BiHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras

A BiHom-associative algebra is a (nonassociative) algebra $A$ endowed with two commuting multiplicative linear maps $\alpha,\beta\colon A\rightarrow A$ such that $\alpha (a)(bc)=(ab)\beta (c)$, for all $a, b, c\in A$. This concept arose in the study of algebras in so-called group Hom-categories. In this 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 properties and constructions (representations, twisted tensor products, smash products etc).


Introduction
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 instance [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 identity. 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 endomorphism with respect to the bracket satisfying the so-called Hom-Jacobi identity Since any associative algebra becomes a Lie algebra by taking the commutator [a, b] = ab − ba, it was natural to look for a Hom-analogue of this property. This was accomplished in [24], where the concept of Hom-associative algebra was introduced, as being a linear space A endowed with a multiplication µ : A ⊗ A → A, µ(a ⊗ b) = ab, and a linear map α : A → A satisfying the arXiv:1505.00469v2 [math.RA] 25 Oct 2015 so-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 multiplicativity of α.
In subsequent literature (see for instance [30]) were studied subclasses of these classes of algebras where the linear maps α involved in the definition of a Hom-Lie algebra or Hom-associative algebra are required to be multiplicative, that is α([x, y]) = [α(x), α(y)] for all x, y ∈ L, respectively α(ab) = α(a)α(b) for all a, b ∈ A, and these subclasses were called multiplicative Hom-Lie algebras, respectively multiplicative Hom-associative algebras. Since we will always assume multiplicativity of the maps α and to simplify terminology, we will call Hom-Lie or Hom-associative algebras what was called above multiplicative Hom-Lie or Hom-associative algebras.
The Hom-analogues of coalgebras, bialgebras and Hopf algebras have been introduced in [25,26]. The original definition of a Hom-bialgebra involved two linear maps, one twisting the associativity condition and the other one the coassociativity condition. Later, two directions of study on Hom-bialgebras were developed, one in which the two maps coincide (these are still called Hom-bialgebras) and another one, started in [8], where the two maps are assumed to be inverse to each other (these are called monoidal Hom-bialgebras).
The main tool for constructing examples of Hom-type algebras is the so-called "twisting principle" introduced by D. Yau for Hom-associative algebras and extended afterwards to other types of Hom-algebras. For instance, if A is an associative algebra and α : A → A is an algebra map, then A with the new multiplication defined by a * b = α(a)α(b) is a Hom-associative algebra, called the Yau twist of A.
A categorical interpretation of Hom-associative algebras has been given by Caenepeel and Goyvaerts in [8]. First, to any monoidal category C they associate a new monoidal category H(C), called a Hom-category, whose objects are pairs consisting of an object of C and an automorphism of this object ( H(C) has nontrivial associativity constraint even if the one of C is trivial). By taking C to be k M, the category of linear spaces over a base field k, it turns out that an algebra in the (symmetric) monoidal category H( k M) is the same thing as a Hom-associative algebra (A, µ, α) with bijective α. The bialgebras in H( k M) are the monoidal Hom-bialgebras we mentioned before.
In [14], the first author extended the construction of the Hom-category H(C) to include the action 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 homomorphism. 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 also defined on H c,d,ν (G, C) (see Claim 2.7 and Theorem 2.8) turning it into a braided category which is symmetric whenever C is. When G = Z, c = d = 1 Z and ν = id 1 one gets the category H(C) from [8], while for c = 1 Z , d = −1 Z and ν = id 1 one gets the category H(C).
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 associative algebra and α, β : A → A are two commuting algebra maps, then A with the new multiplication defined by a * b = α(a)β(b) is a BiHom-associative algebra, called the Yau twist of A. As a matter of fact, although we arrived at the concept of BiHom-associative algebra via the categorical machinery presented above, it is the possibility of twisting the multiplication of an associative algebra by two commuting algebra endomorphisms that led us to believe that BiHomassociative algebras are interesting objects in their own. One can think of this as follows. Take again an associative algebra A and α, β : A → A two commuting algebra endomorphisms; define a 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 enough to 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 twisting of an associative algebra by two maps should thus be considered as the "natural" example of a BiHom-associative algebra. We would like to emphasize that for this operation the two maps are not assumed to be bijective, so the resulting BiHom-associative algebra has possibly non bijective structure maps and as such it cannot be regarded, to our knowledge, as an algebra in a monoidal category.
Take now the group G to be arbitrary. It is natural to describe how an algebra in the monoidal category H c,d,ν (G, k M) looks like. By writing down the axioms, it turns out (see Claim 3.1 and Remark 3.5) that an algebra in such a category is a BiHom-associative algebra with bijective structure maps (the associativity of the algebra in the category is equivalent to the BiHom-associativity condition) having some extra structure (like an action of the group on the algebra). So, morally, the group G = Z × Z leads to BiHom-associative algebras but any other group 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 structure we 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 we need the structure maps α and β to be bijective; the commutator is defined by the formula [a, b] = ab − α −1 β(b)αβ −1 (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 linear maps 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 find some of their basic properties. Then we introduce BiHom-coassociative coalgebras and BiHombialgebras together with some of the usual ingredients (comodules, duality, convolution product, primitive elements, module and comodule algebras). We define antipodes for a certain class of BiHom-bialgebras, called monoidal BiHom-bialgebras, leading thus to the concept of monoidal BiHom-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 call a BiHom-pseudotwistor. We write down explicitly such a smash product, obtained from an action of a Yau twist of the quantum group U q (sl 2 ) on a Yau twist of the quantum plane A 2|0 q . As a final remark, let us note that one could introduce a less restrictive concept of BiHomassociative algebra by dropping the assumptions that α and β are multiplicative and/or that they commute (note that all the examples of q-deformations of Witt or Virasoro algebras are not multiplicative). Unfortunately, by dropping any of these assumptions, one loses the main class of 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 general is not a BiHom-associative algebra even in this more general sense.

The category H(G, C)
Our aim in this section is to introduce so-called group Hom-categories; proofs of the results in this 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 is a group homomorphism G → Aut C (A). A morphism ξ : Definition 2.2. A monoidal category (see [17,Chapter XI]) is a category C endowed with an object 1 ∈ C (called unit), a functor ⊗ : C × C → C (called tensor product) and functorial isomorphisms a X,Y,Z : (X ⊗ Y ) ⊗ Z → X ⊗ (Y ⊗ Z), l X : 1 ⊗ X → X, r X : X ⊗ 1 → X, for every X, Y , Z in C. The functorial isomorphisms a are called the associativity constraints and satisfy the pentagon axiom, that is The isomorphisms l and r are called the unit constraints and they obey the Triangle Axiom, that is A monoidal functor (F, φ 2 , φ 0 ) : (C, ⊗, 1, a, l, r) → (C , ⊗ , 1 , a , l , r ) between two monoidal categories consists of a functor F : is commutative, and the following conditions are satisfied Claim 2.3. Let G be a group and let (C, ⊗, 1, a, l, r) be a monoidal category. Given any pair of objects (A, for all g ∈ G.
Then f A ⊗ f B is a group homomorphism and hence Let Z(G) be the center of G and let c ∈ Z(G). Then we can consider the functorial isomorphism ϕ(c) : Id H(G,C) → Id H(G,C) defined by setting Also, let Id 1 : G → Aut C (1) denote the constant map equal to Id 1 . Let c, d ∈ Z(G) and let ν ∈ Aut C (1). We set From now on, when (C, ⊗, 1, a, l, r) is a monoidal category, G is a group, c, d ∈ Z(G) and ν ∈ Aut C (1), we will indicate the monoidal category defined in Theorem 2.4 by H c,d,ν (G, C). In the case when c = d = 1 G and ν = Id 1 , we will simply write H(G, C).
Theorem 2.5. Let (C, ⊗, 1, a, l, r) be a monoidal category and G a group. Then the identity functor I : H c,d,ν (G, C) → H(G, C) is a monoidal isomorphism via Definition 2.6 (see [17]). A braided monoidal category (C, ⊗, 1, a, l, r, γ) is a monoidal category (C, ⊗, 1,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 braided monoidal category is called symmetric if we further have γ V,U • γ U,V = Id U ⊗V for every U, V ∈ C. A braided monoidal functor is a monoidal functor F : C → C such that for every U, V ∈ C.
Claim 2.7. Let G be a group and let (C, ⊗, 1, a, l, r, γ) be a braided monoidal category. Let c, d ∈ Z(G) and let ν ∈ Aut C (1). 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), Theorem 2.8. In the setting of Claim 2.7, the category is a braided monoidal category.
From now on, when (C, ⊗, 1, a, l, r, γ) is a braided monoidal category and G is a group, we will still denote the braided monoidal structure defined in Theorem 2.8 with H c,d,ν (G, C). In the case when c = d = 1 G and ν = id 1 , we will simply write respectively H(G, C) instead of H c,d,ν (G, C) and Theorem 2.9. Let G be a group and let (C, ⊗, 1, a, l, r, γ) be a braided monoidal category. Then the identity functor I : H c,d,ν (G, C) → H(G, C) is a braided monoidal isomorphism via Remark 2.10. Let G be a torsion-free abelian group. Corollary 4 in [4] states that, up to a braided monoidal category isomorphism, there is a unique braided monoidal structure (actually symmetric) on the category of representations over the group algebra k[G], considered monoidal via a structure induced by that of vector spaces over the field k. Thus Theorem 2.9 can be deduced from this result whenever G is a torsion-free abelian group. We should remark that this result in [4] stems from the fact that the third Harrison cohomology group H 3 Harr (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 2 then H 3 Harr (G, k, G m ) has exactly two elements and so in this case there are two distinct equivalence classes of braided monoidal structures on the category of representations over the group algebra k[G], considered monoidal via a structure induced by that of vector spaces over the field k. This does not contradict our Theorem 2.9. In fact, there might exist braided monoidal structures different from the ones considered in the statement of Theorem 2.9.
Claim 2.11. Let (C, ⊗, 1, a, l, r) be a monoidal category and G a group, let c, d ∈ Z(G) and Definition 2.12. Given a monoidal category M, a quadruple (A, µ, u, c) is called a braided unital algebra in M if (for simplicity, we will omit to write the associators): • (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 the Yang-Baxter equation • the following conditions hold: A braided unital algebra is called symmetric whenever c 2 = Id A .  In a symmetric monoidal category (C, ⊗, 1, 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 ).

Generalized Hom-structures
Let k be a field and let k M be the category of linear spaces regarded as a braided monoidal category in the usual way. Then, for every group G, the category H(G, k M) identifies with the category 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 is the multiplication by an element of k\{0} that we will also denote by ν. Note that, given and for every x ∈ X.
Note that when c = d = 1 G and ν = 1 k , it turns out that A is simply a k[G]-module algebra.
and the braiding is 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 Inspired by Example 3.2, we introduce the following concept.
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, with notation µ(a ⊗ a ) = aa , satisfying the following conditions, for all a, a , a ∈ A: α(a)(a a ) = (aa )β(a ) (BiHom-associativity).
We call α and β (in this order) the structure maps of A. and Remark 3.4. A Hom-associative algebra (A, µ, α) can be regarded as the BiHom-associative algebra (A, µ, α, α).

Remark 3.5.
A BiHom-associative algebra with bijective structure maps is exactly an algebra in On the other hand, in the setting of Claim 3.1, if we define the maps α, β : A → A by α(a) = c · a and β(a) = d −1 · a, for all a ∈ A, the axiom 2) in Claim 3.1 implies 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-associative algebras, that are obtained by a computer algebra system. Let {e 1 , e 2 } be a basis; for i = 1, 2 the maps α i , β i and the multiplication µ i are defined by and where a, b are parameters in k, with b = 1 in the first case and a = 0 in the second. In both cases, the unit is e 1 .
In view of Claim 3.7, a BiHom-associative algebra with bijective structure maps is a Yau twist 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).
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 expressed as a Hom-associative algebra. Let k be a field and A = k[X]. Let α : A → A be the algebra map defined by setting α(X) = X 2 and let β = Id k[X] . Then we can consider the BiHom-associative algebra Let us assume that there exists θ ∈ End(k[X]) such that (A, µ • (α ⊗ β), θ) is a Hom-associative algebra. Then we should have that where a i ∈ k for every i = 0, 1, . . . , n and a n = 0. Since which implies that 2n + 3 = 6 + n, i.e., n = 3, and hence Let us set c = a 3 and let us check the equality The left-hand side is The right-hand side is Thus the equality does not hold.
Remark 3.10. Given two algebras (A, µ A , 1 A ) and (B, µ B , 1 B ) in a braided monoidal category (C, ⊗, 1, a, l, r, c), it is well known that A ⊗ B becomes also an algebra in the category, with multiplication µ A⊗B defined by In the case of our category H c,d,ν (G, k M), we have, for every x, y ∈ A, x , y ∈ B: is a BiHom-associative algebra (called the tensor product of A and B), where µ A⊗B is the usual multiplication: (a ⊗ b)(a ⊗ b ) = aa ⊗ bb . If A and B are unital with units 1 A and respectively 1 B then A ⊗ B is also unital with unit 1 A ⊗ 1 B . This is consistent with Remark 3.10.
) and τ is the usual flip.
We will write down 4) explicitly. We have

Thus 4) is equivalent to
which is equivalent to ]] = 0, for every x, y, z ∈ L (Jacobi condition).
In particular, a Lie algebra in Inspired by Example 3.12, we introduce the following concept.
We call α and β (in this order) the structure maps of L.  In view of Claim 2.14, we have: If (A, µ, α, β) is a BiHom-associative algebra with bijective α and β, then, for every a, a ∈ A, we can set By Theorem 2.9, the identity functor I : Enveloping algebras of Hom-Lie algebras where introduced in [29] (see also [8,Section 8]).

Representations
From now on, we will always work over a base field k. All algebras, linear spaces etc. will be over k; unadorned ⊗ means ⊗ k . For a comultiplication ∆ : C → C ⊗ C on a linear space C, we use a Sweedler-type notation ∆(c) = c 1 ⊗ c 2 , 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: 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 .    If (A, µ, α, β) is a BiHom-associative algebra, then (A, α, β) is a left A-module with action defined by a · b = ab, for all a, b ∈ A. Lemma 4.3. Let (E, µ, 1 E ) be an associative unital algebra and u, v ∈ E two invertible elements such that uv = vu. Define the linear mapsα,β : E → E,α(a) = uau −1 ,β(a) = vav −1 , for all a ∈ E, and the linear mapμ : Then (E,μ,α,β) is a unital BiHom-associative algebra with unit v, denoted by E(u, v).
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 a)), for all a ∈ A. Thus, for all a, b ∈ A, we havẽ Hence, we have ⇐⇒ ϕ(ab)(n) = (ϕ(a) * ϕ(b))(n), ∀ a, b ∈ A, n ∈ 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 the module M is unital.
We recall the following concept from [27] (see also [7] on this subject).  Inspired by this remark, we can introduce now the following concept: A first indication that this is indeed the appropriate concept of representation for BiHom-Lie algebras is provided by the following result (extending the corresponding one for Hom-associative algebras in [6]), whose proof is straightforward and left to the reader. A second indication is provided by the fact that, under certain circumstances, we can construct the semidirect product (the Hom-case is done in [27]).

BiHom-coassociative coalgebras and BiHom-bialgebras
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 linear space, ψ, ω : C → C and ∆ : C → C ⊗ C are linear maps, such that We call ψ and ω (in this order) the structure maps of C.
A BiHom-coassociative coalgebra (C, ∆, ψ, ω) is called counital if there exists a linear map ε : C → k (called a counit) such that 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.
We discuss now the duality between BiHom-associative and BiHom-coassociative structures.
Proof . The product µ = ∆ * is defined from C * ⊗ C * to C * by where ·, · 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 Therefore, the BiHom-associativity condition µ • (µ ⊗ ψ * − ω * ⊗ µ) = 0 follows from the BiHom- Moreover, if C has a counit ε then for f ∈ C * and x ∈ C we have which shows that ε is the unit of C * .
The dual of a BiHom-associative algebra (A, µ, α, β) is not always a BiHom-coassociative coalgebra, because (A ⊗ A) * A * ⊗ A * . Nevertheless, it is the case if the BiHom-associative algebra is finite-dimensional, since (A ⊗ A) * = A * ⊗ A * in this case.
More generally, we can define the finite dual of A by where a cofinite ideal I is an ideal I ⊂ A such that A/I is finite-dimensional and where we say that 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 two elements of A • is again in A • because the intersection of two cofinite ideals is again a cofinite ideal. If A is finite-dimensional, of course A • = A * . As in the classical case, one can show that if A and B are two BiHom-associative algebras and f : A → B is a morphism of BiHomassociative algebras, then the dual map f * : 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 provided 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 is counital whenever A is unital, with counit ε : We can now define the notion of BiHom-bialgebra.
We see now that analogues of Yau's twisting principle hold for the BiHom-structures we defined (proofs are straightforward and left to the reader): Proposition 5.9.
Proof . By the counit property, we have ω( Since α and β are comultiplicative maps and α p β q (1) = 1, it follows that α p β q (x) is a primitive element whenever x is a primitive element.
Consequently, the set of all primitive elements of H, denoted by Prim(H), has a structure of BiHom-Lie algebra.
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 Hmodule BiHom-algebra if (A, α A , β A ) is a left H-module, with action denoted by H ⊗ A → A, h ⊗ a → h · a, such that the following condition is satisfied Remark 5.15. This concept contains as particular cases the concepts of module algebras over a 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: 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 above sense, with action

Monoidal BiHom-Hopf algebras and BiHom-Hopf algebras
In this section, we introduce the concept of monoidal BiHom-Hopf algebra and discuss a possible generalization of Hom-Hopf algebras to BiHom-Hopf algebras. We begin with a lemma whose proof is obvious. Then (Hom(C, A), , φ, γ) is a BiHom-associative algebra. Moreover, if A is unital with unit 1 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 the linear maps f : then (Hom(C, A), , η • ε) is an associative unital algebra.
Proof . Let f, g, h ∈ Hom(C, A). We have Similarly, The BiHom-associativity of µ and the BiHom-coassociativity of ∆ lead to the BiHom-associativity of the convolution product .
The map η • ε is the unit for the convolution product. Indeed, for f ∈ Hom(C, A) and x ∈ C, we have The last statement follows from Lemma 6.1.
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-Hopf algebra as the 8-tuple (H, µ, ∆, α, β, 1 H , ε H , S). Proof . A straightforward computation. Let us only note that α, β being bialgebra maps, they automatically commute with S.
We state now the basic properties of the antipode.  (ii) S(β(a)α(b)) = S(β(b))S(α(a)), for all a, b ∈ H; (ii) We define the linear maps R, L, m : H ⊗ H → H by the formulae (for all a, b ∈ H): One can easily check that R, L, m ∈ Hom(H ⊗ H, H) (where H ⊗ H is the tensor product BiHom-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). We compute finishing the proof.
(iii) similar to the proof of (ii), by defining the linear maps L, R, δ : H → H ⊗ H, for all h ∈ H, and proving that L (respectively R) is a left (respectively right) convolution inverse of δ in Hom(H, H ⊗ H).
Remark 6.7. We had to restrict the definition of the antipode to the class of monoidal BiHombialgebras because, if H is a Hopf algebra with antipode S and we make an arbitrary Yau twist of H, then in general S will not satisfy the defining property of an antipode for the Yau twist, as the next example shows. = α(S(X)) so that we get α(S(1))X = X 4 α(S(1)), which implies that α(S(1)) = 0. On the other hand, we have and this is a contradiction.
In view of all the above, we propose the following definition for what might be a BiHom-Hopf algebra, that is moreover invariant under Yau twisting: Definition 6.9. Let (H, µ, ∆, α, β, ψ, ω) be a unital and counital BiHom-bialgebra with a unit 1 H and a counit ε H . A linear map S : H → H is called an antipode if it commutes with all the maps α, β, ψ, ω and it satisfies the following relation: 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.
Proposition 7.2. Let (D, µ,α,β) be a BiHom-associative algebra and α, β : D → D two multiplicative linear maps such that any two of the mapsα,β, α, β commute. Define the maps Then T is an (α, β)-BiHom-pseudotwistor with companionsT 1 ,T 2 and the BiHom-associative algebras D T α,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: The condition (7.6) is similar, so we check (7.7): It is obvious that D T α,β and D (α,β) coincide. We have the following multiplicative linear maps α, β defined with respect to the basis B by α(e 1 ) = e 1 , α(e 2 ) = ae 1 + (1 − a)e 2 , β(e 1 ) = e 1 , β(e 2 ) = be 1 where a, b are parameters in k. One can easily see that any two of the mapsα,β, α, β commute. By the previous proposition, we can construct the BiHom-associative algebra D (α,β) = (D, 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.
We introduce now twisted tensor products of BiHom-associative algebras.
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 ∈ A and b, b ∈ B) as follows 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 where we use the standard notation for T 13 . The BiHom-associative algebra (A ⊗ B) T is denoted by A ⊗ R B and is called the BiHom-twisted tensor product of A and B; its multiplication is defined by (a ⊗ b)(a ⊗ b ) = aa R ⊗ b R b , 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: This relation is equivalent to which, by using (7.13) and (7.14), is equivalent to which is obviously true.
We need to prove the relations (7.1)-(7.7) (withα = 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. We compute (r and R are two more copies of R) and the two terms are equal because of the relation (7.17).
Define the linear map 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 denoting by p another copy of P and by u another copy of U ) finishing the proof.