BiHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras
Giacomo Graziani, Abdenacer Makhlouf, Claudia Menini, Florin Panaite
SSymmetry, Integrability and Geometry: Methods and Applications SIGMA (2015), 086, 34 pages BiHom-Associative Algebras, BiHom-Lie Algebrasand BiHom-Bialgebras
Giacomo GRAZIANI † , Abdenacer MAKHLOUF † , Claudia MENINI † and Florin PANAITE † † Universit´e Joseph Fourier Grenoble I Institut Fourier,100, Rue des Maths BP74 38402 Saint-Martin-d’H`eres, France
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[email protected] † Universit´e de Haute Alsace, Laboratoire de Math´ematiques, Informatique et Applications,4, Rue des fr`eres Lumi`ere, F-68093 Mulhouse, France
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[email protected] † University of Ferrara, Department of Mathematics,Via Machiavelli 30, Ferrara, I-44121, Italy
E-mail: [email protected] † Institute of Mathematics of the Romanian Academy,PO-Box 1-764, RO-014700 Bucharest, Romania
E-mail: fl[email protected]
Received May 12, 2015, in final form October 13, 2015; Published online October 25, 2015http://dx.doi.org/10.3842/SIGMA.2015.086
Abstract.
A BiHom-associative algebra is a (nonassociative) algebra A endowed with twocommuting multiplicative linear maps α, β : A → A such that α ( a )( bc ) = ( ab ) β ( c ), for all a, b, c ∈ A . This concept arose in the study of algebras in so-called group Hom-categories. Inthis paper, we introduce as well BiHom-Lie algebras (also by using the categorical approach)and BiHom-bialgebras. We discuss these new structures by presenting some basic propertiesand constructions (representations, twisted tensor products, smash products etc). Key words:
BiHom-associative algebra; BiHom-Lie algebra; BiHom-bialgebra; representa-tion; twisting; smash product
The origin of Hom-structures may be found in the physics literature around 1990, concerning q -deformations of algebras of vector fields, especially Witt and Virasoro algebras, see for in-stance [1, 10, 12, 19]. Hartwig, Larsson and Silvestrov studied this kind of algebras in [15, 18]and called them Hom-Lie algebras because they involve a homomorphism in the defining iden-tity. More precisely, a Hom-Lie algebra is a linear space L endowed with two linear maps[ − ] : L ⊗ L → L and α : L → L such that [ − ] is skew-symmetric and α is an algebra endomor-phism with respect to the bracket satisfying the so-called Hom-Jacobi identity[ α ( x ) , [ y, z ]] + [ α ( y ) , [ z, x ]] + [ α ( z ) , [ x, y ]] = 0 , ∀ x, y, z ∈ L. Since any associative algebra becomes a Lie algebra by taking the commutator [ a, b ] = ab − ba , itwas natural to look for a Hom-analogue of this property. This was accomplished in [24], wherethe concept of Hom-associative algebra was introduced, as being a linear space A endowed witha multiplication µ : A ⊗ A → A , µ ( a ⊗ b ) = ab , and a linear map α : A → A satisfying the a r X i v : . [ m a t h . R A ] O c t G. Graziani, A. Makhlouf, C. Menini and F. Panaiteso-called Hom-associativity condition α ( a )( bc ) = ( ab ) α ( c ) , ∀ a, b, c ∈ A. If A is Hom-associative then ( A, [ a, b ] = ab − ba, α ) becomes a Hom-Lie algebra, denoted by L ( A ).Notice that Hom-Lie algebras, in this paper, were considered without the assumption of multi-plicativity of α .In subsequent literature (see for instance [30]) were studied subclasses of these classes of alge-bras where the linear maps α involved in the definition of a Hom-Lie algebra or Hom-associativealgebra are required to be multiplicative, that is α ([ x, y ]) = [ α ( x ) , α ( y )] for all x, y ∈ L , respec-tively α ( ab ) = α ( a ) α ( b ) for all a, b ∈ A , and these subclasses were called multiplicative Hom-Liealgebras, respectively multiplicative Hom-associative algebras. Since we will always assume mul-tiplicativity of the maps α and to simplify terminology, we will call Hom-Lie or Hom-associativealgebras what was called above multiplicative Hom-Lie or Hom-associative algebras.The Hom-analogues of coalgebras, bialgebras and Hopf algebras have been introducedin [25, 26]. The original definition of a Hom-bialgebra involved two linear maps, one twistingthe associativity condition and the other one the coassociativity condition. Later, two directionsof study on Hom-bialgebras were developed, one in which the two maps coincide (these are stillcalled Hom-bialgebras) and another one, started in [8], where the two maps are assumed to beinverse to each other (these are called monoidal Hom-bialgebras).In the last years, many concepts and properties from classical algebraic theories have beenextended to the framework of Hom-structures, see for instance [2, 3, 7, 8, 11, 16, 20, 22, 25, 26,27, 31, 30].The main tool for constructing examples of Hom-type algebras is the so-called “twistingprinciple” introduced by D. Yau for Hom-associative algebras and extended afterwards to othertypes of Hom-algebras. For instance, if A is an associative algebra and α : A → A is an algebramap, then A with the new multiplication defined by a ∗ b = α ( a ) α ( b ) is a Hom-associativealgebra, called the Yau twist of A .A categorical interpretation of Hom-associative algebras has been given by Caenepeel andGoyvaerts in [8]. First, to any monoidal category C they associate a new monoidal category (cid:101) H ( C ),called a Hom-category, whose objects are pairs consisting of an object of C and an automorphismof this object ( (cid:101) H ( C ) has nontrivial associativity constraint even if the one of C is trivial). Bytaking C to be k M , the category of linear spaces over a base field k , it turns out that analgebra in the (symmetric) monoidal category (cid:101) H ( k M ) is the same thing as a Hom-associativealgebra ( A, µ, α ) with bijective α . The bialgebras in (cid:101) H ( k M ) are the monoidal Hom-bialgebraswe mentioned before.In [14], the first author extended the construction of the Hom-category (cid:101) H ( C ) to include theaction of a given group G . Namely, given a monoidal category C , a group G , two elements c, d ∈ Z ( G ) and ν an automorphism of the unit object of C , the group Hom-category H c,d,ν ( G , C )has as objects pairs ( A, f A ), where A is an object in C and f A : G →
Aut C ( A ) is a group homo-morphism. The associativity constraint of H c,d,ν ( G , C ) is naturally defined by means of c , d , ν (see Claim 2.3 and Theorem 2.4) and it is, in general, non trivial. A braided structure is alsodefined on H c,d,ν ( G , C ) (see Claim 2.7 and Theorem 2.8) turning it into a braided category whichis symmetric whenever C is. When G = Z , c = d = 1 Z and ν = id one gets the category H ( C )from [8], while for c = 1 Z , d = − Z and ν = id one gets the category (cid:101) H ( C ).We first look at the case when G = Z × Z , c = (1 , d = (0 , ν = id and C = k M .If M ∈ k M , a group homomorphism f M : Z × Z → Aut k ( M ) is completely determined by f M ((1 , α M and f M ((0 , β − M . Thus, an object in H ( Z × Z , k M ) identifies with a triple ( M, α M , β M ), where α M , β M ∈ Aut k ( M )and α M ◦ β M = β M ◦ α M . For ( X, α X , β X ), ( Y, α Y , β Y ), ( Z, α Z , β Z ) objects in the categoryiHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras 3 H (1 , , (0 , , ( Z × Z , k M ), the associativity constraint in H (1 , , (0 , , ( Z × Z , k M ) is given by (cid:0) a c,d,ν (cid:1) ( X,α X ,β X ) , ( Y,α Y ,β Y ) , ( Z,α Z ,β Z ) = a X,Y,Z ◦ (cid:2) ( α X ⊗ Y ) ⊗ β − Z (cid:3) , and the braiding is γ c,d,ν ( X,α X ,β X ) , ( Y,α Y ,β Y ) = τ (cid:2)(cid:0) α X β − X (cid:1) ⊗ (cid:0) α − Y β Y (cid:1)(cid:3) , where τ : X ⊗ Y → Y ⊗ X denotes the usual flip in the category of linear spaces. Note that γ is a symmetric braiding. Being H (1 , , (0 , , ( Z × Z , k M ) an additive braided monoidal category,all the concepts of algebra, Lie algebra and so on, can be introduced in this case.By writing down the axioms for an algebra in H (1 , , (0 , , ( Z × Z , k M ) and discarding theinvertibility of α and β if not needed, we arrived at the following concept. A BiHom-associativealgebra over k is a linear space A endowed with a multiplication µ : A ⊗ A → A , µ ( a ⊗ b ) = ab ,and two commuting multiplicative linear maps α, β : A → A satisfying what we call the BiHom-associativity condition α ( a )( bc ) = ( ab ) β ( c ) , ∀ a, b, c ∈ A. Thus, a BiHom-associative algebra with bijective structure maps is exactly an algebra in H (1 , , (0 , , ( Z × Z , k M ).Obviously, a BiHom-associative algebra for which α = β is just a Hom-associative algebra.The remarkable fact is that the twisting principle may be also applied: if A is an associativealgebra and α, β : A → A are two commuting algebra maps, then A with the new multiplicationdefined by a ∗ b = α ( a ) β ( b ) is a BiHom-associative algebra, called the Yau twist of A . Asa matter of fact, although we arrived at the concept of BiHom-associative algebra via thecategorical machinery presented above, it is the possibility of twisting the multiplication of anassociative algebra by two commuting algebra endomorphisms that led us to believe that BiHom-associative algebras are interesting objects in their own. One can think of this as follows. Takeagain an associative algebra A and α, β : A → A two commuting algebra endomorphisms; definea new multiplication on A by a ∗ b = α ( a ) β ( b ). Then it is natural to ask the following question:what kind of structure is ( A, ∗ )? Example 3.9 in this paper shows that, in general, ( A, ∗ ) is not a Hom-associative algebra, so the theory of Hom-associative algebras is not general enoughto cover this natural operation of twisting the multiplication of an associative algebra by two maps; but this operation fits in the framework of BiHom-associative algebras. The Yau twistingof an associative algebra by two maps should thus be considered as the “natural” example ofa BiHom-associative algebra. We would like to emphasize that for this operation the two mapsare not assumed to be bijective, so the resulting BiHom-associative algebra has possibly nonbijective structure maps and as such it cannot be regarded, to our knowledge, as an algebra ina monoidal category.Take now the group G to be arbitrary. It is natural to describe how an algebra in themonoidal category H c,d,ν ( G , k M ) looks like. By writing down the axioms, it turns out (seeClaim 3.1 and Remark 3.5) that an algebra in such a category is a BiHom-associative algebrawith bijective structure maps (the associativity of the algebra in the category is equivalent to theBiHom-associativity condition) having some extra structure (like an action of the group on thealgebra). So, morally, the group G = Z × Z leads to BiHom-associative algebras but any othergroup would not lead to something like a “higher” structure than BiHom-associative algebras(for instance, one cannot have something like TriHom-associative algebras).We initiate in this paper the study of what we will call BiHom-structures. The next structurewe introduce is that of a BiHom-Lie algebra; for this, we use also a categorical approach.Unlike the Hom case, to obtain a BiHom-Lie algebra from a BiHom-associative algebra weneed the structure maps α and β to be bijective; the commutator is defined by the formula G. Graziani, A. Makhlouf, C. Menini and F. Panaite[ a, b ] = ab − α − β ( b ) αβ − ( a ). Nevertheless, just as in the Hom-case, the Yau twist works: if( L, [ − ]) is a Lie algebra over a field k and α, β : L → L are two commuting multiplicative linearmaps and we define the linear map {−} : L ⊗ L → L , { a, b } = [ α ( a ) , β ( b )], for all a, b ∈ L , then L ( α,β ) := ( L, {−} , α, β ) is a BiHom-Lie algebra, called the Yau twist of ( L, [ − ]).We define representations of BiHom-associative algebras and BiHom-Lie algebras and findsome of their basic properties. Then we introduce BiHom-coassociative coalgebras and BiHom-bialgebras together with some of the usual ingredients (comodules, duality, convolution product,primitive elements, module and comodule algebras). We define antipodes for a certain class ofBiHom-bialgebras, called monoidal BiHom-bialgebras, leading thus to the concept of monoidalBiHom-Hopf algebras. We define smash products, as particular cases of twisted tensor products,introduced in turn as a particular case of twisting a BiHom-associative algebra by what we calla BiHom-pseudotwistor. We write down explicitly such a smash product, obtained from anaction of a Yau twist of the quantum group U q ( sl ) on a Yau twist of the quantum plane A | q .As a final remark, let us note that one could introduce a less restrictive concept of BiHom-associative algebra by dropping the assumptions that α and β are multiplicative and/or thatthey commute (note that all the examples of q -deformations of Witt or Virasoro algebras are notmultiplicative). Unfortunately, by dropping any of these assumptions, one loses the main classof examples, the Yau twists, in the sense that if A is an associative algebra and α, β : A → A are two arbitrary linear maps, and we define as before a ∗ b = α ( a ) β ( b ), then ( A, ∗ ) in generalis not a BiHom-associative algebra even in this more general sense. H ( G , C ) Our aim in this section is to introduce so-called group Hom-categories; proofs of the results inthis section may be found in [14].
Definition 2.1.
Let G be a group and let C be a category. The group Hom-category H ( G , C )associated to G and C is the category having as objects pairs ( A, f A ), where A ∈ C and f A isa group homomorphism G →
Aut C ( A ). A morphism ξ : ( A, f A ) → ( B, f B ) in H ( G , C ) is a mor-phism ξ : A → B in C such that f B ( g ) ◦ ξ = ξ ◦ f A ( g ), for all g ∈ G . Definition 2.2. A monoidal category (see [17, Chapter XI]) is a category C endowed withan object ∈ C (called unit ), a functor ⊗ : C × C → C (called tensor product ) and functorialisomorphisms a X,Y,Z : ( X ⊗ Y ) ⊗ Z → X ⊗ ( Y ⊗ Z ), l X : ⊗ X → X , r X : X ⊗ → X , for every X , Y , Z in C . The functorial isomorphisms a are called the associativity constraints and satisfythe pentagon axiom, that is( U ⊗ a V,W,X ) ◦ a U,V ⊗ W,X ◦ ( a U,V,W ⊗ X ) = a U,V,W ⊗ X ◦ a U ⊗ V,W,X holds true, for every U , V , W , X in C . The isomorphisms l and r are called the unit constraints and they obey the Triangle Axiom, that is( V ⊗ l W ) ◦ a V, ,W = r V ⊗ W, for every V , W in C . A monoidal functor ( F, φ , φ ) : ( C , ⊗ , , a, l, r ) → ( C (cid:48) , ⊗ (cid:48) , (cid:48) , a (cid:48) , l (cid:48) , r (cid:48) ) between two monoidalcategories consists of a functor F : C → C (cid:48) , an isomorphism φ ( U, V ) : F ( U ) ⊗ (cid:48) F ( V ) → F ( U ⊗ V ),natural in U, V ∈ C , and an isomorphism φ : (cid:48) → F ( ) such that the diagram( F ( U ) ⊗ (cid:48) F ( V )) ⊗ (cid:48) F ( W ) a (cid:48) F ( U ) ,F ( V ) ,F ( W ) (cid:15) (cid:15) φ ( U,V ) ⊗ (cid:48) F ( W ) (cid:47) (cid:47) F ( U ⊗ V ) ⊗ (cid:48) F ( W ) φ ( U ⊗ V,W ) (cid:47) (cid:47) F (( U ⊗ V ) ⊗ W ) F ( a U,V,W ) (cid:15) (cid:15) F ( U ) ⊗ (cid:48) ( F ( V ) ⊗ (cid:48) F ( W )) F ( U ) ⊗ (cid:48) φ ( V,W ) (cid:47) (cid:47) F ( U ) ⊗ (cid:48) F ( V ⊗ W ) φ ( U,V ⊗ W ) (cid:47) (cid:47) F ( U ⊗ ( V ⊗ W ))iHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras 5is commutative, and the following conditions are satisfied F ( l U ) ◦ φ ( , U ) ◦ ( φ ⊗ (cid:48) F ( U )) = l (cid:48) F ( U ) , F ( r U ) ◦ φ ( U, ) ◦ ( F ( U ) ⊗ (cid:48) φ ) = r (cid:48) F ( U ) . Claim 2.3.
Let G be a group and let ( C , ⊗ , , a, l, r ) be a monoidal category. Given any pairof objects ( A, f A ) , ( B, f B ) ∈ H ( G , C ), consider the map f A ⊗ f B : G →
Aut C ( A ⊗ B ) defined bysetting( f A ⊗ f B )( g ) = f A ( g ) ⊗ f B ( g ) , for all g ∈ G . Then f A ⊗ f B is a group homomorphism and hence( A ⊗ B, f A ⊗ f B ) ∈ H ( G , C ) . Moreover, if φ : ( A, f A ) → ( ˜ A, f ˜ A ) and ξ : ( B, f B ) → ( ˜ B, f ˜ B ) are morphisms in H ( G , C ), then φ ⊗ ξ : ( A ⊗ B, f A ⊗ f B ) → (cid:0) ˜ A ⊗ ˜ B, f ˜ A ⊗ f ˜ B (cid:1) is a morphism in H ( G , C ).Let Z ( G ) be the center of G and let c ∈ Z ( G ). Then we can consider the functorial isomor-phism ϕ ( c ) : Id H ( G , C ) → Id H ( G , C ) defined by setting ϕ ( c )( A, f A ) = f A ( c ) , for every ( A, f A ) in H ( G , C ) . Also, let (cid:99) Id : G →
Aut C ( ) denote the constant map equal to Id .Let c, d ∈ Z ( G ) and let ν ∈ Aut C ( ). We set a c,d,ν = a ◦ (cid:2)(cid:0) ϕ ( c ) ⊗ Id H ( G , C ) (cid:1) ⊗ ϕ ( d ) (cid:3) , l c,d,ν = ϕ (cid:0) d − (cid:1) ◦ l ◦ (cid:0) ν ⊗ Id H ( G , C ) (cid:1) ,r c,d,ν = ϕ ( c ) ◦ r ◦ (cid:0) Id H ( G , C ) ⊗ ν (cid:1) . Theorem 2.4.
In the setting of Claim , the category H c,d,ν ( G , C ) = (cid:0) H ( G , C ) , ⊗ , (cid:0) , (cid:99) Id (cid:1) , a c,d,ν , l c,d,ν , r c,d,ν (cid:1) is monoidal. From now on, when ( C , ⊗ , , a, l, r ) is a monoidal category, G is a group, c, d ∈ Z ( G ) and ν ∈ Aut C ( ), we will indicate the monoidal category defined in Theorem 2.4 by H c,d,ν ( G , C ). Inthe case when c = d = G and ν = Id , we will simply write H ( G , C ). Theorem 2.5.
Let ( C , ⊗ , , a, l, r ) be a monoidal category and G a group. Then the identityfunctor I : H c,d,ν ( G , C ) → H ( G , C ) is a monoidal isomorphism via φ = ν − : (cid:0) , (cid:99) Id (cid:1) → (cid:0) , (cid:99) Id (cid:1) and φ (( A, f A ) , ( B, f B )) = f A (cid:0) c − (cid:1) ⊗ f B ( d ) , for every ( A, f A ) , ( B, f B ) ∈ H c,d,ν ( G , C ) . Definition 2.6 (see [17]) . A braided monoidal category ( C , ⊗ , , a, l, r, γ ) is a monoidal category( C , ⊗ , ,a, l, r ) equipped with a braiding γ , that is, an isomorphism γ U,V : U ⊗ V → V ⊗ U ,natural in U, V ∈ C , satisfying, for all
U, V, W ∈ C , the hexagon axioms a V,W,U ◦ γ U,V ⊗ W ◦ a U,V,W = ( V ⊗ γ U,W ) ◦ a V,U,W ◦ ( γ U,V ⊗ W ) ,a − W,U,V ◦ γ U ⊗ V,W ◦ a − U,V,W = ( γ U,W ⊗ V ) ◦ a − U,W,V ◦ ( U ⊗ γ V,W ) . A braided monoidal category is called symmetric if we further have γ V,U ◦ γ U,V = Id U ⊗ V forevery U, V ∈ C . A braided monoidal functor is a monoidal functor F : C → C (cid:48) such that F ( γ U,V ) ◦ φ ( U, V ) = φ ( V, U ) ◦ γ (cid:48) F ( U ) ,F ( V ) , for every U, V ∈ C . G. Graziani, A. Makhlouf, C. Menini and F. Panaite
Claim 2.7.
Let G be a group and let ( C , ⊗ , , a, l, r, γ ) be a braided monoidal category. Let c, d ∈ Z ( G ) and let ν ∈ Aut C ( ). We will introduce a braided structure on the monoidal category H c,d,ν ( G , C ) by setting, for every ( A, f A ) and ( B, f B ) in H ( G , C ), γ c,d,ν ( A,f A ) , ( B,f B ) = γ A,B ◦ (cid:0) f A ( cd ) ⊗ f B (cid:0) c − d − (cid:1)(cid:1) . Theorem 2.8.
In the setting of Claim , the category (cid:0) H ( G , C ) , ⊗ , (cid:0) , (cid:99) Id (cid:1) , a c,d,ν , l c,d,ν , r c,d,ν , γ c,d,ν (cid:1) is a braided monoidal category. From now on, when ( C , ⊗ , , a, l, r, γ ) is a braided monoidal category and G is a group, wewill still denote the braided monoidal structure defined in Theorem 2.8 with H c,d,ν ( G , C ). Inthe case when c = d = G and ν = id , we will simply write respectively H ( G , C ) instead of H c,d,ν ( G , C ) and γ ( A,f A ) , ( B,f B ) instead of γ c,d,ν ( A,f A ) , ( B,f B ) . Theorem 2.9.
Let G be a group and let ( C , ⊗ , , a, l, r, γ ) be a braided monoidal category. Thenthe identity functor I : H c,d,ν ( G , C ) → H ( G , C ) is a braided monoidal isomorphism via φ = ν − : (cid:0) , (cid:99) Id (cid:1) → ( , (cid:99) Id ) and φ (( A, f A ) , ( B, f B )) = f A (cid:0) c − (cid:1) ⊗ f B ( d ) , for every ( A, f A ) , ( B, f B ) ∈ H c,d,ν ( G , C ) . Remark 2.10.
Let G be a torsion-free abelian group. Corollary 4 in [4] states that, up toa braided monoidal category isomorphism, there is a unique braided monoidal structure (actuallysymmetric) on the category of representations over the group algebra k [ G ], considered monoidalvia a structure induced by that of vector spaces over the field k . Thus Theorem 2.9 can bededuced from this result whenever G is a torsion-free abelian group. We should remark that thisresult in [4] stems from the fact that the third Harrison cohomology group H ( G , k , G m ) has,in this case, just one element. If G is not a torsion-free abelian group then this might not happen.As one of the referees pointed out, in the case when k = C and G = C then H ( G , k , G m )has exactly two elements and so in this case there are two distinct equivalence classes of braidedmonoidal structures on the category of representations over the group algebra k [ G ], consideredmonoidal via a structure induced by that of vector spaces over the field k . This does notcontradict our Theorem 2.9. In fact, there might exist braided monoidal structures differentfrom the ones considered in the statement of Theorem 2.9. Claim 2.11.
Let ( C , ⊗ , , a, l, r ) be a monoidal category and G a group, let c, d ∈ Z ( G ) and ν ∈ Aut C ( ). A unital algebra in H c,d,ν ( G , C ) is a triple (( A, f A ) , µ, u ) where1) ( A, f A ) ∈ H ( G , C );2) µ : ( A ⊗ A, f A ⊗ f A ) → ( A, f A ) is a morphism in H ( G , C );3) u : ( , (cid:99) Id ) → ( A, f A ) is a morphism in H ( G , C );4) µ ◦ ( µ ⊗ A ) = µ ◦ ( A ⊗ µ ) ◦ a c,d,νA,A,A ;5) µ ◦ ( u ⊗ A ) ◦ ( l c,d,νA ) − = Id A ;6) Id A = µ ◦ ( A ⊗ u ) ◦ ( r c,d,νA ) − . Definition 2.12.
Given a monoidal category M , a quadruple ( A, µ, u, c ) is called a braidedunital algebra in M if (for simplicity, we will omit to write the associators):iHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras 7 • ( A, µ, u ) is a unital algebra in M ; • ( A, c ) is a braided object in M , i.e., c : A ⊗ A → A ⊗ A is invertible and satisfies theYang–Baxter equation( c ⊗ A )( A ⊗ c )( c ⊗ A ) = ( A ⊗ c )( c ⊗ A )( A ⊗ c ); • the following conditions hold: c ( µ ⊗ A ) = ( A ⊗ µ )( c ⊗ A )( A ⊗ c ) , c ( A ⊗ µ ) = ( µ ⊗ A )( A ⊗ c )( c ⊗ A ) ,c ( u ⊗ A ) l − A = ( A ⊗ u ) r − A , c ( A ⊗ u ) r − A = ( u ⊗ A ) l − A . A braided unital algebra is called symmetric whenever c = Id A . Definition 2.13.
Given an additive monoidal category M , a braided Lie algebra in M consistsof a triple ( L, c, [ − ] : L ⊗ L → L ) where ( L, c ) is a braided object and the following equalitieshold true:[ − ] = − [ − ] ◦ c (skew-symmetry);[ − ] ◦ ( L ⊗ [ − ]) ◦ (cid:2) Id L ⊗ ( L ⊗ L ) +( L ⊗ c ) a L,L,L ( c ⊗ L ) a − L,L,L + a L,L,L ( c ⊗ L ) a − L,L,L ( L ⊗ c ) (cid:3) = 0 (Jacobi condition); c ◦ ( L ⊗ [ − ]) a L,L,L = ([ − ] ⊗ L ) a − L,L,L ( L ⊗ c ) a L,L,L ( c ⊗ L ); (2.1) c ◦ ([ − ] ⊗ L ) a − L,L,L = ( L ⊗ [ − ]) a L,L,L ( c ⊗ L ) a − L,L,L ( L ⊗ c ) . (2.2)Let M be an additive braided monoidal category. A Lie algebra in M consists of a pair( L, [ − ] : L ⊗ L → L ) such that ( L, c
L,L , [ − ]) is a braided Lie algebra in the additive monoidalcategory M , where c L,L is the braiding c of M evaluated on L (note that in this case theconditions (2.1) and (2.2) are automatically satisfied). Claim 2.14.
Given a symmetric algebra (
A, µ, u, c ), one has that [ − ] := µ ◦ (Id A ⊗ A − c ) definesa braided Lie algebra structure on A (see [13, Construction 2.16]).In a symmetric monoidal category ( C , ⊗ , , a, l, r, c ), it is well known that any unital algebra( A, µ, u ) gives rise to a braided unital algebra (
A, µ, u, c
A,A ). Let k be a field and let k M be the category of linear spaces regarded as a braided monoidalcategory in the usual way. Then, for every group G , the category H ( G , k M ) identifies with thecategory k [ G ]-Mod of left modules over the group algebra k [ G ].Let c, d ∈ Z ( G ) and ν an automorphism of k regarded as linear space over k , that is ν isthe multiplication by an element of k \{ } that we will also denote by ν . Note that, given X, Y, Z ∈ k [ G ]-Mod, we have a c,d,νX,Y,Z (( x ⊗ y ) ⊗ z ) = c · x ⊗ ( y ⊗ d · z ) , for every x ∈ X, y ∈ Y, z ∈ Z,l c,d,νX ( t ⊗ x ) = d − · ( νtx ) and r c,d,νX ( x ⊗ t ) = c · ( νtx ) , for every t ∈ k and x ∈ X, so that (cid:0) l c,d,νA (cid:1) − ( x ) = ( ν − ⊗ d · x ) and (cid:0) r c,d,ν (cid:1) − ( x ) = (cid:0) c − · x ⊗ ν − (cid:1) , for every x ∈ X. The unit object of H c,d,ν ( G , k M ) is { k } regarded as a left k [ G ]-module in the trivial way. G. Graziani, A. Makhlouf, C. Menini and F. Panaite Claim 3.1.
In view of 2.11, a unital algebra in H c,d,ν ( G , k M ) is a triple (( A, f A ) , µ, u ), where1) A ∈ k [ G ]-Mod;2) µ : A ⊗ A → A is a morphism in k [ G ]-Mod, i.e., g · ( ab ) = ( g · a )( g · b ), for every g ∈ G , a, b ∈ A ;3) u : { k } → A is a morphisms in k [ G ]-Mod, i.e., g · u (1 k ) = u (1 k ), for every g ∈ G ;4) ( x · y ) · z = ( c · x ) · [ y · ( d · z )], for every x, y, z ∈ A , which is equivalent to( c · x ) · ( y · z ) = ( x · y ) · (cid:0) d − · z (cid:1) , ∀ x, y, z ∈ A ;5) u ( ν − ) · ( d · x ) = x , for every x ∈ A ;6) ( c − · x ) · u ( ν − ) = x , for every x ∈ A .Note that when c = d = 1 G and ν = 1 k , it turns out that A is simply a k [ G ]-module algebra. Example 3.2.
Let M be a k -linear space and G = Z × Z . Then a group morphism f M : Z × Z → Aut k ( M ) is completely determined by f M ((1 , α M and f M ((0 , β − M . Thus an object in H ( Z × Z , k M ) identifies with a triple ( M, α M , β M ), where α M , β M ∈ Aut k ( M )and α M ◦ β M = β M ◦ α M . Also, a morphism f : ( M, α M , β M ) → ( N, α N , β N ) is just a linear map f : M → N such that f ◦ α M = α N ◦ f and f ◦ β M = β N ◦ f . Moreover, the tensor product, in thecategory, of the objects ( M, α M , β M ) and ( N, α N , β N ) is the object ( M ⊗ N, α M ⊗ α N , β M ⊗ β N ).We set c = (1 , d = (0 ,
1) and ν = 1 k . For ( X, α X , β X ), ( Y, α Y , β Y ), ( Z, α Z , β Z ) objects in H (1 , , (0 , , ( Z × Z , k M ), the associativity constraints in H (1 , , (0 , , ( Z × Z , k M ) are given by (cid:0) a c,d,ν (cid:1) ( X,α X ,β X ) , ( Y,α Y ,β Y ) , ( Z,α Z ,β Z ) : ( X ⊗ Y ) ⊗ Z → X ⊗ ( Y ⊗ Z ) , (cid:0) a c,d,ν (cid:1) ( X,α X ,β X ) , ( Y,α Y ,β Y ) , ( Z,α Z ,β Z ) = a X,Y,Z ◦ (cid:2) ( α X ⊗ Y ) ⊗ β − Z (cid:3) , and the braiding is γ c,d,ν ( X,α X ,β X ) , ( Y,α Y ,β Y ) = τ (cid:2)(cid:0) α X β − X (cid:1) ⊗ (cid:0) α Y β − Y (cid:1) − (cid:3) = τ (cid:2)(cid:0) α X β − X (cid:1) ⊗ (cid:0) α − Y β Y (cid:1)(cid:3) , where τ : X ⊗ Y → Y ⊗ X denotes the usual flip in the category of linear spaces. Note that γ is a symmetric braiding.Then, in view of 3.1, an algebra in H (1 , , (0 , , ( Z × Z , k M ) is a triple (( A, α, β ) , µ, u ), where1) α, β ∈ Aut k ( A ) and α ◦ β = β ◦ α ;2) µ : ( A ⊗ A, α ⊗ α, β ⊗ β ) → ( A, α, β ) is a morphism in k [ Z × Z ]-Mod, i.e., α ( a · b ) = α ( a ) · α ( b )and β ( a · b ) = β ( a ) · β ( b ) for every a, b ∈ A ;3) u : { k } → ( A, α, β ) is a morphisms in k [ Z × Z ]-Mod, i.e., α ( u (1 k )) = u (1 k ) and β ( u (1 k )) = u (1 k );4) α ( x ) · ( y · z ) = ( x · y ) · β ( z ), for every x, y, z ∈ A ;5) u (1 k ) · ( β − ( x )) = x , for every x ∈ A , which is equivalent to u (1 k ) · x = β ( x ), for every x ∈ A ;6) ( α − ( x )) · u (1 k ) = x , for every x ∈ A , which is equivalent to x · u (1 k ) = α ( x ), for every x ∈ A .Inspired by Example 3.2, we introduce the following concept.iHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras 9 Definition 3.3.
Let k be a field. A BiHom-associative algebra over k is a 4-tuple ( A, µ, α, β ),where A is a k -linear space, α : A → A , β : A → A and µ : A ⊗ A → A are linear maps, withnotation µ ( a ⊗ a (cid:48) ) = aa (cid:48) , satisfying the following conditions, for all a, a (cid:48) , a (cid:48)(cid:48) ∈ A : α ◦ β = β ◦ α,α ( aa (cid:48) ) = α ( a ) α ( a (cid:48) ) and β ( aa (cid:48) ) = β ( a ) β ( a (cid:48) ) (multiplicativity) ,α ( a )( a (cid:48) a (cid:48)(cid:48) ) = ( aa (cid:48) ) β ( a (cid:48)(cid:48) ) (BiHom-associativity) . We call α and β (in this order) the structure maps of A .A morphism f : ( A, µ A , α A , β A ) → ( B, µ B , α B , β B ) of BiHom-associative algebras is a linearmap f : A → B such that α B ◦ f = f ◦ α A , β B ◦ f = f ◦ β A and f ◦ µ A = µ B ◦ ( f ⊗ f ).A BiHom-associative algebra ( A, µ, α, β ) is called unital if there exists an element 1 A ∈ A (called a unit ) such that α (1 A ) = 1 A , β (1 A ) = 1 A and a A = α ( a ) and 1 A a = β ( a ) , ∀ a ∈ A. A morphism of unital BiHom-associative algebras f : A → B is called unital if f (1 A ) = 1 B . Remark 3.4.
A Hom-associative algebra (
A, µ, α ) can be regarded as the BiHom-associativealgebra (
A, µ, α, α ). Remark 3.5.
A BiHom-associative algebra with bijective structure maps is exactly an algebrain H (1 , , (0 , , ( Z × Z , k M ). On the other hand, in the setting of Claim 3.1, if we define themaps α, β : A → A by α ( a ) = c · a and β ( a ) = d − · a , for all a ∈ A , the axiom 2) in Claim 3.1implies that α and β are multiplicative and then the axiom 4) in Claim 3.1 says that ( A, µ, α, β )is a BiHom-associative algebra.
Example 3.6.
We give now two families of examples of 2-dimensional unital BiHom-associativealgebras, that are obtained by a computer algebra system. Let { e , e } be a basis; for i = 1 , α i , β i and the multiplication µ i are defined by α ( e ) = e , α ( e ) = 2 ab − e − e ,β ( e ) = e , β ( e ) = − ae + be ,µ ( e , e ) = e , µ ( e , e ) = − ae + be ,µ ( e , e ) = 2 ab − e − e , µ ( e , e ) = − a ( b − b − e + ae , and α ( e ) = e , α ( e ) = b (1 − a ) a e + ae ,β ( e ) = e , β ( e ) = be + (1 − a ) e ,µ ( e , e ) = e , µ ( e , e ) = be + (1 − a ) e ,µ ( e , e ) = b (1 − a ) a e + ae , µ ( e , e ) = ba e , where a , b are parameters in k , with b (cid:54) = 1 in the first case and a (cid:54) = 0 in the second. In bothcases, the unit is e . Claim 3.7.
In view of Theorem 2.5, if (
A, µ, α, β ) is a BiHom-associative algebra, and α and β are invertible, then ( A, µ ◦ ( α − ⊗ β − ) , Id A , Id A ) is a BiHom-associative algebra, i.e., the mul-tiplication µ ◦ ( α − ⊗ β − ) is associative in the usual sense.0 G. Graziani, A. Makhlouf, C. Menini and F. PanaiteOn the other hand, if ( A, µ : A ⊗ A → A ) is an associative algebra and α, β : A → A arecommuting algebra endomorphisms, then one can easily check that ( A, µ ◦ ( α ⊗ β ) , α, β ) isa BiHom-associative algebra, denoted by A ( α,β ) and called the Yau twist of (
A, µ ).In view of Claim 3.7, a BiHom-associative algebra with bijective structure maps is a Yautwist of an associative algebra.The Yau twisting procedure for BiHom-associative algebras admits a more general form,which we state in the next result (the proof is straightforward and left to the reader).
Proposition 3.8.
Let ( D, µ, ˜ α, ˜ β ) be a BiHom-associative algebra and α, β : D → D two multi-plicative linear maps such that any two of the maps ˜ α , ˜ β , α , β commute. Then ( D, µ ◦ ( α ⊗ β ) , ˜ α ◦ α, ˜ β ◦ β ) is also a BiHom-associative algebra, denoted by D ( α,β ) . Example 3.9.
We present an example of a BiHom-associative algebra that cannot be expressedas a Hom-associative algebra. Let k be a field and A = k [ X ]. Let α : A → A be the algebra mapdefined by setting α ( X ) = X and let β = Id k [ X ] . Then we can consider the BiHom-associativealgebra A ( α,β ) = ( A, µ ◦ ( α ⊗ β ) , α, β ), where µ : A ⊗ A → A is the usual multiplication. Forevery a, a (cid:48) ∈ A set a ∗ a (cid:48) = µ ◦ ( α ⊗ β )( a ⊗ a (cid:48) ) = α ( a ) a (cid:48) . Let us assume that there exists θ ∈ End( k [ X ]) such that ( A, µ ◦ ( α ⊗ β ) , θ ) is a Hom-associativealgebra. Then we should have that θ ( X ) ∗ ( X ∗ X ) = ( X ∗ X ) ∗ θ ( X ) . (3.1)Write θ ( X ) = n (cid:88) i =0 a i X i , where a i ∈ k for every i = 0 , , . . . , n and a n (cid:54) = 0 . Since X ∗ X = α ( X ) X = X , (3.1) rewrites as n (cid:88) i =0 a i X i ∗ X = X ∗ n (cid:88) i =0 a i X i , and hence as n (cid:88) i =0 a i α ( X ) i X = α ( X ) n (cid:88) i =0 a i X i , i.e., n (cid:88) i =0 a i X i +3 = n (cid:88) i =0 a i X i , which implies that2 n + 3 = 6 + n, i.e. , n = 3 , and hence a X + a X + a X + a X = a X + a X + a X + a X , iHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras 11so that θ ( X ) = a X . Let us set c = a and let us check the equality θ (cid:0) X (cid:1) ∗ ( X ∗ X ) = (cid:0) X ∗ X (cid:1) ∗ θ ( X ) . The left-hand side is θ (cid:0) X (cid:1) ∗ ( X ∗ X ) = c X ∗ X = α (cid:0) c X (cid:1) X = c X . The right-hand side is (cid:0) X ∗ X (cid:1) ∗ θ ( X ) = (cid:0) α (cid:0) X (cid:1) X (cid:1) ∗ θ ( X ) = X ∗ θ ( X ) = cX X = cX . Thus the equality does not hold.
Remark 3.10.
Given two algebras (
A, µ A , A ) and ( B, µ B , B ) in a braided monoidal category( C , ⊗ , , a, l, r, c ), it is well known that A ⊗ B becomes also an algebra in the category, withmultiplication µ A ⊗ B defined by µ A ⊗ B = ( µ A ⊗ µ B ) ◦ a − A,A,B ⊗ B ◦ ( A ⊗ a A,B,B ) ◦ ( A ⊗ ( c B,A ⊗ B )) ◦ (cid:0) A ⊗ a − B,A,B (cid:1) ◦ a A,B,A ⊗ B . In the case of our category H c,d,ν ( G , k M ), we have, for every x, y ∈ A , x (cid:48) , y (cid:48) ∈ B : µ A ⊗ B (( x ⊗ y ) ⊗ ( x (cid:48) ⊗ y (cid:48) )) = (( µ A ⊗ µ B ) ◦ a − A,A,B ⊗ B ◦ ( A ⊗ a A,B,B ) ◦ ( A ⊗ ( c B,A ⊗ B )) ◦ (cid:0) A ⊗ a − B,A,B (cid:1) ◦ a A,B,A ⊗ B )(( x ⊗ y ) ⊗ ( x (cid:48) ⊗ y (cid:48) ))= (cid:0) ( µ A ⊗ µ B ) ◦ a − A,A,B ⊗ B ◦ ( A ⊗ a A,B,B ) ◦ ( A ⊗ ( c B,A ⊗ B )) ◦ (cid:0) A ⊗ a − B,A,B (cid:1)(cid:1) ( cx ⊗ ( y ⊗ ( dx (cid:48) ⊗ dy (cid:48) )))= (cid:0) ( µ A ⊗ µ B ) ◦ a − A,A,B ⊗ B ◦ ( A ⊗ a A,B,B ) (cid:1)(cid:0) cx ⊗ (cid:0)(cid:0) c − x (cid:48) ⊗ dy (cid:1) ⊗ y (cid:48) (cid:1)(cid:1) = (cid:0) ( µ A ⊗ µ B ) ◦ a − A,A,B ⊗ B (cid:1) ( cx ⊗ ( x (cid:48) ⊗ ( dy ⊗ dy (cid:48) )))= ( µ A ⊗ µ B )(( x ⊗ x (cid:48) ) ⊗ ( y ⊗ y (cid:48) )) = ( x · x (cid:48) ) ⊗ ( y · y (cid:48) ) . In particular, if (
A, α A , β A ) and ( B, α B , β B ) are two algebras in H (1 , , (0 , , ( Z × Z , k M ), theirbraided tensor product A ⊗ B in the category is the algebra ( A ⊗ B, α A ⊗ α B , β A ⊗ β B ), whosemultiplication is given by ( a ⊗ b )( a (cid:48) ⊗ b (cid:48) ) = aa (cid:48) ⊗ bb (cid:48) , for all a, a (cid:48) ∈ A and b, b (cid:48) ∈ B . Remark 3.11.
If (
A, µ A , α A , β A ) and ( B, µ B , α B , β B ) are two BiHom-associative algebras overa field k , then ( A ⊗ B, µ A ⊗ B , α A ⊗ α B , β A ⊗ β B ) is a BiHom-associative algebra (called the tensorproduct of A and B ), where µ A ⊗ B is the usual multiplication: ( a ⊗ b )( a (cid:48) ⊗ b (cid:48) ) = aa (cid:48) ⊗ bb (cid:48) . If A and B are unital with units 1 A and respectively 1 B then A ⊗ B is also unital with unit 1 A ⊗ B .This is consistent with Remark 3.10. Example 3.12.
In view of Definition 2.13, a
Lie algebra in H c,d,ν ( G , k M ) is a pair (( L, f L ) , [ − ]),where1) ( L, f L ) ∈ k [ G ]-Mod;2) [ − ] : L ⊗ L → L is a morphism in k [ G ]-Mod;3) [ − ] = − [ − ] ◦ γ L,L ;2 G. Graziani, A. Makhlouf, C. Menini and F. Panaite4) [ − ] ◦ ( L ⊗ [ − ]) + [ − ] ◦ ( L ⊗ [ − ]) ◦ ( L ⊗ γ L,L ) a L,L,L ( γ L,L ⊗ L ) a − L,L,L + [ − ] ◦ ( L ⊗ [ − ]) a L,L,L ( γ L,L ⊗ L ) a − L,L,L ( L ⊗ γ L,L ) = 0 , where γ L,L = τ ◦ ( f L ( cd ) ⊗ f L ( c − d − )) and τ is the usual flip.We will write down 4) explicitly. We have (cid:0)(cid:0) L ⊗ γ L,L (cid:1) a L,L,L (cid:0) γ L,L ⊗ L (cid:1) a − L,L,L (cid:1) ( x ⊗ ( y ⊗ z ))= (cid:0) L ⊗ γ L,L (cid:1) a L,L,L (cid:0) γ L,L ⊗ L (cid:1)(cid:0)(cid:0) c − x ⊗ y (cid:1) ⊗ d − z (cid:1) = (cid:0) L ⊗ γ L,L (cid:1) a L,L,L (cid:0)(cid:0) c − d − y ⊗ cdc − x (cid:1) ⊗ d − z (cid:1) = (cid:0) L ⊗ γ L,L (cid:1)(cid:0) cc − d − y ⊗ (cid:0) cdc − x ⊗ dd − z (cid:1)(cid:1) = cc − d − y ⊗ (cid:0) c − d − dd − z ⊗ cdcdc − x (cid:1) = d − y ⊗ (cid:0) c − d − z ⊗ dcdx (cid:1) , therefore[ − ] ◦ ( L ⊗ [ − ]) (cid:0)(cid:0) L ⊗ γ L,L (cid:1) a L,L,L (cid:0) γ L,L ⊗ L (cid:1) a − L,L,L (cid:1) ( x ⊗ ( y ⊗ z ))= (cid:2) d − y, (cid:2) c − d − z, cd x (cid:3)(cid:3) , and (cid:0) a L,L,L (cid:0) γ L,L ⊗ L (cid:1) a − L,L,L (cid:0) L ⊗ γ L,L (cid:1)(cid:1) ( x ⊗ ( y ⊗ z ))= a L,L,L (cid:0) γ L,L ⊗ L (cid:1) a − L,L,L (cid:0) x ⊗ (cid:0) c − d − z ⊗ cdy (cid:1)(cid:1) = a L,L,L (cid:0) γ L,L ⊗ L (cid:1)(cid:0) c − x ⊗ (cid:0) c − d − z ⊗ d − cdy (cid:1)(cid:1) = a L,L,L (cid:0) γ L,L ⊗ L (cid:1)(cid:0)(cid:0) c − x ⊗ c − d − z (cid:1) ⊗ cy (cid:1) = a L,L,L (cid:0)(cid:0) c − d − z ⊗ cdc − x (cid:1) ⊗ cy (cid:1) = (cid:0)(cid:0) c − d − z ⊗ dx (cid:1) ⊗ cdy (cid:1) , hence[ − ] ◦ ( L ⊗ [ − ]) (cid:0) a L,L,L (cid:0) γ L,L ⊗ L (cid:1) a − L,L,L (cid:0) L ⊗ γ L,L (cid:1)(cid:1) ( x ⊗ ( y ⊗ z )) = (cid:2) c − d − z, [ dx, cdy ] (cid:3) . Thus 4) is equivalent to[ x, [ y, z ]] + (cid:2) d − y, (cid:2) c − d − z, cd x (cid:3)(cid:3) + (cid:2) c − d − z, [ dx, cdy ] (cid:3) = 0 , for every x, y, z ∈ L, which is equivalent to (cid:2) d − x, (cid:2) d − y, cz (cid:3)(cid:3) + (cid:2) d − y, (cid:2) d − z, cx (cid:3)(cid:3) + (cid:2) d − z, (cid:2) d − x, cy (cid:3)(cid:3) = 0 , for every x, y, z ∈ L. Thus, a Lie algebra in H c,d,ν ( G , k M ) is a pair ( L, [ − ]), where1) L ∈ k [ G ]-Mod;2) g [ x, y ] = [ gx, gy ], for every x, y ∈ L ;3) [ x, y ] = − [ c − d − y, cdx ], for every x, y ∈ L , i.e., [ x, cdy ] = − [ y, cdx ], for every x, y ∈ L (skew-symmetry);iHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras 134) [ d − x, [ d − y, cz ]] + [ d − y, [ d − z, cx ]] + [ d − z, [ d − x, cy ]] = 0, for every x, y, z ∈ L (Jacobicondition).In particular, a Lie algebra in H (1 , , (0 , , ( Z × Z , k M ) is a pair (( L, α, β ) , [ − ]), where1) α, β ∈ Aut k ( L ) and α ◦ β = β ◦ α ;2) [ − ] : ( L ⊗ L, α ⊗ α, β ⊗ β ) → ( L, α, β ) is a morphism in k [ Z × Z ]-Mod, i.e., α [ a, b ] =[ α ( a ) , α ( b )] and β [ a, b ] = [ β ( a ) , β ( b )], for every a, b ∈ L ;3) [ a, αβ − ( b )] = − [ b, αβ − ( a )], for every a, b ∈ L , which is equivalent to [ β ( a ) , α ( b )] = − [ β ( b ) , α ( a )], for every a, b ∈ L ;4) [ β x, [ βy, αz ]] + [ β y, [ βz, αx ]] + [ β z, [ βx, αy ]] = 0, for every x, y, z ∈ L .Inspired by Example 3.12, we introduce the following concept. Definition 3.13. A BiHom-Lie algebra over a field k is a 4-tuple ( L, [ − ] , α, β ), where L isa k -linear space, α : L → L , β : L → L and [ − ] : L ⊗ L → L are linear maps, with notation[ − ]( a ⊗ a (cid:48) ) = [ a, a (cid:48) ], satisfying the following conditions, for all a, a (cid:48) , a (cid:48)(cid:48) ∈ L : α ◦ β = β ◦ α,α ([ a (cid:48) , a (cid:48)(cid:48) ]) = [ α ( a (cid:48) ) , α ( a (cid:48)(cid:48) )] and β ([ a (cid:48) , a (cid:48)(cid:48) ]) = [ β ( a (cid:48) ) , β ( a (cid:48)(cid:48) )] , [ β ( a ) , α ( a (cid:48) )] = − [ β ( a (cid:48) ) , α ( a )] (skew-symmetry) , (cid:2) β ( a ) , [ β ( a (cid:48) ) , α ( a (cid:48)(cid:48) )] (cid:3) + (cid:2) β ( a (cid:48) ) , [ β ( a (cid:48)(cid:48) ) , α ( a )] (cid:3) + (cid:2) β ( a (cid:48)(cid:48) ) , [ β ( a ) , α ( a (cid:48) )] (cid:3) = 0(BiHom-Jacobi condition) . We call α and β (in this order) the structure maps of L . A morphism f : ( L, [ − ] , α, β ) → ( L (cid:48) , [ − ] (cid:48) , α (cid:48) , β (cid:48) ) of BiHom-Lie algebras is a linear map f : L → L (cid:48) such that α (cid:48) ◦ f = f ◦ α , β (cid:48) ◦ f = f ◦ β and f ([ x, y ]) = [ f ( x ) , f ( y )] (cid:48) , for all x, y ∈ L .Thus, a Lie algebra in H (1 , , (0 , , ( Z × Z , k M ) is exactly a BiHom-Lie algebra with bijective structure maps. Remark 3.14.
Obviously, a Hom-Lie algebra ( L, [ − ] , α ) is a particular case of a BiHom-Liealgebra, namely ( L, [ − ] , α, α ). Conversely, a BiHom-Lie algebra ( L, [ − ] , α, α ) with bijective α isthe Hom-Lie algebra ( L, [ − ] , α ).In view of Claim 2.14, we have: Proposition 3.15. If ( A, µ, α, β ) is a BiHom-associative algebra with bijective α and β , then,for every a, a (cid:48) ∈ A , we can set [ a, a (cid:48) ] = aa (cid:48) − (cid:0) α − β ( a (cid:48) ) (cid:1)(cid:0) αβ − ( a ) (cid:1) . Then ( A, [ − ] , α, β ) is a BiHom-Lie algebra, denoted by L ( A ) . The proofs of the following three results are straightforward and left to the reader.
Proposition 3.16.
Let ( L, [ − ]) be an ordinary Lie algebra over a field k and let α, β : L → L two commuting linear maps such that α ([ a, a (cid:48) ]) = [ α ( a ) , α ( a (cid:48) )] and β ([ a, a (cid:48) ]) = [ β ( a ) , β ( a (cid:48) )] , forall a, a (cid:48) ∈ L . Define the linear map {−} : L ⊗ L → L , { a, b } = [ α ( a ) , β ( b )] , for all a, b ∈ L. Then L ( α,β ) := ( L, {−} , α, β ) is a BiHom-Lie algebra, called the Yau twist of ( L, [ − ]) . Claim 3.17.
More generally, let ( L, [ − ] , α, β ) be a BiHom-Lie algebra and α (cid:48) , β (cid:48) : L → L linearmaps such that α (cid:48) ([ a, b ]) = [ α (cid:48) ( a ) , α (cid:48) ( b )] and β (cid:48) ([ a, b ]) = [ β (cid:48) ( a ) , β (cid:48) ( b )] for all a, b ∈ L , and anytwo of the maps α, β, α (cid:48) , β (cid:48) commute. Then ( L, [ − ] ( α (cid:48) ,β (cid:48) ) := [ − ] ◦ ( α (cid:48) ⊗ β (cid:48) ) , α ◦ α (cid:48) , β ◦ β (cid:48) ) isa BiHom-Lie algebra. Proposition 3.18.
Let ( A, µ ) be an associative algebra and α, β : A → A two commuting algebraisomorphisms. Then L ( A ( α,β ) ) = L ( A ) ( α,β ) , as BiHom-Lie algebras. Remark 3.19.
Let G be a group and c, d ∈ Z ( G ), ν ∈ Aut C ( ). It is straightforward toprove that the category H c,d,ν ( G , k M ) fulfills the assumption of [5, Theorem 6.4]. Hence, forany Lie algebra ( L, [ − ]) in H c,d,ν ( G , k M ), we can consider the universal enveloping bialgebra U (( L, [ − ])) as introduced in [5]. By [5, Remark 6.5], U (( L, [ − ])) as a bialgebra is a quotientof the tensor bialgebra T L . The morphism giving the projection is induced by the canonicalprojection p : T L → U ( L, [ − ]) defining the universal enveloping algebra. At algebra level wehave U ( L, [ − ]) = T L (cid:0) [ x, y ] − x ⊗ y + γ L,L ( x ⊗ y ) | x, y ∈ L (cid:1) = T L (cid:0) [ x, y ] − x ⊗ y + (cid:0) f L ( c − d − )( y ) (cid:1) ⊗ f L ( cd )( x ) | x, y ∈ L (cid:1) . By Theorem 2.9, the identity functor I : H c,d,ν ( G , k M ) → H ( G , k M ) is a braided monoidalisomorphism. Let F : H ( G , k M ) → k M be the forgetful functor. Then F ◦ I is a monoidalfunctor H c,d,ν ( G , k M ) → k M to which we can apply [5, Theorem 8.5] to get that H c,d,ν ( G , k M )is what is called in [5] a Milnor–Moore category. This implies that, by [5, Theorem 7.2], wehave an isomorphism ( L, [ − ]) → PU (( L, [ − ])), where PU (( L, [ − ])) denotes the primitive part of U (( L, [ − ])). That is, half of the Milnor–Moore theorem holds.The case G = Z can be found in [5, Remark 9.10].In the particular case of a Lie algebra (( L, α, β ) , [ − ]) in H (1 , , (0 , , ( Z × Z , k M ) we have that U ( L, [ − ]) = T L (cid:0) [ x, y ] − x ⊗ y + ( α − β )( y ) ⊗ ( αβ − )( x ) | x, y ∈ L (cid:1) . Enveloping algebras of Hom-Lie algebras where introduced in [29] (see also [8, Section 8]).
From now on, we will always work over a base field k . All algebras, linear spaces etc. will beover k ; unadorned ⊗ means ⊗ k . For a comultiplication ∆ : C → C ⊗ C on a linear space C ,we use a Sweedler-type notation ∆( c ) = c ⊗ c , for c ∈ C . Unless otherwise specified, the(co)algebras ((co)associative or not) that will appear in what follows are not supposed to be(co)unital, and a multiplication µ : V ⊗ V → V on a linear space V is denoted by juxtaposition: µ ( v ⊗ v (cid:48) ) = vv (cid:48) . For the composition of two maps f and g , we will write either g ◦ f or simply gf .For the identity map on a linear space V we will use the notation id V . Definition 4.1.
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 are linear maps and we have a linearmap A ⊗ M → M , a ⊗ m (cid:55)→ a · m , such that, for all a, a (cid:48) ∈ A , m ∈ M , we have α M ◦ β M = β M ◦ α M , (4.1) α M ( a · m ) = α A ( a ) · α M ( m ) , (4.2) β M ( a · m ) = β A ( a ) · β M ( m ) , (4.3)iHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras 15 α A ( a ) · ( a (cid:48) · m ) = ( aa (cid:48) ) · β M ( m ) . (4.4)If ( M, α M , β M ) and ( N, α N , β N ) are left A -modules (both A -actions denoted by · ), a mor-phism of left A -modules f : M → N is a linear map satisfying the conditions α N ◦ f = f ◦ α M , β N ◦ f = f ◦ β M and f ( a · m ) = a · f ( m ), for all a ∈ A and m ∈ M .If ( A, µ A , α A , β A , A ) is a unital BiHom-associative algebra and ( M, α M , β M ) is a left A -module, them M is called unital if 1 A · m = β M ( m ), for all m ∈ M . Remark 4.2.
If (
A, µ, α, β ) is a BiHom-associative algebra, then (
A, α, β ) is a left A -modulewith action defined by a · b = ab , for all a, b ∈ A . Lemma 4.3.
Let ( E, µ, E ) be an associative unital algebra and u, v ∈ E two invertible elementssuch that uv = vu . Define the linear maps ˜ α, ˜ β : E → E , ˜ α ( a ) = uau − , ˜ β ( a ) = vav − , for all a ∈ E , and the linear map ˜ µ : E ⊗ E → E , ˜ µ ( a ⊗ b ) := a ∗ b = uau − bv − , for all a, b ∈ E . Then ( E, ˜ µ, ˜ α, ˜ β ) is a unital BiHom-associative algebra with unit v , denoted by E ( u, v ) . Proof .
Obviously ˜ α ◦ ˜ β = ˜ β ◦ ˜ α because uv = vu . Then, for all a, b, c ∈ E :˜ α ( a ) ∗ ˜ α ( b ) = (cid:0) uau − (cid:1) ∗ (cid:0) ubu − (cid:1) = uuau − u − ubu − v − = uuau − bu − v − = ˜ α (cid:0) uau − bv − (cid:1) = ˜ α ( a ∗ b ) , ˜ β ( a ) ∗ ˜ β ( b ) = (cid:0) vav − (cid:1) ∗ (cid:0) vbv − (cid:1) = uvav − u − vbv − v − = uvau − bv − v − = ˜ β (cid:0) uau − bv − (cid:1) = ˜ β ( a ∗ b ) , ˜ α ( a ) ∗ ( b ∗ c ) = (cid:0) uau − (cid:1) ∗ (cid:0) ubu − cv − (cid:1) = uuau − u − ubu − cv − v − = uuau − bv − u − vcv − v − = (cid:0) uau − bv − (cid:1) ∗ (cid:0) vcv − (cid:1) = ( a ∗ b ) ∗ ˜ β ( c ) , so ( E, ˜ µ, ˜ α, ˜ β ) is indeed a BiHom-associative algebra. To prove that v is the unit, we compute˜ α ( v ) = uvu − = v, ˜ β ( v ) = vvv − = v,a ∗ v = uau − vv − = uau − = ˜ α ( a ) , v ∗ a = uvu − av − = vav − = ˜ β ( a ) , finishing the proof. (cid:4) Proposition 4.4.
Let ( A, µ A , α A , β A ) be a BiHom-associative algebra, M a linear space and α M , β M : M → M two commuting linear isomorphisms. Consider the associative unital algebra E = End( M ) with its usual structure, denote u := α M , v := β M , and construct the BiHom-associative algebra ( E, ˜ µ, ˜ α, ˜ β ) = End( M )( α M , β M ) as in Lemma . Then setting a structureof a left A -module on ( M, α M , β M ) is equivalent to giving a morphism of BiHom-associativealgebras ϕ : ( A, µ A , α A , β A ) → ( E, ˜ µ, ˜ α, ˜ β ) . If A is moreover unital with unit A , then the module ( M, α M , β M ) is unital if and only if the morphism ϕ is unital. Proof .
The correspondence is given as follows: the module structure A ⊗ M → M is definedby setting a ⊗ m (cid:55)→ a · m if and only if a · m = ϕ ( a )( m ), for all a ∈ A , m ∈ M . It is easy to seethat conditions (4.2) and (4.3) are equivalent to ˜ α ◦ ϕ = ϕ ◦ α A and respectively ˜ β ◦ ϕ = ϕ ◦ β A .We prove that, assuming (4.2) and (4.3), we have that (4.4) is equivalent to ϕ ◦ µ A = ˜ µ ◦ ( ϕ ⊗ ϕ ).Note first that (4.2) may be written as α M ◦ ϕ ( a ) = ϕ ( α A ( a )) ◦ α M , for all a ∈ A , or equivalently α M ◦ ϕ ( a ) ◦ α − M = ϕ ( α A ( a )), for all a ∈ A . Thus, for all a, b ∈ A , we have˜ µ ◦ ( ϕ ⊗ ϕ )( a ⊗ b ) = ϕ ( a ) ∗ ϕ ( b ) = α M ◦ ϕ ( a ) ◦ α − M ◦ ϕ ( b ) ◦ β − M = ϕ ( α A ( a )) ◦ ϕ ( b ) ◦ β − M . Hence, we have ϕ ◦ µ A = ˜ µ ◦ ( ϕ ⊗ ϕ ) ⇐⇒ ϕ ( ab ) = ϕ ( a ) ∗ ϕ ( b ) , ∀ a, b ∈ A, ⇐⇒ ϕ ( ab )( n ) = ( ϕ ( a ) ∗ ϕ ( b ))( n ) , ∀ a, b ∈ A, n ∈ M, ⇐⇒ ( ab ) · n = ( ϕ ( α A ( a )) ◦ ϕ ( b ) ◦ β − M )( n ) , ∀ a, b ∈ A, n ∈ M, ⇐⇒ ( ab ) · β M ( m ) = ( ϕ ( α A ( a )) ◦ ϕ ( b ))( m ) , ∀ a, b ∈ A, m ∈ M, ⇐⇒ ( ab ) · β M ( m ) = α A ( a ) · ( b · m ) , ∀ a, b ∈ A, m ∈ M, which is exactly (4.4).Assume that A is unital with unit 1 A . The fact that ϕ is unital is equivalent to ϕ (1 A ) = β M ,which is equivalent to 1 A · m = β M ( m ), for all m ∈ M , which is equivalent to saying that themodule M is unital. (cid:4) We recall the following concept from [27] (see also [7] on this subject).
Definition 4.5 ([27]) . Let ( L, [ − ] , α ) be a Hom-Lie algebra. A representation of L is a triple( M, ρ, A ), where M is a linear space, A : M → M and ρ : L → End( M ) are linear maps suchthat, for all x, y ∈ L , the following conditions are satisfied: ρ ( α ( x )) ◦ A = A ◦ ρ ( x ) , ρ ([ x, y ]) ◦ A = ρ ( α ( x )) ◦ ρ ( y ) − ρ ( α ( y )) ◦ ρ ( x ) . Remark 4.6.
Let ( L, [ − ] , α ) be a Hom-Lie algebra, M a linear space, A : M → M and ρ : L → End( M ) linear maps such that A is bijective. We can consider the Hom-associative algebraEnd( M )( A, A ) as in Lemma 4.3, and then the Hom-Lie algebra L (End( M )( A, A )). Then onecan check that (
M, ρ, A ) is a representation of L if and only if ρ is a morphism of Hom-Liealgebras from L to L (End( M )( A, A )).Inspired by this remark, we can introduce now the following concept:
Definition 4.7.
Let ( L, [ − ] , α, β ) be a BiHom-Lie algebra. A representation of L is a 4-tuple( M, ρ, α M , β M ), where M is a linear space, α M , β M : M → M are two commuting linear mapsand ρ : L → End( M ) is a linear map such that, for all x, y ∈ L , we have ρ ( α ( x )) ◦ α M = α M ◦ ρ ( x ) , (4.5) ρ ( β ( x )) ◦ β M = β M ◦ ρ ( x ) , (4.6) ρ ([ β ( x ) , y ]) ◦ β M = ρ ( αβ ( x )) ◦ ρ ( y ) − ρ ( β ( y )) ◦ ρ ( α ( x )) . (4.7)A first indication that this is indeed the appropriate concept of representation for BiHom-Liealgebras is provided by the following result (extending the corresponding one for Hom-associativealgebras in [6]), whose proof is straightforward and left to the reader. Proposition 4.8.
Let ( A, µ A , α A , β A ) be a BiHom-associative algebra with bijective structuremaps and ( M, α M , β M ) a left A -module, with action A ⊗ M → M , a ⊗ m (cid:55)→ a · m . Then we havea representation ( M, ρ, α M , β M ) of the BiHom-Lie algebra L ( A ) , where ρ : L ( A ) → End( M ) isthe linear map defined by ρ ( a )( m ) = a · m , for all a ∈ A , m ∈ M . A second indication is provided by the fact that, under certain circumstances, we can con-struct the semidirect product (the Hom-case is done in [27]).
Proposition 4.9.
Let ( L, [ − ] , α, β ) be a BiHom-Lie algebra and ( M, ρ, α M , β M ) a representationof L , with notation ρ ( x )( a ) = x · a , for all x ∈ L , a ∈ M . Assume that the maps α and β M are bijective. Then L (cid:110) M := ( L ⊕ M, [ − ] , α ⊕ α M , β ⊕ β M ) is a BiHom-Lie algebra ( called thesemidirect product ) , where α ⊕ α M , β ⊕ β M : L ⊕ M → L ⊕ M are defined by ( α ⊕ α M )( x, a ) =( α ( x ) , α M ( a )) and ( β ⊕ β M )( x, a ) = ( β ( x ) , β M ( a )) , and, for all x, y ∈ L and a, b ∈ M , thebracket [ − ] is defined by [( x, a ) , ( y, b )] = (cid:0) [ x, y ] , x · b − α − β ( y ) · α M β − M ( a ) (cid:1) . iHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras 17 Proof .
Follows by a direct computation that is left to the reader. (cid:4)
Proposition 4.10.
Let ( L, [ − ] , α, β ) be a BiHom-Lie algebra such that the map β is surjective, M a linear space, α M , β M : M → M two commuting linear isomorphisms and ρ : L → End( M ) a linear map. Then ( M, ρ, α M , β M ) is a representation of L if and only if ρ is a morphism ofBiHom-Lie algebras from L to L (End( M )( α M , β M )) . Proof .
Obviously, (4.5) and (4.6) are respectively equivalent to ˜ α ◦ ρ = ρ ◦ α and ˜ β ◦ ρ = ρ ◦ β , sowe only need to prove that, assuming (4.5) and (4.6), (4.7) is equivalent to ρ ([ x, y ]) = [ ρ ( x ) , ρ ( y )]for all x, y ∈ L . First we write down explicitly the bracket of L (End( M )( α M , β M )). In view ofProposition 3.15, this bracket looks as follows, for f, g ∈ End( M ):[ f, g ] = f ∗ g − (cid:0) ˜ α − ˜ β ( g ) (cid:1) ∗ (cid:0) ˜ α ˜ β − ( f ) (cid:1) = f ∗ g − (cid:0) ˜ α − (cid:0) β M ◦ g ◦ β − M (cid:1)(cid:1) ∗ (cid:0) ˜ α (cid:0) β − M ◦ f ◦ β M (cid:1)(cid:1) = f ∗ g − (cid:0) α − M ◦ β M ◦ g ◦ β − M ◦ α M (cid:1) ∗ (cid:0) α M ◦ β − M ◦ f ◦ β M ◦ α − M (cid:1) = α M ◦ f ◦ α − M ◦ g ◦ β − M − α M ◦ α − M ◦ β M ◦ g ◦ β − M ◦ α M ◦ α − M ◦ α M ◦ β − M ◦ f ◦ β M ◦ α − M ◦ β − M = α M ◦ f ◦ α − M ◦ g ◦ β − M − β M ◦ g ◦ β − M ◦ α M ◦ β − M ◦ f ◦ β M ◦ α − M ◦ β − M . Let x, y ∈ L ; we take f = ρ ( β ( x )), g = ρ ( y ). We obtain[ ρ ( β ( x )) , ρ ( y )] ◦ β M = α M ◦ ρ ( β ( x )) ◦ α − M ◦ ρ ( y ) − β M ◦ ρ ( y ) ◦ β − M ◦ α M ◦ β − M ◦ ρ ( β ( x )) ◦ β M ◦ α − M (4.5) , (4.6) = ρ ( αβ ( x )) ◦ ρ ( y ) − ρ ( β ( y )) ◦ α M ◦ ρ ( x ) ◦ α − M (4.5) = ρ ( αβ ( x )) ◦ ρ ( y ) − ρ ( β ( y )) ◦ ρ ( α ( x )) , which is the right-hand side of (4.7). So, we have that (4.7) holds if and only if ρ ([ β ( x ) , y ]) =[ ρ ( β ( x )) , ρ ( y )] for all x, y ∈ L , which is equivalent to ρ ([ a, b ]) = [ ρ ( a ) , ρ ( b )], for all a, b ∈ L ,because β is surjective. (cid:4) Proposition 4.11.
Let ( L, [ − ] , α, β ) be a BiHom-Lie algebra and define the linear map ad : L → End( L ) , ad( x )( y ) = [ x, y ] , for all x, y ∈ L . If the maps α and β are bijective, then ( L, ad , α, β ) is a representation of L . Proof .
The conditions (4.5) and (4.6) are equivalent to α ([ a, b ]) = [ α ( a ) , α ( b )] and β ([ a, b ]) =[ β ( a ) , β ( b )] for all a, b ∈ L , so we only need to prove (4.7). Note first that the skew-symmetrycondition impliesad( x )( y ) = − (cid:2) α − β ( y ) , αβ − ( x ) (cid:3) , ∀ x, y ∈ L. We compute the left-hand side of (4.7) applied to z ∈ L :(ad([ β ( x ) , y ]) ◦ β )( z ) = ad([ β ( x ) , y ])( β ( z )) = − (cid:2) α − β ( z ) , αβ − ([ β ( x ) , y ]) (cid:3) = − (cid:2) β ( α − ( z )) , (cid:2) α ( x ) , αβ − ( y ) (cid:3)(cid:3) = − (cid:2) β ( α − ( z )) , (cid:2) β ( αβ − ( x )) , α ( β − ( y )) (cid:3)(cid:3) . We compute the right-hand side of (4.7) applied to z ∈ L :(ad( αβ ( x )) ◦ ad( y ))( z ) − (ad( β ( y )) ◦ ad( α ( x ))( z )= ad( αβ ( x )) (cid:0) − (cid:2) α − β ( z ) , αβ − ( y ) (cid:3)(cid:1) − ad( β ( y )) (cid:0) − (cid:2) α − β ( z ) , α β − ( x ) (cid:3)(cid:1) (cid:2) α − β (cid:0)(cid:2) α − β ( z ) , αβ − ( y ) (cid:3)(cid:1) , αβ − αβ ( x ) (cid:3) − (cid:2) α − β (cid:0)(cid:2) α − β ( z ) , α β − ( x ) (cid:3)(cid:1) , αβ − β ( y ) (cid:3) = (cid:2) β (cid:0)(cid:2) α − β ( z ) , β − ( y ) (cid:3)(cid:1) , α ( x ) (cid:3) − (cid:2) β (cid:0)(cid:2) α − β ( z ) , αβ − ( x ) (cid:3)(cid:1) , α ( y ) (cid:3) skew-symmetry = − (cid:2) βα ( x ) , (cid:2) α − β ( z ) , αβ − ( y ) (cid:3)(cid:3) + (cid:2) β ( y ) , (cid:2) α − β ( z ) , α β − ( x ) (cid:3)(cid:3) = [ βα ( x ) , [ y, z ]] + (cid:2) β ( y ) , (cid:2) βα − ( z ) , α β − ( x ) (cid:3)(cid:3) = (cid:2) β (cid:0) αβ − ( x ) (cid:1) , (cid:2) β (cid:0) β − ( y ) (cid:1) , α (cid:0) α − ( z ) (cid:1)(cid:3)(cid:3) + (cid:2) β ( β − ( y )) , (cid:2) β (cid:0) α − ( z ) (cid:1) , α (cid:0) αβ − ( x ) (cid:1)(cid:3)(cid:3) , and (4.7) holds because of the BiHom-Jacobi identity applied to the elements a = αβ − ( x ), a (cid:48) = β − ( y ) and a (cid:48)(cid:48) = α − ( z ). (cid:4) We introduce now the dual concept to the one of BiHom-associative algebra.
Definition 5.1. A BiHom-coassociative coalgebra is a 4-tuple ( C, ∆ , ψ, ω ), in which C is a linearspace, ψ, ω : C → C and ∆ : C → C ⊗ C are linear maps, such that ψ ◦ ω = ω ◦ ψ, ( ψ ⊗ ψ ) ◦ ∆ = ∆ ◦ ψ, ( ω ⊗ ω ) ◦ ∆ = ∆ ◦ ω, (∆ ⊗ ψ ) ◦ ∆ = ( ω ⊗ ∆) ◦ ∆ . We call ψ and ω (in this order) the structure maps of C .A morphism g : ( C, ∆ C , ψ C , ω C ) → ( D, ∆ D , ψ D , ω D ) of BiHom-coassociative coalgebras isa linear map g : C → D such that ψ D ◦ g = g ◦ ψ C , ω D ◦ g = g ◦ ω C and ( g ⊗ g ) ◦ ∆ C = ∆ D ◦ g .A BiHom-coassociative coalgebra ( C, ∆ , ψ, ω ) is called counital if there exists a linear map ε : C → k (called a counit ) such that ε ◦ ψ = ε, ε ◦ ω = ε, (id C ⊗ ε ) ◦ ∆ = ω and ( ε ⊗ id C ) ◦ ∆ = ψ. A morphism of counital BiHom-coassociative coalgebras g : C → D is called counital if ε D ◦ g = ε C , where ε C and ε D are the counits of C and D , respectively. Remark 5.2.
If ( C, ∆ C , ψ C , ω C ) and ( D, ∆ D , ψ D , ω D ) are two BiHom-coassociative coalgebras,then ( C ⊗ D, ∆ C ⊗ D , ψ C ⊗ ψ D , ω C ⊗ ω D ) is also a BiHom-coassociative coalgebra (called thetensor product of C and D ), where ∆ C ⊗ D : C ⊗ D → C ⊗ D ⊗ C ⊗ D is defined by ∆( c ⊗ d ) = c ⊗ d ⊗ c ⊗ d , for all c ∈ C , d ∈ D . If C and D are counital with counits ε C and respectively ε D ,then C ⊗ D is also counital with counit ε C ⊗ ε D . Definition 5.3.
Let ( C, ∆ C , ψ C , ω C ) be a BiHom-coassociative coalgebra. A right C -comodule is a triple ( M, ψ M , ω M ), where M is a linear space, ψ M , ω M : M → M are linear maps and wehave a linear map (called a coaction) ρ : M → M ⊗ C , with notation ρ ( m ) = m (0) ⊗ m (1) , forall m ∈ M , such that the following conditions are satisfied ψ M ◦ ω M = ω M ◦ ψ M , ( ψ M ⊗ ψ C ) ◦ ρ = ρ ◦ ψ M , ( ω M ⊗ ω C ) ◦ ρ = ρ ◦ ω M , ( ω M ⊗ ∆ C ) ◦ ρ = ( ρ ⊗ ψ C ) ◦ ρ. If (
M, ψ M , ω M ) and ( N, ψ N , ω N ) are right C -comodules with coactions ρ M and respective-ly ρ N , a morphism of right C -comodules f : M → N is a linear map satisfying the conditions ψ N ◦ f = f ◦ ψ M , ω N ◦ f = f ◦ ω M and ρ N ◦ f = ( f ⊗ id C ) ◦ ρ M .If ( C, ∆ C , ψ C , ω C , ε C ) is a counital BiHom-coassociative coalgebra and ( M, ψ M , ω M ) is a right C -comodule with coaction ρ , then M is called counital if (id M ⊗ ε C ) ◦ ρ = ω M .iHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras 19 Remark 5.4.
If ( C, ∆ , ψ, ω ) is a BiHom-coassociative coalgebra, then ( C, ψ, ω ) is a right C -comodule, with coaction ρ = ∆.We discuss now the duality between BiHom-associative and BiHom-coassociative structures. Theorem 5.5.
Let ( C, ∆ , ψ, ω ) be a BiHom-coassociative coalgebra. Then its dual linear spaceis provided with a structure of BiHom-associative algebra ( C ∗ , ∆ ∗ , ω ∗ , ψ ∗ ) , where ∆ ∗ , ψ ∗ , ω ∗ are the transpose maps. Moreover, the BiHom-associative algebra C ∗ is unital whenever theBiHom-coassociative coalgebra C is counital. Proof .
The product µ = ∆ ∗ is defined from C ∗ ⊗ C ∗ to C ∗ by( f g )( x ) = ∆ ∗ ( f, g )( x ) = (cid:104) ∆( x ) , f ⊗ g (cid:105) = ( f ⊗ g )(∆( x )) = f ( x ) g ( x ) , ∀ x ∈ C, where (cid:104)· , ·(cid:105) is the natural pairing between the linear space C ⊗ C and its dual linear space. For f, g, h ∈ C ∗ and x ∈ C , we have( f g ) ψ ∗ ( h )( x ) = (cid:104) (∆ ⊗ ψ ) ◦ ∆( x ) , f ⊗ g ⊗ h (cid:105) ,ω ∗ ( f )( gh )( x ) = (cid:104) ( ω ⊗ ∆) ◦ ∆( x ) , f ⊗ g ⊗ h (cid:105) . Therefore, the BiHom-associativity condition µ ◦ ( µ ⊗ ψ ∗ − ω ∗ ⊗ µ ) = 0 follows from the BiHom-coassociativity condition (∆ ⊗ ψ − ω ⊗ ∆) ◦ ∆ = 0.Moreover, if C has a counit ε then for f ∈ C ∗ and x ∈ C we have( εf )( x ) = ε ( x ) f ( x ) = f ( ε ( x ) x ) = f ( ψ ( x )) = ψ ∗ ( f )( x ) , ( f ε )( x ) = f ( x ) ε ( x ) = f ( x ε ( x )) = f ( ω ( x )) = ω ∗ ( f )( x ) , which shows that ε is the unit of C ∗ . (cid:4) The dual of a BiHom-associative algebra (
A, µ, α, β ) is not always a BiHom-coassociativecoalgebra, because ( A ⊗ A ) ∗ (cid:41) A ∗ ⊗ A ∗ . Nevertheless, it is the case if the BiHom-associativealgebra is finite-dimensional, since ( A ⊗ A ) ∗ = A ∗ ⊗ A ∗ in this case.More generally, we can define the finite dual of A by A ◦ = { f ∈ A ∗ /f ( I ) = 0 for some cofinite ideal I of A } , where a cofinite ideal I is an ideal I ⊂ A such that A/I is finite-dimensional and where we saythat I is an ideal of A if for x ∈ I and y ∈ A we have xy ∈ I , yx ∈ I and α ( x ) ∈ I , β ( x ) ∈ I . A ◦ is a subspace of A ∗ since it is closed under multiplication by scalars and the sum of twoelements of A ◦ is again in A ◦ because the intersection of two cofinite ideals is again a cofiniteideal. If A is finite-dimensional, of course A ◦ = A ∗ . As in the classical case, one can showthat if A and B are two BiHom-associative algebras and f : A → B is a morphism of BiHom-associative algebras, then the dual map f ∗ : B ∗ → A ∗ satisfies f ∗ ( B ◦ ) ⊂ A ◦ .Therefore, a similar proof to the one of the previous theorem leads to: Theorem 5.6.
Let ( A, µ, α, β ) be a BiHom-associative algebra. Then its finite dual is pro-vided with a structure of BiHom-coassociative coalgebra ( A ◦ , ∆ , β ◦ , α ◦ ) , where ∆ = µ ◦ = µ ∗ | A ◦ and β ◦ , α ◦ are the transpose maps on A ◦ . Moreover, the BiHom-coassociative coalgebra iscounital whenever A is unital, with counit ε : A ◦ → k defined by ε ( f ) = f (1 A ) . We can now define the notion of BiHom-bialgebra.0 G. Graziani, A. Makhlouf, C. Menini and F. Panaite
Definition 5.7. A BiHom-bialgebra is a 7-tuple (
H, µ, ∆ , α, β, ψ, ω ), with the property that( H, µ, α, β ) is a BiHom-associative algebra, ( H, ∆ , ψ, ω ) is a BiHom-coassociative coalgebra andmoreover the following relations are satisfied, for all h, h (cid:48) ∈ H :∆( hh (cid:48) ) = h h (cid:48) ⊗ h h (cid:48) , (5.1) α ◦ ψ = ψ ◦ α, α ◦ ω = ω ◦ α, β ◦ ψ = ψ ◦ β, β ◦ ω = ω ◦ β, ( α ⊗ α ) ◦ ∆ = ∆ ◦ α, ( β ⊗ β ) ◦ ∆ = ∆ ◦ β,ψ ( hh (cid:48) ) = ψ ( h ) ψ ( h (cid:48) ) , ω ( hh (cid:48) ) = ω ( h ) ω ( h (cid:48) ) . We say that H is a unital and counital BiHom-bialgebra if, in addition, it admits a unit 1 H and a counit ε H such that∆(1 H ) = 1 H ⊗ H , ε H (1 H ) = 1 , ψ (1 H ) = 1 H , ω (1 H ) = 1 H ,ε H ◦ α = ε H , ε H ◦ β = ε H , ε H ( hh (cid:48) ) = ε H ( h ) ε H ( h (cid:48) ) , ∀ h, h (cid:48) ∈ H. Let us record the formula expressing the BiHom-coassociativity of ∆:∆( h ) ⊗ ψ ( h ) = ω ( h ) ⊗ ∆( h ) , ∀ h ∈ H. (5.2) Remark 5.8.
Obviously, a BiHom-bialgebra (
H, µ, ∆ , α, β, ψ, ω ) with α = β = ψ = ω reducesto a Hom-bialgebra, as used for instance in [22, 23], while a BiHom-bialgebra for which ψ = ω = α − = β − reduces to a monoidal Hom-bialgebra, in the terminology of [8].We see now that analogues of Yau’s twisting principle hold for the BiHom-structures wedefined (proofs are straightforward and left to the reader): Proposition 5.9. ( i ) Let ( A, µ ) be an associative algebra and α, β : A → A two commuting algebra endomor-phisms. Define a new multiplication µ ( α,β ) : A ⊗ A → A , by µ ( α,β ) := µ ◦ ( α ⊗ β ) . Then ( A, µ ( α,β ) , α, β ) is a BiHom-associative algebra, denoted by A ( α,β ) . If A is unital withunit A , then A ( α,β ) is also unital with unit A . ( ii ) Let ( C, ∆) be a coassociative coalgebra and ψ, ω : C → C two commuting coalgebra endo-morphisms. Define a new comultiplication ∆ ( ψ,ω ) : C → C ⊗ C , by ∆ ( ψ,ω ) := ( ω ⊗ ψ ) ◦ ∆ .Then ( C, ∆ ( ψ,ω ) , ψ, ω ) is a BiHom-coassociative coalgebra, denoted by C ( ψ,ω ) . If C is couni-tal with counit ε C , then C ( ψ,ω ) is also counital with counit ε C . ( iii ) Let ( H, µ, ∆) be a bialgebra and α, β, ψ, ω : H → H bialgebra endomorphisms such that anytwo of them commute. If we define µ ( α,β ) and ∆ ( ψ,ω ) as in ( i ) and ( ii ) , then H ( α,β,ψ,ω ) :=( H, µ ( α,β ) , ∆ ( ψ,ω ) , α, β, ψ, ω ) is a BiHom-bialgebra. More generally, a BiHom-bialgebra (
H, µ, ∆ , α, β, ψ, ω ) and multiplicative and comultiplica-tive linear maps α (cid:48) , β (cid:48) , ψ (cid:48) , ω (cid:48) such that any two of the maps α , β , ψ , ω , α (cid:48) , β (cid:48) , ψ (cid:48) , ω (cid:48) commute,give rise to a new BiHom-bialgebra ( H, µ ◦ ( α (cid:48) ⊗ β (cid:48) ) , ( ω (cid:48) ⊗ ψ (cid:48) ) ◦ ∆ , α ◦ α (cid:48) , β ◦ β (cid:48) , ψ ◦ ψ (cid:48) , ω ◦ ω (cid:48) ).Hence, if the maps α , β , ψ , ω are invertible, one can untwist the BiHom-bialgebra and geta bialgebra by taking α (cid:48) = α − , β (cid:48) = β − , ψ (cid:48) = ψ − , ω (cid:48) = ω − . Proposition 5.10.
Let ( A, µ A ) be an associative algebra and α A , β A : A → A two commutingalgebra endomorphisms. Assume that M is a left A -module, with action A ⊗ M → M , a ⊗ m (cid:55)→ a · m . Let α M , β M : M → M be two commuting linear maps such that α M ( a · m ) = α A ( a ) · α M ( m ) and β M ( a · m ) = β A ( a ) · β M ( m ) , for all a ∈ A , m ∈ M . Then ( M, α M , β M ) becomes a left moduleover A ( α A ,β A ) , with action A ( α A ,β A ) ⊗ M → M , a ⊗ m (cid:55)→ a (cid:46) m := α A ( a ) · β M ( m ) . iHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras 21 Proposition 5.11.
Let ( C, ∆ C ) be a coassociative coalgebra and ψ C , ω C : C → C two commutingcoalgebra endomorphisms. Assume that M is a right C -comodule, with coaction ρ : M → M ⊗ C , ρ ( m ) = m (0) ⊗ m (1) , for all m ∈ M . Let ψ M , ω M : M → M be two commuting linear maps suchthat ( ψ M ⊗ ψ C ) ◦ ρ = ρ ◦ ψ M and ( ω M ⊗ ω C ) ◦ ρ = ρ ◦ ω M . Then ( M, ψ M , ω M ) becomes a rightcomodule over the BiHom-coassociative coalgebra C ( ψ C ,ω C ) , with coaction M → M ⊗ C ( ψ C ,ω C ) , m (cid:55)→ m (cid:104) (cid:105) ⊗ m (cid:104) (cid:105) := ω M ( m (0) ) ⊗ ψ C ( m (1) ) . We describe in what follows primitive elements of a BiHom-bialgebra.Let (
H, µ, ∆ , α, β, ψ, ω ) be a unital and counital BiHom-bialgebra with a unit 1 = η (1) anda counit ε . We assume that α and β are bijective.An element x ∈ H is called primitive if ∆( x ) = 1 ⊗ x + x ⊗ Lemma 5.12.
Let x be a primitive element in H . Then ε ( x )1 = ω ( x ) − x = ψ ( x ) − x , andtherefore ω ( x ) = ψ ( x ) . Moreover, α p β q ( x ) is also a primitive element for any p, q ∈ Z . Proof .
By the counit property, we have ω ( x ) = (id H ⊗ ε )(1 ⊗ x + x ⊗
1) = ε ( x )1 + ε (1) x = ε ( x )1 + x , and similarly ψ ( x ) = ε ( x )1 + x .Since α and β are comultiplicative maps and α p β q (1) = 1, it follows that α p β q ( x ) is a primitiveelement whenever x is a primitive element. (cid:4) Proposition 5.13.
Let ( H, µ, ∆ , α, β, ψ, ω ) be a unital and counital BiHom-bialgebra, with unit η (1) and counit ε . Assume that α and β are bijective. If x and y are two primitive elementsin H , then the commutator [ x, y ] = xy − α − β ( y ) αβ − ( x ) is also a primitive element.Consequently, the set of all primitive elements of H , denoted by Prim( H ) , has a structure ofBiHom-Lie algebra. Proof .
We compute∆( xy ) = ∆( x )∆( y ) = (1 ⊗ x + x ⊗ ⊗ y + y ⊗ ⊗ xy + β ( y ) ⊗ α ( x ) + α ( x ) ⊗ β ( y ) + xy ⊗ , ∆ (cid:0) α − β ( y ) αβ − ( x ) (cid:1) = ∆ (cid:0) α − β ( y ) (cid:1) ∆ (cid:0) αβ − ( x ) (cid:1) = (cid:0) ⊗ α − β ( y ) + α − β ( y ) ⊗ (cid:1)(cid:0) ⊗ αβ − ( x ) + αβ − ( x ) ⊗ (cid:1) = 1 ⊗ α − β ( y ) αβ − ( x ) + β (cid:0) αβ − ( x ) (cid:1) ⊗ α (cid:0) α − β ( y ) (cid:1) + α (cid:0) α − β ( y ) (cid:1) ⊗ β (cid:0) αβ − ( x ) (cid:1) + α − β ( y ) αβ − ( x ) ⊗
1= 1 ⊗ α − β ( y ) αβ − ( x ) + α ( x ) ⊗ β ( y )+ β ( y ) ⊗ α ( x ) + α − β ( y ) αβ − ( x ) ⊗ . Therefore, we have∆([ x, y ]) = ∆( xy ) − ∆ (cid:0) α − β ( y ) αβ − ( x ) (cid:1) = 1 ⊗ [ x, y ] + [ x, y ] ⊗ , which means that P rim ( H ) is closed under the bracket multiplication [ · , · ]. Hence, Prim( H ) isa BiHom-Lie algebra by Proposition 3.15. (cid:4) Now, we introduce the notion of H -module BiHom-algebra, where H is a BiHom-bialgebra. Definition 5.14.
Let (
H, µ H , ∆ H , α H , β H , ψ H , ω H ) be a BiHom-bialgebra for which the maps α H , β H , ψ H , ω H are bijective. A BiHom-associative algebra ( A, µ A , α A , β A ) is called a left H -module BiHom-algebra if ( A, α A , β A ) is a left H -module, with action denoted by H ⊗ A → A , h ⊗ a (cid:55)→ h · a , such that the following condition is satisfied h · ( aa (cid:48) ) = (cid:2) α − H (cid:0) ω − H ( h ) (cid:1) · a (cid:3) [ β − H (cid:0) ψ − H ( h ) (cid:1) · a (cid:48) ] , ∀ h ∈ H, a, a (cid:48) ∈ A. (5.3)2 G. Graziani, A. Makhlouf, C. Menini and F. Panaite Remark 5.15.
This concept contains as particular cases the concepts of module algebras overa Hom-bialgebra, respectively monoidal Hom-bialgebra, introduced in [31], respectively [11].The choice of (5.3) is motivated by the following result, whose proof is also left to the reader:
Proposition 5.16.
Let ( H, µ H , ∆ H ) be a bialgebra and ( A, µ A ) a left H -module algebra in theusual sense, with action denoted by H ⊗ A → A , h ⊗ a (cid:55)→ h · a . Let α H , β H , ψ H , ω H : H → H bebialgebra endomorphisms of H such that any two of them commute; let α A , β A : A → A be twocommuting algebra endomorphisms such that, for all h ∈ H and a ∈ A , we have α A ( h · a ) = α H ( h ) · α A ( a ) and β A ( h · a ) = β H ( h ) · β A ( a ) . If we consider the BiHom-bialgebra H ( α H ,β H ,ψ H ,ω H ) and the BiHom-associative algebra A ( α A ,β A ) as defined before, then A ( α A ,β A ) is a left H ( α H ,β H ,ψ H ,ω H ) -module BiHom-algebra in the abovesense, with action H ( α H ,β H ,ψ H ,ω H ) ⊗ A ( α A ,β A ) → A ( α A ,β A ) , h ⊗ a (cid:55)→ h (cid:46) a := α H ( h ) · β A ( a ) . In this section, we introduce the concept of monoidal BiHom-Hopf algebra and discuss a possiblegeneralization of Hom-Hopf algebras to BiHom-Hopf algebras.We begin with a lemma whose proof is obvious.
Lemma 6.1.
Let ( A, µ, α, β ) be a BiHom-associative algebra. Define A := { a ∈ A/α ( a ) = β ( a ) = a } . Then ( A, µ ) is an associative algebra. If A is unital with unit A , then A is alsothe unit of A ( in particular, it follows that the unit of a BiHom-associative algebra, if it exists,is unique ) . Proposition 6.2.
Let ( A, µ, α, β ) be a BiHom-associative algebra and ( C, ∆ , ψ, ω ) a BiHom-coassociative coalgebra. Set, for f, g ∈ Hom(
C, A ) , f (cid:63) g = µ ◦ ( f ⊗ g ) ◦ ∆ . Define the linear maps φ, γ : Hom( C, A ) → Hom(
C, A ) by φ ( f ) = α ◦ f ◦ ω and γ ( f ) = β ◦ f ◦ ψ , for all f ∈ Hom(
C, A ) .Then (Hom( C, A ) , (cid:63), φ, γ ) is a BiHom-associative algebra.Moreover, if A is unital with unit A and C is counital with counit ε , then Hom(
C, A ) is a unital BiHom-asssociative algebra with unit η ◦ ε , where we denote by η the linear map η : k → A , η (1) = 1 A .In particular, if we denote by Hom(
C, A ) the linear subspace of Hom(
C, A ) consisting of thelinear maps f : C → A such that α ◦ f ◦ ω = f and β ◦ f ◦ ψ = f , then (Hom( C, A ) , (cid:63), η ◦ ε ) isan associative unital algebra. Proof .
Let f, g, h ∈ Hom(
C, A ). We have φ ( f ) (cid:63) ( g (cid:63) h ) = µ ◦ ( φ ( f ) ⊗ ( g (cid:63) h ))∆ = µ ◦ ( φ ( f ) ⊗ ( µ ◦ ( g ⊗ h ) ◦ ∆))∆= µ ◦ (( α ⊗ µ ) ◦ ( f ⊗ g ⊗ h ) ◦ ( ω ⊗ ∆))∆ . Similarly,( f (cid:63) g ) (cid:63) γ ( h ) = µ ◦ (( µ ⊗ β ) ◦ ( f ⊗ g ⊗ h ) ◦ (∆ ⊗ ψ ))∆ . The BiHom-associativity of µ and the BiHom-coassociativity of ∆ lead to the BiHom-associati-vity of the convolution product (cid:63) .iHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras 23The map η ◦ ε is the unit for the convolution product. Indeed, for f ∈ Hom(
C, A ) and x ∈ C ,we have( f (cid:63) ( η ◦ ε ))( x ) = µ ◦ ( f ⊗ η ◦ ε ) ◦ ∆( x ) = µ ( f ( x ) ⊗ η ◦ ε ( x )) = ε ( x ) µ ( f ( x ) ⊗ η (1))= ε ( x )( α ◦ f )( x ) = ( α ◦ f )( x ε ( x )) = α ◦ f ◦ ω ( x ) . A similar calculation shows that ( η ◦ ε ) (cid:63) f = β ◦ f ◦ ψ .The last statement follows from Lemma 6.1. (cid:4) Definition 6.3.
Let (
H, µ, ∆ , α, β, ψ, ω ) be a unital and counital BiHom-bialgebra. We saythat H is a monoidal BiHom-bialgebra if α , β , ψ , ω are bijective and ω = α − and ψ = β − .We will refer to a monoidal BiHom-bialgebra as the 5-tuple ( H, µ, ∆ , α, β ).If ( H, µ, ∆ , α, β ) is a monoidal BiHom-bialgebra, we can consider the associative unital alge-bra Hom( H, H ), and since ω = α − and ψ = β − , it follows that id H ∈ Hom(
H, H ). Definition 6.4.
Let (
H, µ, ∆ , α, β ) be a monoidal BiHom-bialgebra with a unit 1 H and a co-unit ε H . A linear map S : H → H is called an antipode if α ◦ S = S ◦ α and β ◦ S = S ◦ β (i.e., S ∈ Hom(
H, H )) and S is the convolution inverse of id H in Hom( H, H ), that is S ( h ) h = ε H ( h )1 H = h S ( h ) , ∀ h ∈ H. A monoidal BiHom-Hopf algebra is a monoidal BiHom-bialgebra endowed with an antipode.Obviously, if the antipode exists, it is unique; we will refer to the monoidal BiHom-Hopfalgebra as the 8-tuple ( H, µ, ∆ , α, β, H , ε H , S ). Proposition 6.5.
Let ( H, µ, ∆ , H , ε H ) be a Hopf algebra ( in the usual sense ) with antipo-de S . Let α, β : H → H be two unital and counital commuting bialgebra automorphisms. Then ( H, µ ◦ ( α ⊗ β ) , ( α − ⊗ β − ) ◦ ∆ , α, β, H , ε H , S ) is a monoidal BiHom-bialgebra. Proof .
A straightforward computation. Let us only note that α , β being bialgebra maps, theyautomatically commute with S . (cid:4) We state now the basic properties of the antipode.
Proposition 6.6.
Let ( H, µ, ∆ , α, β, H , ε H , S ) be a monoidal BiHom-Hopf algebra. Then ( i ) S (1 H ) = 1 H and ε H ◦ S = ε H ; ( ii ) S ( β ( a ) α ( b )) = S ( β ( b )) S ( α ( a )) , for all a, b ∈ H ; ( iii ) α ( S ( h ) ) ⊗ β ( S ( h ) ) = β ( S ( h )) ⊗ α ( S ( h )) , for all h ∈ H . Proof . (i) By ∆(1 H ) = 1 H ⊗ H we obtain S (1 H )1 H = ε H (1 H )1 H , so α ( S (1 H )) = 1 H , and since α ◦ S = S ◦ α and α (1 H ) = 1 H we obtain S (1 H ) = 1 H . Then, if h ∈ H , we apply ε H to the equality h S ( h ) = ε H ( h )1 H , and we obtain ε H ( h ) ε H ( S ( h )) = ε H ( h ), so ε H ( S ( ε H ( h ) h )) = ε H ( h ),hence ε H ( S ( β − ( h ))) = ε H ( h ), and since S ◦ β = β ◦ S and ε H ◦ β = ε H we obtain ε H ◦ S = ε H .(ii) We define the linear maps R, L, m : H ⊗ H → H by the formulae (for all a, b ∈ H ): R ( a ⊗ b ) = S ( β ( b )) S ( α ( a )) , L ( a ⊗ b ) = S ( β ( a ) α ( b )) , m ( a ⊗ b ) = β ( a ) α ( b ) . One can easily check that
R, L, m ∈ Hom( H ⊗ H, H ) (where H ⊗ H is the tensor productBiHom-coassociative coalgebra). Thus, to prove that R = L , it is enough to prove that L (respectively R ) is a left (respectively right) convolution inverse of m in Hom( H ⊗ H, H ). Wecompute(
L (cid:63) m )( a ⊗ b ) = L ( a ⊗ b ) m ( a ⊗ b ) = S ( β ( a ) α ( b ))( β ( a ) α ( b ))4 G. Graziani, A. Makhlouf, C. Menini and F. Panaite= S ( β ( a ) α ( b ) )( β ( a ) α ( b ) ) = S (( β ( a ) α ( b )) )( β ( a ) α ( b )) = ε H ( β ( a ) α ( b ))1 H = ε H ( a ) ε H ( b )1 H , ( m (cid:63) R )( a ⊗ b ) = m ( a ⊗ b ) R ( a ⊗ b ) = ( β ( a ) α ( b ))( S ( β ( b )) S ( α ( a )))= α (cid:0) α − β ( a ) b (cid:1) ( β ( S ( b )) α ( S ( a ))) = (cid:0)(cid:0) α − β ( a ) b (cid:1) β ( S ( b ))) αβ ( S ( a ) (cid:1) = ( β ( a )( b S ( b ))) αβ ( S ( a )) = ( β ( a ) ε H ( b )1 H ) αβ ( S ( a ))= ε H ( b ) αβ ( a ) αβ ( S ( a )) = ε H ( b ) αβ ( a S ( a ))= ε H ( b ) αβ ( ε H ( a )1 H ) = ε H ( a ) ε H ( b )1 H , finishing the proof.(iii) similar to the proof of (ii), by defining the linear maps L , R , δ : H → H ⊗ H , L ( h ) = α ( S ( h ) ) ⊗ β ( S ( h ) ) , R ( h ) = β ( S ( h )) ⊗ α ( S ( h )) , δ ( h ) = α ( h ) ⊗ β ( h ) , for all h ∈ H , and proving that L (respectively R ) is a left (respectively right) convolutioninverse of δ in Hom( H, H ⊗ H ). (cid:4) Remark 6.7.
We had to restrict the definition of the antipode to the class of monoidal BiHom-bialgebras because, if H is a Hopf algebra with antipode S and we make an arbitrary Yau twistof H , then in general S will not satisfy the defining property of an antipode for the Yau twist,as the next example shows. Example 6.8.
Let k be a field and let H = k [ X ], regarded as a Hopf algebra in the usual way.Let α : H → H be the algebra map defined by setting α ( X ) = X and let β = ω = ψ = Id H .Then we can consider the BiHom-bialgebra H ( α,β,ψ,ω ) := ( H, µ ( α,β ) , ∆ ( ψ,ω ) , α, β, ψ, ω ), where µ : H ⊗ H → H is the usual multiplication and ∆ : H → H ⊗ H is the usual comultiplication.Moreover H ( α,β,ψ,ω ) has unit 1 H = η H (1 k ) and counit ε H that coincide with the ones of H .Assume that there exists a linear map S : H → H such that S (cid:63)
Id = Id (cid:63)S = η H ◦ ε H , i.e., µ ( α,β ) ◦ ( S ⊗ Id) ◦ ∆ ( ψ,ω ) = µ ( α,β ) ◦ (Id ⊗ S ) ◦ ∆ ( ψ,ω ) = η H ◦ ε H . (6.1)Then we compute (cid:0) µ ( α,β ) ◦ (Id ⊗ S ) ◦ ∆ ( ψ,ω ) (cid:1) ( X ) = α ( X ) S (1) + α (1) S ( X ) = X S (1) + S ( X ) , ( µ ( α,β ) ◦ ( S ⊗ Id) ◦ ∆ ( ψ,ω ) )( X ) = α ( S ( X ))1 + α ( S (1)) X, and ( η H ◦ ε H )( X ) = 01 H , so that from (6.1) we get S ( X ) = − X S (1) (6.2)and α ( S ( X )) = − α ( S (1)) X, (6.3)and hence − α ( S (1)) X (6.3) = α ( S ( X )) (6.2) = α (cid:0) − X S (1)) def. α = − α (cid:0) X (cid:1) α ( S (1)) def. α = − X α ( S (1)) , so that we get α ( S (1)) X = X α ( S (1)), which implies that α ( S (1)) = 0.On the other hand, we have1 = ( η H ◦ ε H )(1) (6.1) = (cid:0) µ ( α,β ) ◦ ( S ⊗ Id) ◦ ∆ ( ψ,ω ) (cid:1) (1) = α ( S (1))1 = 0 , and this is a contradiction.iHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras 25In view of all the above, we propose the following definition for what might be a BiHom-Hopfalgebra, that is moreover invariant under Yau twisting: Definition 6.9.
Let (
H, µ, ∆ , α, β, ψ, ω ) be a unital and counital BiHom-bialgebra witha unit 1 H and a counit ε H . A linear map S : H → H is called an antipode if it commuteswith all the maps α , β , ψ , ω and it satisfies the following relation: βψ ( S ( h )) αω ( h ) = ε H ( h )1 H = βψ ( h ) αω ( S ( h )) , ∀ h ∈ H. A BiHom-Hopf algebra is a unital and counital BiHom-bialgebra with an antipode.We hope to make a more detailed analysis of these structures in a forthcoming paper.
Inspired by Proposition 3.8, by the concept of pseudotwistor for associative algebras introducedin [21] and its generalization for Hom-associative algebras introduced in [23], we arrive at thefollowing concept and result:
Theorem 7.1.
Let ( D, µ, ˜ α, ˜ β ) be a BiHom-associative algebra and α, β : D → D two multiplica-tive linear maps such that any two of the maps ˜ α , ˜ β , α , β commute. Let T : D ⊗ D → D ⊗ D a linear map and assume that there exist two linear maps ˜ T , ˜ T : D ⊗ D ⊗ D → D ⊗ D ⊗ D suchthat the following relations hold: ( α ⊗ α ) ◦ T = T ◦ ( α ⊗ α ) , (7.1)( β ⊗ β ) ◦ T = T ◦ ( β ⊗ β ) , (7.2)( ˜ α ⊗ ˜ α ) ◦ T = T ◦ ( ˜ α ⊗ ˜ α ) , (7.3)( ˜ β ⊗ ˜ β ) ◦ T = T ◦ ( ˜ β ⊗ ˜ β ) , (7.4) T ◦ ( ˜ α ⊗ µ ) = ( ˜ α ⊗ µ ) ◦ ˜ T ◦ ( T ⊗ id D ) , (7.5) T ◦ ( µ ⊗ ˜ β ) = ( µ ⊗ ˜ β ) ◦ ˜ T ◦ (id D ⊗ T ) , (7.6)˜ T ◦ ( T ⊗ id D ) ◦ ( α ⊗ T ) = ˜ T ◦ (id D ⊗ T ) ◦ ( T ⊗ β ) . (7.7) Then D Tα,β := (
D, µ ◦ T, ˜ α ◦ α, ˜ β ◦ β ) is also a BiHom-associative algebra. The map T is calledan ( α, β ) -BiHom-pseudotwistor and the two maps ˜ T , ˜ T are called the companions of T . Inthe particular case α = β = id D , we simply call T a BiHom-pseudotwistor and we denote D Tα,β by D T . Proof .
The fact that ˜ α ◦ α and ˜ β ◦ β are multiplicative with respect to µ ◦ T follows immediatelyfrom (7.1)–(7.4) and the fact that α , β , ˜ α , ˜ β are multiplicative with respect to µ . Now we provethe BiHom-associativity of µ ◦ T :( µ ◦ T ) ◦ (( µ ◦ T ) ⊗ ( ˜ β ◦ β )) = µ ◦ T ◦ ( µ ⊗ ˜ β ) ◦ ( T ⊗ β ) (7.6) = µ ◦ ( µ ⊗ ˜ β ) ◦ ˜ T ◦ (id D ⊗ T ) ◦ ( T ⊗ β ) (7.7) = µ ◦ ( µ ⊗ ˜ β ) ◦ ˜ T ◦ ( T ⊗ id D ) ◦ ( α ⊗ T ) = µ ◦ ( ˜ α ⊗ µ ) ◦ ˜ T ◦ ( T ⊗ id D ) ◦ ( α ⊗ T ) (7.5) = µ ◦ T ◦ ( ˜ α ⊗ µ ) ◦ ( α ⊗ T ) = ( µ ◦ T ) ◦ (( ˜ α ◦ α ) ⊗ ( µ ◦ T )) , finishing the proof. (cid:4) D, µ ) is an associative algebra and ˜ α = ˜ β = α = β = id D , an ( α, β )-BiHom-pseudotwistor reduces to a pseudotwistor (as defined in [21]) and the BiHom-associative alge-bra D Tα,β is actually associative.We show now that Proposition 3.8 is a particular case of Theorem 7.1.
Proposition 7.2.
Let ( D, µ, ˜ α, ˜ β ) be a BiHom-associative algebra and α, β : D → D two multi-plicative linear maps such that any two of the maps ˜ α , ˜ β , α , β commute. Define the maps T : D ⊗ D → D ⊗ D, T = α ⊗ β, ˜ T : D ⊗ D ⊗ D → D ⊗ D ⊗ D, ˜ T = id D ⊗ id D ⊗ β, ˜ T : D ⊗ D ⊗ D → D ⊗ D ⊗ D, ˜ T = α ⊗ id D ⊗ id D . Then T is an ( α, β ) -BiHom-pseudotwistor with companions ˜ T , ˜ T and the BiHom-associativealgebras D Tα,β and D ( α,β ) coincide. Proof .
The conditions (7.1)–(7.4) are obviously satisfied. We check (7.5), for a, b, c ∈ D : (cid:0) ( ˜ α ⊗ µ ) ◦ ˜ T ◦ ( T ⊗ id D ) (cid:1) ( a ⊗ b ⊗ c ) = (cid:0) ( ˜ α ⊗ µ ) ◦ ˜ T (cid:1) ( α ( a ) ⊗ β ( b ) ⊗ c )= ( ˜ α ⊗ µ )( α ( a ) ⊗ β ( b ) ⊗ β ( c )) = ( ˜ α ◦ α )( a ) ⊗ β ( bc )= T ( ˜ α ( a ) ⊗ bc ) = ( T ◦ ( ˜ α ⊗ µ ))( a ⊗ b ⊗ c ) . The condition (7.6) is similar, so we check (7.7): (cid:0) ˜ T ◦ ( T ⊗ id D ) ◦ ( α ⊗ T ) (cid:1) ( a ⊗ b ⊗ c ) = (cid:0) ˜ T ◦ ( T ⊗ id D ) (cid:1) ( α ( a ) ⊗ α ( b ) ⊗ β ( c ))= ˜ T (cid:0) α ( a ) ⊗ βα ( b ) ⊗ β ( c ) (cid:1) = α ( a ) ⊗ βα ( b ) ⊗ β ( c ) = ˜ T (cid:0) α ( a ) ⊗ αβ ( b ) ⊗ β ( c ) (cid:1) = (cid:0) ˜ T ◦ (id D ⊗ T ) (cid:1) ( α ( a ) ⊗ β ( b ) ⊗ β ( c )) = (cid:0) ˜ T ◦ (id D ⊗ T (cid:1) ◦ ( T ⊗ β ))( a ⊗ b ⊗ c ) . It is obvious that D Tα,β and D ( α,β ) coincide. (cid:4) Example 7.3.
We consider the 2-dimensional BiHom-associative algebra (
D, µ, ˜ α, ˜ β ) definedwith respect to a basis B = { e , e } by µ ( e , e ) = µ ( e , e ) = e , µ ( e , e ) = µ ( e , e ) = e , ˜ α ( e ) = e , ˜ α ( e ) = e , ˜ β ( e ) = e , ˜ β ( e ) = e . We have the following multiplicative linear maps α, β defined with respect to the basis B by α ( e ) = e , α ( e ) = ae + (1 − a ) e , β ( e ) = e , β ( e ) = be + (1 − b ) e , where a , b are parameters in k . One can easily see that any two of the maps ˜ α , ˜ β , α , β commute.By the previous proposition, we can construct the BiHom-associative algebra D ( α,β ) = ( D, µ T = µ ◦ ( α ⊗ β ) , α T = ˜ α ◦ α, β T = ˜ β ◦ β ) defined on the basis B by µ T ( e , e ) = e , µ T ( e , e ) = e , µ T ( e , e ) = ae + (1 − a ) e ,µ T ( e , e ) = ae + (1 − a ) e , α T ( e ) = e , α T ( e ) = ae + (1 − a ) e ,β T ( e ) = e , β T ( e ) = e . Definition 7.4 ([9, 28]) . Let (
A, µ A ), ( B, µ B ) be two associative algebras. A twisting map between A and B is a linear map R : B ⊗ A → A ⊗ B satisfying the conditions R ◦ (id B ⊗ µ A ) = ( µ A ⊗ id B ) ◦ (id A ⊗ R ) ◦ ( R ⊗ id A ) , (7.8) R ◦ ( µ B ⊗ id A ) = (id A ⊗ µ B ) ◦ ( R ⊗ id B ) ◦ (id B ⊗ R ) . If this is the case, the map µ R = ( µ A ⊗ µ B ) ◦ (id A ⊗ R ⊗ id B ) is an associative product on A ⊗ B ;the associative algebra ( A ⊗ B, µ R ) is denoted by A ⊗ R B and called the twisted tensor product of A and B afforded by R .iHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras 27We introduce now twisted tensor products of BiHom-associative algebras. Definition 7.5.
Let (
A, µ A , α A , β A ) and ( B, µ B , α B , β B ) be two BiHom-associative algebrassuch that the maps α A , β A , α B , β B are bijective. A linear map R : B ⊗ A → A ⊗ B is calleda BiHom-twisting map between A and B if the following conditions are satisfied( α A ⊗ α B ) ◦ R = R ◦ ( α B ⊗ α A ) , (7.9)( β A ⊗ β B ) ◦ R = R ◦ ( β B ⊗ β A ) , (7.10) R ◦ ( α B ⊗ µ A ) = ( µ A ⊗ β B ) ◦ (id A ⊗ R ) ◦ (cid:0) id A ⊗ α B β − B ⊗ id A (cid:1) ◦ ( R ⊗ id A ) , (7.11) R ◦ ( µ B ⊗ β A ) = ( α A ⊗ µ B ) ◦ ( R ⊗ id B ) ◦ (cid:0) id B ⊗ α − A β A ⊗ id B (cid:1) ◦ (id B ⊗ R ) . (7.12)If we use the standard Sweedler-type notation R ( b ⊗ a ) = a R ⊗ b R = a r ⊗ b r , for a ∈ A , b ∈ B ,then the above conditions may be rewritten (for all a, a (cid:48) ∈ A and b, b (cid:48) ∈ B ) as follows α A ( a R ) ⊗ α B ( b R ) = α A ( a ) R ⊗ α B ( b ) R , (7.13) β A ( a R ) ⊗ β B ( b R ) = β A ( a ) R ⊗ β B ( b ) R , (7.14)( aa (cid:48) ) R ⊗ α B ( b ) R = a R a (cid:48) r ⊗ β B (cid:0)(cid:2) α B β − B ( b R ) (cid:3) r (cid:1) , (7.15) β A ( a ) R ⊗ ( bb (cid:48) ) R = α A ([ α − A β A ( a R )] r ) ⊗ b r b (cid:48) R . (7.16) Proposition 7.6.
Let ( A, µ A , α A , β A ) and ( B, µ B , α B , β B ) be two BiHom-associative algebraswith bijective structure maps, R : B ⊗ A → A ⊗ B a BiHom-twisting map. Define the linear map T : ( A ⊗ B ) ⊗ ( A ⊗ B ) → ( A ⊗ B ) ⊗ ( A ⊗ B ) ,T (( a ⊗ b ) ⊗ ( a (cid:48) ⊗ b (cid:48) )) = ( a ⊗ b R ) ⊗ ( a (cid:48) R ⊗ b (cid:48) ) . Then T is a BiHom-pseudotwistor for the tensor product ( A ⊗ B, µ A ⊗ B , α A ⊗ α B , β A ⊗ β B ) of A and B , with companions ˜ T = (cid:0) id A ⊗ α − B β B ⊗ id A ⊗ id B ⊗ id A ⊗ id B (cid:1) ◦ T ◦ (id A ⊗ α B β − B ⊗ id A ⊗ id B ⊗ id A ⊗ id B ) , ˜ T = (cid:0) id A ⊗ id B ⊗ id A ⊗ id B ⊗ α A β − A ⊗ id B (cid:1) ◦ T ◦ (id A ⊗ id B ⊗ id A ⊗ id B ⊗ α − A β A ⊗ id B ) , where we use the standard notation for T . The BiHom-associative algebra ( A ⊗ B ) T is denotedby A ⊗ R B and is called the BiHom-twisted tensor product of A and B ; its multiplication isdefined by ( a ⊗ b )( a (cid:48) ⊗ b (cid:48) ) = aa (cid:48) R ⊗ b R b (cid:48) , and the structure maps are α A ⊗ α B and β A ⊗ β B . Proof .
We begin by proving the following relation, for all a ∈ A , b ∈ B : α − B β B (cid:0)(cid:2) α B β − B ( b ) (cid:3) R (cid:1) ⊗ a R = b R ⊗ α A β − A (cid:0)(cid:2) α − A β A ( a ) (cid:3) R (cid:1) . (7.17)This relation is equivalent to β B (cid:0)(cid:2) α B β − B ( b ) (cid:3) R (cid:1) ⊗ β A ( a R ) = α B ( b R ) ⊗ α A (cid:0)(cid:2) α − A β A ( a ) (cid:3) R (cid:1) , which, by using (7.13) and (7.14), is equivalent to α B ( b ) R ⊗ β A ( a ) R = α B ( b ) R ⊗ β A ( a ) R , which is obviously true.8 G. Graziani, A. Makhlouf, C. Menini and F. PanaiteWe need to prove the relations (7.1)–(7.7) (with ˜ α = α A ⊗ α B , ˜ β = β A ⊗ β B , α = β =id A ⊗ id B ). We will prove only (7.7), while (7.1)–(7.6) are very easy and left to the reader. Wecompute ( r and R are two more copies of R )˜ T ◦ ( T ⊗ id) ◦ (id ⊗ T )( a ⊗ b ⊗ a (cid:48) ⊗ b (cid:48) ⊗ a (cid:48)(cid:48) ⊗ b (cid:48)(cid:48) ) = ˜ T ( a ⊗ b r ⊗ a (cid:48) r ⊗ b (cid:48) R ⊗ a (cid:48)(cid:48) R ⊗ b (cid:48)(cid:48) )= a ⊗ α − B β B (cid:0)(cid:2) α B β − B ( b r ) (cid:3) R (cid:1) ⊗ a (cid:48) r ⊗ b (cid:48) R ⊗ ( a (cid:48)(cid:48) R ) R ⊗ b (cid:48)(cid:48) , ˜ T ◦ (id ⊗ T ) ◦ ( T ⊗ id)( a ⊗ b ⊗ a (cid:48) ⊗ b (cid:48) ⊗ a (cid:48)(cid:48) ⊗ b (cid:48)(cid:48) ) = ˜ T ( a ⊗ b r ⊗ a (cid:48) r ⊗ b (cid:48) R ⊗ a (cid:48)(cid:48) R ⊗ b (cid:48)(cid:48) )= a ⊗ ( b r ) R ⊗ a (cid:48) r ⊗ b (cid:48) R ⊗ α A β − A (cid:0)(cid:2) α − A β A ( a (cid:48)(cid:48) R ) (cid:3) R (cid:1) ⊗ b (cid:48)(cid:48) , and the two terms are equal because of the relation (7.17). (cid:4) Remark 7.7.
Let (
A, µ A , α A , β A ) and ( B, µ B , α B , β B ) be two BiHom-associative algebras withbijective structure maps. Then obviously the linear map R : B ⊗ A → A ⊗ B , R ( b ⊗ a ) = a ⊗ b ,is a BiHom-twisting map and the BiHom-twisted tensor product A ⊗ R B coincides with theordinary tensor product A ⊗ B . Proposition 7.8.
Let ( A, µ A ) and ( B, µ B ) be two associative algebras, α A , β A : A → A twocommuting algebra isomorphisms of A and α B , β B : B → B two commuting algebra isomorphismsof B . Let P : B ⊗ A → A ⊗ B be a twisting map satisfying the conditions ( α A ⊗ α B ) ◦ P = P ◦ ( α B ⊗ α A ) , (7.18)( β A ⊗ β B ) ◦ P = P ◦ ( β B ⊗ β A ) . (7.19) Define the linear map U : B ⊗ A → A ⊗ B, U ( b ⊗ a ) = β − A ( β A ( a ) P ) ⊗ α − B ( α B ( b ) P ) . Then U is a BiHom-twisting map between the BiHom-associative algebras A ( α A ,β A ) and B ( α B ,β B ) and the BiHom-associative algebras A ( α A ,β A ) ⊗ U B ( α B ,β B ) and ( A ⊗ P B ) ( α A ⊗ α B ,β A ⊗ β B ) coincide. Proof .
We only prove (7.15) for U and leave the rest to the reader. We compute (by denotingby p another copy of P and by u another copy of U )( aa (cid:48) ) U ⊗ α B ( b ) U = [ α A ( a ) β A ( a (cid:48) )] U ⊗ α B ( b ) U = β − A (cid:0)(cid:2) β A α A ( a ) β A ( a (cid:48) ) (cid:3) P (cid:1) ⊗ α − B ( α B ( b ) P ) (7.8) = β − A ( β A α A ( a ) P ) β − A (cid:0) β A ( a (cid:48) ) p (cid:1) ⊗ α − B (( α B ( b ) P ) p )= β − A ( α A ( β A ( a )) P ) β − A (cid:0) β A ( a (cid:48) ) p (cid:1) ⊗ α − B ([ α B ( α B ( b )) P ] p ) (7.18) = β − A α A ( β A ( a ) P ) β − A (cid:0) β A ( a (cid:48) ) p (cid:1) ⊗ α − B ( α B ( α B ( b ) P ) p ) ,a U a (cid:48) u ⊗ β B (cid:0)(cid:2) α B β − B ( b U ) (cid:3) u (cid:1) = α A ( a U ) β A ( a (cid:48) u ) ⊗ β B (cid:0)(cid:2) α B β − B ( b U ) (cid:3) u (cid:1) = α A β − A ( β A ( a ) P ) β A ( a (cid:48) u ) ⊗ β B (cid:0)(cid:2) β − B ( α B ( b ) P ) (cid:3) u (cid:1) = α A β − A ( β A ( a ) P ) β A ( a (cid:48) ) p ⊗ α − B β B (cid:0)(cid:2) α B β − B ( α B ( b ) P ) (cid:3) p (cid:1) = α A β − A ( β A ( a ) P ) β A ( a (cid:48) ) p ⊗ α − B β B (cid:0) β − B ( α B ( α B ( b ) P )) p (cid:1) = α A β − A ( β A ( a ) P ) β − A (cid:0) β A ( a (cid:48) ) (cid:1) p ⊗ α − B β B (cid:0) β − B ( α B ( α B ( b ) P )) p (cid:1) (7.19) = α A β − A ( β A ( a ) P ) β − A (cid:0) β A ( a (cid:48) ) p (cid:1) ⊗ α − B ( α B ( α B ( b ) P ) p ) , finishing the proof. (cid:4) iHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras 29 We construct first a large family of BiHom-twisting maps.
Theorem 8.1.
Let ( H, µ H , ∆ H , α H , β H , ψ H , ω H ) be a BiHom-bialgebra, ( A, µ A , α A , β A ) a left H -module BiHom-algebra, with action denoted by H ⊗ A → A , h ⊗ a (cid:55)→ h · a , and assume thatall structure maps α H , β H , ψ H , ω H , α A , β A are bijective. Let m, n, p ∈ Z . Define the linear map R m,n,p : H ⊗ A → A ⊗ H, R m,n,p ( h ⊗ a ) = α mH β nH ω pH ( h ) · β − A ( a ) ⊗ ψ − H ( h ) . Then R m,n,p is a BiHom-twisting map between A and H . Proof .
The relations (7.9) and (7.10) are very easy to prove and left to the reader.Proof of (7.11):( µ A ⊗ β H ) ◦ (id A ⊗ R m,n,p ) ◦ (cid:0) id A ⊗ α H β − H ⊗ id A (cid:1) ◦ ( R m,n,p ⊗ id A )( h ⊗ a ⊗ a (cid:48) )= ( µ A ⊗ β H ) ◦ (id A ⊗ R m,n,p ) (cid:0) α mH β nH ω pH ( h ) · β − A ( a ) ⊗ α H β − H ψ − H ( h ) ⊗ a (cid:48) (cid:1) = ( µ A ⊗ β H ) (cid:0) α mH β nH ω pH ( h ) · β − A ( a ) ⊗ α mH β nH ω pH (cid:0)(cid:2) α H β − H ψ − H ( h ) (cid:3) (cid:1) · β − A ( a (cid:48) ) ⊗ ψ − H (cid:0)(cid:2) α H β − H ψ − H ( h ) (cid:3) (cid:1)(cid:1) = ( µ A ⊗ β H ) (cid:0) α mH β nH ω pH ( h ) · β − A ( a ) ⊗ α m +1 H β n − H ω pH ψ − H (( h ) ) · β − A ( a (cid:48) ) ⊗ α H β − H ψ − H (( h ) ) (cid:1) = (cid:2) α mH β nH ω pH ( h ) · β − A ( a ) (cid:3)(cid:2) α m +1 H β n − H ψ − H ω pH (( h ) ) · β − A ( a (cid:48) ) (cid:3) ⊗ α H ψ − H (( h ) ) (5 . = (cid:2) α mH β nH ω p − H (( h ) ) · β − A ( a ) (cid:3)(cid:2) α m +1 H β n − H ψ − H ω pH (( h ) ) · β − A ( a (cid:48) ) (cid:3) ⊗ α H ψ − H ( h )= (cid:2) α − H ω − H (cid:0) α m +1 H β nH ω pH (( h ) ) (cid:1) · β − A ( a ) (cid:3)(cid:2) β − H ψ − H (cid:0) α m +1 H β nH ω pH (( h ) ) (cid:1) · β − A ( a (cid:48) ) (cid:3) ⊗ α H ψ − H ( h )= (cid:8) α − H ω − H (cid:0)(cid:2) α m +1 H β nH ω pH ( h ) (cid:3) (cid:1) · β − A ( a ) (cid:9)(cid:8) β − H ψ − H (cid:0)(cid:2) α m +1 H β nH ω pH ( h ) (cid:3) (cid:1) · β − A ( a (cid:48) ) (cid:9) ⊗ α H ψ − H ( h ) (5 . = α m +1 H β nH ω pH ( h ) · β − A ( aa (cid:48) ) ⊗ α H ψ − H ( h ) = ( R m,n,p ◦ ( α H ⊗ µ A ))( h ⊗ a ⊗ a (cid:48) ) . Proof of (7.12):( α A ⊗ µ H ) ◦ ( R m,n,p ⊗ id H ) ◦ (cid:0) id H ⊗ α − A β A ⊗ id H (cid:1) ◦ (id H ⊗ R m,n,p )( h ⊗ h (cid:48) ⊗ a )= ( α A ⊗ µ H ) ◦ ( R m,n,p ⊗ id H )( h ⊗ α − A β A (cid:0) α mH β nH ω pH ( h (cid:48) ) · β − A ( a ) (cid:1) ⊗ ψ − H ( h (cid:48) ))= ( α A ⊗ µ H ) ◦ ( R m,n,p ⊗ id H ) (cid:0) h ⊗ α m − H β n +1 H ω pH ( h (cid:48) ) · α − A ( a ) ⊗ ψ − H ( h (cid:48) ) (cid:1) = ( α A ⊗ µ H ) (cid:0) α mH β nH ω pH ( h ) · (cid:0) α m − H β nH ω pH ( h (cid:48) ) · α − A β − A ( a ) (cid:1) ⊗ ψ − H ( h ) ⊗ ψ − H ( h (cid:48) ) (cid:1) = α m +1 H β nH ω pH ( h ) · (cid:0) α mH β nH ω pH ( h (cid:48) ) · β − A ( a ) (cid:1) ⊗ ψ − H ( h h (cid:48) ) (4 . = (cid:8)(cid:2) α mH β nH ω pH ( h ) (cid:3)(cid:2) α mH β nH ω pH ( h (cid:48) ) (cid:3)(cid:9) · a ⊗ ψ − H ( h h (cid:48) )= α mH β nH ω pH ( h h (cid:48) ) · a ⊗ ψ − H ( h h (cid:48) ) (5 . = α mH β nH ω pH (( hh (cid:48) ) ) · a ⊗ ψ − H (( hh (cid:48) ) ) = ( R m,n,p ◦ ( µ H ⊗ β A ))( h ⊗ h (cid:48) ⊗ a ) , finishing the proof. (cid:4) Definition 8.2.
Let (
H, µ H , ∆ H , α H , β H , ψ H , ω H ) be a BiHom-bialgebra and ( A, µ A , α A , β A )a left H -module BiHom-algebra, with left H -module structure H ⊗ A → A , h ⊗ a (cid:55)→ h · a , such0 G. Graziani, A. Makhlouf, C. Menini and F. Panaitethat all structure maps α H , β H , ψ H , ω H , α A , β A are bijective. Consider the BiHom-twistingmap R = R , − , − : H ⊗ A → A ⊗ H, R ( h ⊗ a ) = β − H ω − H ( h ) · β − A ( a ) ⊗ ψ − H ( h ) . (8.1)We denote the BiHom-associative algebra A ⊗ R H by A H (we denote a ⊗ h := a h , for a ∈ A , h ∈ H ) and call it the BiHom-smash product of A and H . Its structure maps are α A ⊗ α H and β A ⊗ β H , and its multiplication is( a h )( a (cid:48) h (cid:48) ) = a (cid:0) β − H ω − H ( h ) · β − A ( a (cid:48) ) (cid:1) ψ − H ( h ) h (cid:48) . Remark 8.3. If H is a Hom-bialgebra, i.e., α H = β H = ψ H = ω H , and A is a Hom-associativealgebra, the multiplication of A H becomes( a h )( a (cid:48) h (cid:48) ) = a (cid:0) α − H ( h ) · α − A ( a (cid:48) ) (cid:1) α − H ( h ) h (cid:48) , which is the formula introduced in [23]. If H is a monoidal Hom-bialgebra, i.e., ψ H = ω H = α − H = β − H , and A is a Hom-associative algebra, the multiplication of A H becomes( a h )( a (cid:48) h (cid:48) ) = a (cid:0) h · α − A ( a (cid:48) ) (cid:1) α H ( h ) h (cid:48) , which is the formula introduced in [11], used also in [20] for defining the Radford biproduct formonoidal Hom-bialgebras. Proposition 8.4.
In the same setting as in Proposition , and assuming moreover thatthe maps α A and β A are bijective, if we denote by A H the usual smash product between A and H , then α A ⊗ α H and β A ⊗ β H are commuting algebra endomorphisms of A H and theBiHom-associative algebras ( A H ) ( α A ⊗ α H ,β A ⊗ β H ) and A ( α A ,β A ) H ( α H ,β H ,ψ H ,ω H ) coincide. Proof .
We will apply Proposition 7.8. In our situation, we have the twisting map P : H ⊗ A → A ⊗ H , P ( h ⊗ a ) = h · a ⊗ h , for which A H = A ⊗ P H . Obviously P satisfies the condi-tions (7.18) and (7.19), so, by Proposition 7.8, we obtain the map U : H ⊗ A → A ⊗ H, U ( h ⊗ a ) = β − A ( β A ( a ) P ) ⊗ α − H ( α H ( h ) P ) , which is a BiHom-twisting map between A ( α A ,β A ) and H ( α H ,β H ) and we have( A H ) ( α A ⊗ α H ,β A ⊗ β H ) = A ( α A ,β A ) ⊗ U H ( α H ,β H ) . Thus, the proof will be finished if we prove that the map U coincides with the map R affordingthe BiHom-smash product A ( α A ,β A ) H ( α H ,β H ,ψ H ,ω H ) . We compute U ( h ⊗ a ) = β − A ( α H ( h ) · β A ( a )) ⊗ α − H ( α H ( h ) )= β − A ( α H ( h ) · β A ( a )) ⊗ α − H ( α H ( h )) = α H β − H ( h ) · a ⊗ h ,R ( h ⊗ a ) = β − H ω − H ( ω H ( h )) (cid:46) β − A ( a ) ⊗ ψ − H ( ψ H ( h ))= β − H ( h ) (cid:46) β − A ( a ) ⊗ h = α H β − H ( h ) · a ⊗ h , finishing the proof. (cid:4) Example 8.5.
We construct a class of examples of U q ( sl ) ( α,β,ψ,ω ) -module BiHom-algebra struc-tures on A | q,α,β , generalizing examples of U q ( sl ) α -module Hom-algebra structures on A | q,γ givenin [32, Example 5.7] (here we take the base field k = C ). The quantum group U q ( sl ) is generatedas a unital associative algebra by 4 generators { E, F, K, K − } with relations KK − = 1 = K − K, KE = q EK, KF = q − F K, EF − F E = K − K − q − q − , iHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras 31where q ∈ C with q (cid:54) = 0, q (cid:54) = ±
1. The comultiplication is defined by∆( E ) = 1 ⊗ E + E ⊗ K, ∆( F ) = K − ⊗ F + F ⊗ , ∆( K ) = K ⊗ K, ∆ (cid:0) K − (cid:1) = K − ⊗ K − . We fix λ , λ , λ , λ ∈ C some nonzero elements. The BiHom-bialgebra U q ( sl ) ( α,β,ψ,ω ) =( U q ( sl ) , µ ( α,β ) , ∆ ( ψ,ω ) , α, β, ψ, ω ) is defined (as in Proposition 5.9(iii)) by µ ( α,β ) = µ ◦ ( α ⊗ β )and ∆ ( ψ,ω ) = ( ω ⊗ ψ ) ◦ ∆, where µ and ∆ are respectively the multiplication and comultiplicationof U q ( sl ) and α, β, ψ, ω : U q ( sl ) → U q ( sl ) are bialgebra morphisms such that α ( E ) = λ E, α ( F ) = λ − F, α ( K ) = K, α (cid:0) K − (cid:1) = K − ,β ( E ) = λ E, β ( F ) = λ − F, β ( K ) = K, β (cid:0) K − (cid:1) = K − ,ψ ( E ) = λ E, ψ ( F ) = λ − F, ψ ( K ) = K, ψ (cid:0) K − (cid:1) = K − ,ω ( E ) = λ E, ω ( F ) = λ − F, ω ( K ) = K, ω (cid:0) K − (cid:1) = K − . Note that any two of the maps α , β , ψ , ω commute.Let A | q = k (cid:104) x, y (cid:105) / ( yx − qxy ) be the quantum plane. We fix also some ξ ∈ C , ξ (cid:54) = 0. TheBiHom-quantum plane A | q,α,β = ( A | q , µ A ,α A ,β A , α A , β A ) is the BiHom-associative algebra defined(as in Proposition 5.9(i)) by µ A ,α A ,β A = µ A ◦ ( α A ⊗ β A ), where µ A is the multiplication of A | q and α A , β A : A | q → A | q are the (commuting) algebra morphisms such that α A ( x ) = ξx, α A ( y ) = ξλ − y and β A ( x ) = ξx, β A ( y ) = ξλ − y. We consider A | q as a left U q ( sl )-module algebra as in [32, Example 5.7] (we denote by h ⊗ a (cid:55)→ h · a the U q ( sl )-action on A | q ). By the computations performed in [32, Example 5.7]we know that α A ( h · a ) = α ( h ) · α A ( a ) and β A ( h · a ) = β ( h ) · β A ( a ), for all h ∈ U q ( sl ) and a ∈ A | q . Then, according to Proposition 5.16, there exists a U q ( sl ) ( α,β,ψ,ω ) -module BiHom-algebra structure on A | q,α,β defined by ρ : U q ( sl ) ( α,β,ψ,ω ) ⊗ A | q,α,β → A | q,α,β , ρ ( h ⊗ a ) = h (cid:46) a = α ( h ) · β A ( a ) . By using also the computations performed in [32, Example 5.7] one can see that the map ρ isgiven on generators by ρ (cid:0) E ⊗ x m y n (cid:1) = [ n ] q ξ m + n λ λ − n x m +1 y n − ,ρ (cid:0) F ⊗ x m y n (cid:1) = [ m ] q ξ m + n λ − λ − n x m − y n +1 ,ρ (cid:0) K ± ⊗ P (cid:1) = P (cid:0) q ± ξx, q ∓ ξλ − y (cid:1) , for any monomial x m y n ∈ A | q , where P = P ( x, y ) ∈ A | q and [ n ] q = q n − q − n q − q − .Since ξ (cid:54) = 0 and λ i (cid:54) = 0 for all i = 1 , , ,
4, all the maps α , β , ψ , ω , α A , β A are bijective.According to Theorem 8.1, the map R : U q ( sl ) ( α,β,ψ,ω ) ⊗ A | q,α,β → A | q,α,β ⊗ U q ( sl ) ( α,β,ψ,ω ) definedby (8.1) leads to the smash product A | q,α,β U q ( sl ) ( α,β,ψ,ω ) whose multiplication is defined by( a h )( a (cid:48) h (cid:48) ) = a ∗ (cid:0) β − ω − ( h (1) ) (cid:46) β − A ( a (cid:48) ) (cid:1) ψ − ( h (2) ) • h (cid:48) , where h (1) ⊗ h (2) = ∆ ( ψ,ω ) ( h ) and ∗ (respectively • ) is the multiplication of A | q,α,β (respectively U q ( sl ) ( α,β,ψ,ω ) ).2 G. Graziani, A. Makhlouf, C. Menini and F. PanaiteIn particular, for any G ∈ U q ( sl ) and m, n, r, s ∈ N we have( x m y n K ± )( x r y s G ) = q ± r ∓ s + nr ξ m + n + r + s λ − n λ − s x m + r y n + s K ± β ( G ) , ( x m y n E )( x r y s G ) = q nr ξ m + n + r + s λ − n +11 λ − s x m + r y n + s Eβ ( G )+ [ s ] q q n ( r +1) ξ m + n + r + s λ − n λ − s x m + r +1 y n + s − Kβ ( G ) , ( x m y n F )( x r y s G ) = q s − r + nr ξ m + n + r + s λ − n − λ − s x m + r y n + s F β ( G )+ [ r ] q q n ( r − ξ m + n + r + s λ − n − λ − s x m + r − y n + s +1 β ( G ) , where K ± β ( G ), Eβ ( G ) and F β ( G ) are multiplications in U q ( sl ).We introduce now the BiHom analogue of comodule Hom-algebras defined in [30]. Definition 8.6.
Let (
H, µ H , ∆ H , α H , β H , ψ H , ω H ) be a BiHom-bialgebra. A right H -comoduleBiHom-algebra is a 7-tuple ( D, µ D , α D , β D , ψ D , ω D , ρ D ), where ( D, µ D , α D , β D ) is a BiHom-associative algebra, ( D, ψ D , ω D ) is a right H -comodule via the coaction ρ D : D → D ⊗ H andmoreover ρ D is a morphism of BiHom-associative algebras. Example 8.7.
If (
H, µ H , ∆ H , α H , β H , ψ H , ω H ) is a BiHom-bialgebra, then we have the right H -comodule BiHom-algebra ( H, µ H , α H , β H , ψ H , ω H , ∆ H ).The next result generalizes Proposition 3.6 in [23]. Proposition 8.8.
Let ( H, µ H , ∆ H , α H , β H , ψ H , ω H ) be a BiHom-bialgebra and ( A, µ A , α A , β A ) a left H -module BiHom-algebra, with notation H ⊗ A → A , h ⊗ a (cid:55)→ h · a , such that all structuremaps α H , β H , ψ H , ω H , α A , β A are bijective. Assume that there exist two more linear maps ψ A , ω A : A → A such that any two of the maps α A , β A , ψ A , ω A commute and moreover ω A ( aa (cid:48) ) = ω A ( a ) ω A ( a (cid:48) ) , ∀ a, a (cid:48) ∈ A,ω A ( h · a ) = ω H ( h ) · ω A ( a ) , ∀ a ∈ A, h ∈ H. (8.2) Define the linear map ρ A H : A H → ( A H ) ⊗ H, ρ A H ( a h ) = ( ω A ( a ) h ) ⊗ h . Then ( A H, µ A H , α A ⊗ α H , β A ⊗ β H , ψ A ⊗ ψ H , ω A ⊗ ω H , ρ A H ) is a right H -comodule BiHom-algebra. Proof .
We only prove that ρ A H is multiplicative and leave the other details to the reader: ρ A H (( a h )( a (cid:48) h (cid:48) )) = ω A (cid:0) a (cid:0) β − H ω − H ( h ) · β − A ( a (cid:48) ) (cid:1)(cid:1) (cid:0) ψ − H ( h ) h (cid:48) (cid:1) ⊗ (cid:0) ψ − H ( h ) h (cid:48) (cid:1) = ω A ( a ) ω A (cid:0) β − H ω − H ( h ) · β − A ( a (cid:48) ) (cid:1) ψ − H (( h ) ) h (cid:48) ⊗ ψ − H (( h ) ) h (cid:48) = ω A ( a ) (cid:0) β − H ( h ) · ω A β − A ( a (cid:48) ) (cid:1) ψ − H (( h ) ) h (cid:48) ⊗ ψ − H (( h ) ) h (cid:48) = ω A ( a ) (cid:0) β − H ω − H (( h ) ) · ω A β − A ( a (cid:48) ) (cid:1) ψ − H (( h ) ) h (cid:48) ⊗ h h (cid:48) = ω A ( a ) (cid:0) β − H ω − H (( h ) ) · β − A ω A ( a (cid:48) ) (cid:1) ψ − H (( h ) ) h (cid:48) ⊗ h h (cid:48) = ( ω A ( a ) h )( ω A ( a (cid:48) ) h (cid:48) ) ⊗ h h (cid:48) = ρ A H ( a h ) ρ A H ( a (cid:48) h (cid:48) ) , finishing the proof. (cid:4) iHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras 33 Example 8.9.
Let (
H, µ H , ∆ H , α H , β H , ψ H , ω H ) be a BiHom-bialgebra such that all structuremaps are bijective. Denote by A the linear space H ∗ . Then A becomes a BiHom-associativealgebra with multiplication and structure maps defined by( f • g )( h ) = f (cid:0) α − H ω − H ( h ) (cid:1) g (cid:0) β − H ψ − H ( h ) (cid:1) ,α A : H ∗ → H ∗ , α A ( f )( h ) = f (cid:0) α − H ( h ) (cid:1) ,β A : H ∗ → H ∗ , β A ( f )( h ) = f (cid:0) β − H ( h ) (cid:1) , for all f, g ∈ H ∗ and h ∈ H . Moreover, A becomes a left H -module BiHom-algebra, with action (cid:42) : H ⊗ H ∗ → H ∗ , ( h (cid:42) f )( h (cid:48) ) = f ( α − H β − H ( h (cid:48) ) h ) , for all h, h (cid:48) ∈ H and f ∈ H ∗ . Obviously, α A and β A are bijective maps. Define the linear map ω A : H ∗ → H ∗ , ω A ( f )( h ) = f (cid:0) ω − H ( h ) (cid:1) , ∀ f ∈ H ∗ , h ∈ H, and choose a linear map ψ A : H ∗ → H ∗ that commutes with α A , β A , ω A , for instance one canchoose the map ψ A defined by ψ A ( f )( h ) = f ( ψ − H ( h )), for all f ∈ H ∗ and h ∈ H . Then one cancheck that the hypotheses of Proposition 8.8 are satisfied, and consequently H ∗ H becomesa right H -comodule BiHom-algebra.Note also that, if H is counital with counit ε H such that ε H ◦ α H = ε H and ε H ◦ β H = ε H ,then the BiHom-associative algebra A = H ∗ is unital with unit ε H . Acknowledgements
This paper was written while Claudia Menini was a member of GNSAGA. Florin Panaite wassupported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0635, contract nr. 253/5.10.2011. Parts ofthis paper have been written while Florin Panaite was a visiting professor at University of Fer-rara in September 2014, supported by INdAM, and a visiting scholar at the Erwin SchrodingerInstitute in Vienna in July 2014 in the framework of the “Combinatorics, Geometry and Physics”programme; he would like to thank both these institutions for their warm hospitality.The authors are grateful to the referees for their remarks and questions.
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