( x + v ) ∗ ( x + v )] ≻ ( α ( x ) + β ( v )) , (75)one obtains the conditions of the bimodule of a biHom-dendriform algebra, whichcompletes the proof. ⊓⊔ Proposition 15.
Let (l ≻ , r ≻ , l ≺ , r ≺ , β , β , V ) be a bimodule of a biHom-dendriformalgebra ( A , ≻ , ≺ , α , α ) . Then ( l ≻ , r ≺ , β , β , V ) and ( l ≻ + l ≺ , r ≻ + r ≺ , β , β , V ) are bimodulesof ( A , ∗ , α , α ) ;2) for any bimodule ( l , r , β , β , V ) of ( A , ∗ , α , α ) , ( l , , , r , β , β , V ) is a bi-module of ( A , ≻ , ≺ , α , α ) . ( l ≻ + l ≺ , , , r ≻ + r ≺ , β , β , V ) and ( l ≻ , , , r ≺ , β , β , V ) are bimodules of ( A , ≻ , ≺ , α , α ) ;4) the biHom-dendriform algebras A × l ≻ , r ≻ , l ≺ , r ≺ , α , α , β , β V and A × l ≻ + l ≺ , , , r ≻ + r ≺ , α , α , β , β Vhave the same associated biHom-associative algebra A × l ≻ + l ≺ , r ≻ + r ≺ , α , α , β , β V . Theorem 17.
Let ( A , ≻ A , ≺ A , α , α ) and ( B , ≻ B , ≺ B , β , β ) be two biHom-dendriform algebras. Suppose that there are linear maps l ≻ A , r ≻ A , l ≺ A , r ≺ A : A → gl ( B ) , and l ≻ B , r ≻ B , l ≺ B , r ≺ B : B → gl ( A ) such that (l ≻ A , r ≻ A , l ≺ A , r ≺ A , β , β , B ) is a bimodule of A , and (l ≻ B , r ≻ B , l ≺ B , r ≺ B , α , α , A ) ) is a bimodule of B , satisfy r ≺ A ( α ( x ))( a ≺ B b ) = β ( a ) ≺ B ( r A ( x ) b ) + r ≺ A ( l B ( x ) β ( a )) , (76) l ≺ A ( l ≺ B ( x )) β ( b ) + ( r ≺ A ( x ) a ) ≺ B β ( b ) = β ( a ) ≺ B ( l ≺ A ( x ) b ) + r ≺ A ( r ≺ B ( b ) x ) β ( a ) , (77) l ≺ A ( α ( x ))( a ∗ B b ) = ( l ≺ A ( x ) a ) ∗ B β ( b ) + l ≺ A ( r ≺ A ( a ) x ) β ( b ) , (78) ouble constructions of biHom-Frobenius algebras 41 r ≺ A ( α ( x ))( a ≻ B b ) = r ≻ A ( l ≺ B ( b ) x ) β ( a ) + β ( a ) ≻ B ( r ≺ A ( x ) b ) , (79) l ≺ A ( l ≻ B ( a ) x ) β ( b ) +( r ≻ A ( x ) a ) ≺ B β ( b ) = β ( a ) ≻ B ( l ≺ A ( x ) b ) + r ≻ A ( r ≺ B ( b ) x ) β ( a ) (80) l ≻ A ( α ( x ))( a ≺ B b ) = ( l ≻ A ( x ) a ) ≺ B β ( b ) + l ≺ A ( r ≻ B ( a ) x ) β ( b ) , (81) r ≻ A ( α ( x ))( a ∗ B b ) = β ( a ) ≻ B ( r ≻ A ( x ) b ) + r ≻ A ( l ≻ B ( b ) x ) β ( a ) , (82) β ( a ) ≻ B ( l ≻ A ( x ) b ) + r ≻ A ( r ≻ B ( b ) x ) β ( a ) = l ≻ A ( l B ( a ) x ) β ( b ) + ( r A ( x ) a ) ≻ B β ( b ) , (83) l ≻ A ( α ( x ))( a ≻ B b ) = ( l A ( x ) a ) ≻ B β ( b ) + l ≻ A ( r B ( a ) x ) β ( b ) , (84) r ≺ B ( β ( a ))( x ≺ A y ) = α ( x ) ≺ A ( r B ( a ) y ) + r ≺ B ( l A ( y ) a ) α ( x ) , (85) l ≺ B ( l ≺ A ( x ) a ) α ( y ) +( r ≺ B ( a ) x ) ≺ A α ( y ) = α ( x ) ≺ A ( l B ( a ) y ) + r ≺ B ( r A ( y ) a ) α ( x ) , (86) l ≺ B ( β ( a ))( x ∗ A y ) = ( l ≺ B ( a ) x ) ≺ A α ( y ) + l ≺ B ( r ≺ A ( x ) a ) α ( y ) , (87) r ≺ B ( β ( a ))( x ≻ A y ) = r ≻ B ( l ≺ B ( y ) a ) α ( x ) + α ( x ) ≻ A ( r ≺ B ( a ) y ) , (88) l ≺ B ( l ≻ A ( x ) a ) α ( y ) +( r ≻ B ( a ) x ) ≺ A α ( y ) = α ( x ) ≻ A ( l ≺ B ( a ) y ) + r ≻ B ( r ≺ A ( y ) a ) α ( x ) , (89) l ≻ B ( β ( a ))( x ≺ A y ) = ( l ≻ B ( a ) x ) ≺ A α ( y ) + l ≺ B ( r ≻ A ( x ) a ) α ( y ) , (90) r ≻ B ( β ( a ))( x ∗ A y ) = α ( x ) ≻ A ( r ≻ B ( a ) y ) + r ≻ B ( l ≻ A ( y ) a ) α ( x ) , (91) α ( x ) ≻ A ( l ≻ B ( a ) y ) + r ≻ B ( r ≻ A ( y ) a ) α ( x ) = l ≻ B ( l A ( x ) a ) α ( y ) + ( r B ( a ) x ) ≻ A α ( y ) , (92) l ≻ B ( β ( a ))( x ≻ A y ) = ( l B ( a ) x ) ≻ A α ( y ) + l ≻ B ( r A ( x ) a ) α ( y ) (93) for any x , y ∈ A , a , b ∈ B and l A = l ≻ A + l ≺ A , r A = r ≻ A + r ≺ A , l B = l ≻ B + l ≺ B , r B = r ≻ B + r ≺ B . Then, there is a biHom-dendriform algebra structure on thedirect sum A ⊕ B of the underlying vector spaces of A and B given by ( x + a ) ≻ ( y + b ) = ( x ≻ A y + r ≻ B ( b ) x + l ≻ B ( a ) y )+ ( l ≻ A ( x ) b + r ≻ A ( y ) a + a ≻ B b ) , ( x + a ) ≺ ( y + b ) = ( x ≺ A y + r ≺ B ( b ) x + l ≺ B ( a ) y )+ ( l ≺ A ( x ) b + r ≺ A ( y ) a + a ≺ B b ) for any x , y ∈ A , a , b ∈ B .Proof. The proof is obtained in a similar way as for Theorem 1. ⊓⊔ Let A ⊲⊳ l ≻ A , r ≻ A , l ≺ A , r ≺ A , β , β l ≻ B , r ≻ B , l ≺ B , r ≺ B , α , α B denote this biHom-dendriform algebra. Definition 31.
Let ( A , ≻ A , ≺ A , α , α ) and ( B , ≻ B , ≺ B , β , β ) be two biHom-dendriform algebras. Suppose there exist linear maps l ≻ A , r ≻ A , l ≺ A , r ≺ A : A → gl ( B ) , and l ≻ B , r ≻ B , l ≺ B , r ≺ B : B → gl ( A ) such that ( l ≻ A , r ≻ A , l ≺ A , r ≺ A , β , β ) is a bimodule of A , and ( l ≻ B , r ≻ B , l ≺ B , r ≺ B , α , α ) is a bimodule of B . If (76)- (93) are satisfied, ( A , B , l ≻ A , r ≻ A , l ≺ A , r ≺ A , β , β , l ≻ B , r ≻ B , l ≺ B , r ≺ B , α , α ) iscalled a matched pair of biHom-dendriform algebras . Corollary 8.
Let ( A , B , l ≻ A , r ≻ A , l ≺ A , r ≺ A , β , β , l ≻ B , r ≻ B , l ≺ B , r ≺ B , α , α ) bea matched pair of biHom-dendriform algebras. Then, ( A , B , l ≻ A + l ≺ A , r ≻ A + r ≺ A , l ≻ B + l ≺ B , r ≻ B + r ≺ B , α + β , α + β ) is a matched pair of the associatedbiHom-associative algebras ( A , ∗ A , α , α ) and ( B , ∗ B , β , β ) .Proof. The associated biHom-associative algebra ( A ⊲⊳ B , ∗ , α + β , α + β ) isexactly the biHom-associative algebra obtained from the matched pair of biHom-associative algebras, ( A , B , l A , r A , β , β , l B , r B , α , α ) , with ( x + a ) ∗ ( y + b ) = x ∗ A y + l B ( a ) y + r B ( b ) x + a ∗ B b + l A ( x ) b + r A ( y ) a for all x , y ∈ A , a , b ∈ B , where l A = l ≻ A + l ≺ A , r A = r ≻ A + r ≺ A , l B = l ≻ B + l ≺ B , r B = r ≻ B + r ≺ B . ⊓⊔ In this work, we have constructed a biHom-associative algebra with a decomposi-tion into direct sum of the underlying vector spaces of a biHom-associative algebraand its dual such that both of them are biHom-subalgebras, with either the natu-ral symmetric bilinear form being invariant, or the natural antisymmetric bilinearform being a Connes cocycle. Then, we have performed the double constructions ofbiHom-Frobenius algebras and Connes cocycle, and provided the bialgebra struc-tures.
Aknowledgement
This work is supported by TWAS Research Grant RGA No. 17 - 542 RG / MATHS/ AF / AC _G -FR3240300147. The ICMPA-UNESCO Chair is in partnership withDaniel Iagolnitzer Foundation (DIF), France, and the Association pour la Promo-tion Scientifique de l’Afrique (APSA), supporting the development of mathematicalphysics in Africa. Mahouton Norbert Hounkonnou thanks Professor Sergei D. Sil-vestrov for the invitation to attend the 2nd International Conference on StochasticProcesses and Algebraic Structures (SPAS2019) and the hospitality during his stayas visiting professor at the Mathematics and Applied Mathematics research envi-ronment MAM, Division of Applied Mathematics, Mälardalen University, Västerás,Sweden, where this work has been finalized. Partial support from the Swedish Inter-national Development Agency and International Science Program, (ISP) for capasitybuildning in Mathematics in Africa is also gratefully acknowledged. ouble constructions of biHom-Frobenius algebras 43
References
1. Aizawa, N., Sato, H.: q -deformation of the Virasoro algebra with central extension, Phys. Lett.B , 185-190 (1991) (Hiroshima University preprint, preprint HUPD-9012 (1990))2. Ammar, F., Ejbehi, Z., Makhlouf, A., Cohomology and deformations of Hom-algebras, J. LieTheory , no. 4, 813–836 (2011)3. Armakan, A., Silvestrov, S., Farhangdoost, M.: Enveloping algebras of color hom-Lie alge-bras, Turk. J. Math. , 316-339 (2019). (arXiv:1709.06164 [math.QA], (2017))4. Armakan, A., Silvestrov, S.: Enveloping algebras of certain types of color Hom-Lie alge-bras, In: Silvestrov, S., Malyarenko, A., Ran˘ci´c, M. (eds.), Algebraic Structures and Applica-tions, Springer Proceedings in Mathematics and Statistics, vol. 317, Ch. 10, 257–284, Springer(2020)5. Bai, C.: Double constructions of Frobenius algebras, Connes cocycle and their duality. J. Non-commut. Geom. , 475-530 (2010).6. Bakayoko, I.: Laplacian of Hom-Lie quasi-bialgebras, International Journal of Algebra, (15),713-727 (2014)7. Bakayoko, I.: L -modules, L -comodules and Hom-Lie quasi-bialgebras, African Diaspora Jour-nal of Mathematics, (5)(2015)9. Bakayoko, I., Silvestrov, S.: Multiplicative n -Hom-Lie color algebras, In: Silvestrov,S., Malyarenko, A., Ran˘ci´c, M. (Eds.), Algebraic Structures and Applications, SpringerProceedings in Mathematics and Statistics, Vol 317, Ch. 7, 159-187, Springer (2020).(arXiv:1912.10216[math.QA])10. Bakayoko, I., Silvestrov, S.: Hom-left-symmetric color dialgebras, Hom-tridendriform coloralgebras and Yau’s twisting generalizations, 24pp, arXiv:1912.01441[math.RA] (2019)11. Benayadi, S., Makhlouf, A.: Hom-Lie algebras with symmetric invariant nondegenerate bilin-ear forms, J. Geom. Phys. , 38–60 (2014)12. Ben Abdeljelil, A., Elhamdadi, M., Kaygorodov, I., Makhlouf, A.: Generalized Derivationsof n -BiHom-Lie algebras, In: Silvestrov, S., Malyarenko, A., Ran˘ci´c, M. (Eds.), AlgebraicStructures and Applications, Springer Proceedings in Mathematics and Statistics, Vol 317,Ch. 4, 81-97, Springer (2020). (arXiv:1901.09750[math.RA])13. Bordemann, M., Filk, T., Nowak, C.: Algebraic classification of actions invariant under gen-eralized flip moves of 2-dimensional graph, J. Math. Phys. , 4964-4988 (1994)14. Caenepeel S., Goyvaerts I.: Monoidal Hom-Hopf Algebras, Comm. Algebra no. 6,2216–2240 (2011)15. Chaichian, M., Ellinas, D., Popowicz, Z.: Quantum conformal algebra with central extension,Phys. Lett. B , 95-99 (1990)16. Chaichian, M., Isaev, A. P., Lukierski, J., Popowic, Z., Prešnajder, P.: q -deformations of Vira-soro algebra and conformal dimensions, Phys. Lett. B (1), 32-38 (1991)17. Chaichian, M., Kulish, P., Lukierski, J.: q -deformed Jacobi identity, q -oscillators and q -deformed infinite-dimensional algebras, Phys. Lett. B , 401-406 (1990)18. Chaichian, M., Popowicz, Z., Prešnajder, P.: q -Virasoro algebra and its relation to the q -deformed KdV system, Phys. Lett. B , 63-65 (1990)19. Curtright, T. L., Zachos, C. K.: Deforming maps for quantum algebras, Phys. Lett. B ,237-244 (1990)20. Damaskinsky, E. V., Kulish, P. P.: Deformed oscillators and their applications (in Russian),Zap. Nauch. Semin. LOMI 189, 37-74 (1991) [Engl. transl. in J. Sov. Math., 62, 2963-2986(1992)21. Daskaloyannis, C.: Generalized deformed Virasoro algebras, Modern Phys. Lett. A no. 9,809-816 (1992)22. Graziani, G., Makhlouf, A., Menini, C., Panaite, F.: BiHom-Associative Algebras, BiHom-LieAlgebras and BiHom-Bialgebras, SIGMA 11, 086, 34 pp (2015)4 Mahouton Norbert Hounkonnou, Gbêvèwou Damien Houndedji, Sergei Silvestrov23. Guo, L., Zhang B., Zheng, S.: Universal enveloping algebras and Poincare-Birkhoff-Witt theorem for involutive Hom-Lie algebras, J. Lie Theory, (3), 735-756 (2018).arXiv:1607.05973[math.QA], (2016)24. Hassanzadeh, M., Shapiro, I., Sütlü, S.: Cyclic homology for Hom-associative algebras, J.Geom. Phys. , 40-56 (2015)25. Hartwig, J. T., Larsson, D., Silvestrov, S. D.: Deformations of Lie algebras using σ -derivations, J. Algebra, , 314-361 (2006) (Preprint in Mathematical Sciences 2003:32,LUTFMA-5036-2003, Centre for Mathematical Sciences, Department of Mathematics, LundInstitute of Technology, 52 pp. (2003))26. Hellström, L.: Strong Hom-associativity, In: Silvestrov, S., Malyarenko, A., Ran˘ci´c, M. (eds.),Algebraic Structures and Applications, Springer Proceedings in Mathematics and Statistics,vol. 317, 317–337, Springer (2020)27. Hellström, L., Makhlouf, A., Silvestrov, S. D.: Universal Algebra Applied to Hom-AssociativeAlgebras, and More, In: Makhlouf A., Paal E., Silvestrov S., Stolin A. (eds), Algebra, Geom-etry and Mathematical Physics. Springer Proceedings in Mathematics and Statistics, vol 85.Springer, Berlin, Heidelberg, 157-199 (2014)28. Hu, N.: q -Witt algebras, q -Lie algebras, q -holomorph structure and representations, AlgebraColloq. , no. 1, 51-70 (1999)29. Kassel, C.: Cyclic homology of differential operators, the virasoro algebra and a q -analogue,Comm. Math. Phys. 146 (2), 343-356 (1992)30. Kitouni, A., Makhlouf, A., Silvestrov, S.: On n -ary generalization of BiHom-Lie algebrasand BiHom-associative algebras, In: Silvestrov, S., Malyarenko, A., RancicRan˘ci´c, M. (Eds.),Algebraic Structures and Applications, Springer Proceedings in Mathematics and Statistics,Vol 317, Ch 5, 99-126, Springer (2020)31. Kock, J.: Frobenius algebras and 2D topological quantum field theories. London Math. Soc.Stud. Texts, 59, Cambrige University Press, Cambrige (2004)32. Larsson, D., Sigurdsson, G., Silvestrov, S. D.: Quasi-Lie deformations on the algebra F [ t ] / ( t N ) , J. Gen. Lie Theory Appl. 2, 201-205 (2008)33. Larsson, D., Silvestrov, S. D.: Quasi-Hom-Lie algebras, Central Extensions and 2-cocycle-like identities, J. Algebra , 1473-1478(2005)36. Larsson, D., Silvestrov, S. D.: Quasi-deformations of sl ( F ) using twisted derivations, Comm.Algebra , 4303-4318 (2007) (Preprint in Mathematical Sciences 2004:26, LUTFMA-5047-2004, Centre for Mathematical Sciences, Lund Institute of Technology, Lund University(2004). arXiv:math/0506172 [math.RA] (2005))37. Liu, K. Q.: Quantum central extensions, C. R. Math. Rep. Acad. Sci. Canada (4), 135-140(1991)38. Liu, K. Q.: Characterizations of the Quantum Witt Algebra, Lett. Math. Phys. (4), 257-265(1992)39. Liu, K. Q.: The Quantum Witt Algebra and Quantization of Some Modules over Witt Algebra,PhD Thesis, Department of Mathematics, University of Alberta, Edmonton, Canada (1992)40. Ma, T., Makhlouf, A., Silvestrov, S.: Curved O -operator systems, 17pp, arXiv:1710.05232[math.RA] (2017)41. Ma, T., Makhlouf, A., Silvestrov, S.: Rota-Baxter bisystems and covariant bialgebras, 30 pp,(2017). arXiv:1710.05161[math.RA]ouble constructions of biHom-Frobenius algebras 4542. Ma, T., Makhlouf, A., Silvestrov, S.: Rota-Baxter Cosystems and CoquasitriangularMixed Bialgebras, J. Algebra Appl. Accepted 2019. (Puiblished first online 2020:doi:10.1142/S021949882150064X)43. Makhlouf, A., Silvestrov, S. D.: Hom-algebra structures, J. Gen. Lie Theory Appl. (4), 715-739 (2010). (arXiv:0712.3130 [math.RA])46. Makhlouf, A., Silvestrov, S. D.: Hom-Algebras and Hom-Coalgebras, J. Algebra Appl. (04),553-589 (2010). (arXiv:0811.0400[math.RA])47. Richard, L., Silvestrov, S. D.: Quasi-Lie structure of σ -derivations of C [ t ± ] , J. Algebra σ -Derivationsof C [ z ± , . . ., z ± n ] , In: Silvestrov, S., Paal, E., Abramov, V., Stolin, A. (Eds.), GeneralizedLie Theory in Mathematics, Physics and Beyond, Ch. 22, 257-262, Springer-Verlag, Berlin,Heidelberg (2009)49. Sheng, Y.: Representation of Hom-Lie algebras, Algebr. Reprensent. Theory , no. 6, 1081-1098 (2012)50. Sigurdsson, G., Silvestrov, S.: Graded quasi-Lie algebras of Witt type, Czech. J. Phys. 56:1287-1291 (2006)51. Sigurdsson, G., Silvestrov, S.: Lie color and Hom-Lie algebras of Witt type and their centralextensions, In: Silvestrov, S., Paal, E., Abramov, V., Stolin, A. (Eds.), Generalized Lie Theoryin Mathematics, Physics and Beyond, Ch. 21, 247-255, Springer-Verlag, Berlin, Heidelberg(2009)52. Silvestrov, S.: Paradigm of quasi-Lie and quasi-Hom-Lie algebras and quasi-deformations.In "New techniques in Hopf algebras and graded ring theory", K. Vlaam. Acad. Belgie Wet.Kunsten (KVAB), Brussels, 165-177 (2007)53. Yau, D.: Module Hom-algebras, (2008). arXiv:0812.4695[math.RA]54. Yau, D.: Enveloping algebras of Hom-Lie algebras, J. Gen. Lie Theory Appl. , no. 2, 95-108(2008). arXiv:0709.0849 [math.RA]55. Yau, D.: Hom-algebras and homology, Journal of Lie Theory 19, No. 2, 409-421 (2009)56. Yau, D.: Hom-bialgebras and comodule Hom-algebras, Int. Electron. J. Algebra8