Classification of 3-dimensional Bihom-Associative and Bihom-Bialgebras.
Ahmed Zahari
Université de Haute Alsace, Laboratoire de Mathématiques, Informatique et Applications,
4, rue des Frères Lumière F-68093 Mulhouse, France
[email protected]
Abstract.
The purpose of this paper is to study the structure and the algebraic varieties of BiHom-associative algebras. We provide a classification of n-dimensional BiHom-associative and BiHom-bialgebras and
BiHom Hopf algebras for n≤3.
Key words and phrases:
BiHom-associative algebra, BiHom-Bialgebra, BiHom-Hopf algebra, Classification.
Introduction
The first motivation to study nonassociative BiHom-algebras comes from quasi-deformations of Lie algebras of vector fields, in particular q-deformations of Witt and Virasoro algebras. It was observed in the pioneering works, mainly by physicists, that in these examples a twisted Jacobi identity holds. Motivated by these examples and their generalization on the one hand, and the desire to be able to treat within the same framework such well-known generalizations of Lie algebras as the color and Lie superalgebras on the other hand, quasi-Lie algebras and subclasses of quasi-hom-Lie algebras and hom-Lie algebras were introduced by Hartwig, Larsson and Silvestrov in [5, 6].
The BiHom-associative algebras play the role of associative algebras in the BiHom-Lie setting. Usual functors between the categories of Lie algebras and associative algebras were extended to Hom-setting, see [10] for the construction of the enveloping algebra of a BiHom-Lie algebra.
A Hom-associative algebra (A,μ,α,β) consists of a vector space, a multiplication and two linear endomorphism. It may be viewed as a deformation of an associative algebras, in which the associativity condition is twisted by the linear maps α and β in a certain way such that when α=id and β=id, the BiHom-associative algebras degenerate to exactly associative algebras.
In this paper we aim to study the structure of BiHom-associative algebras.
Let A be an n-dimensional K-linear space and {e1,e2,⋯,en} be a basis of A. A BiHom-algebra structure on A with product μ is determined by n3 structure constants Cijk, were μ(ei,ej)=∑k=1nCijkek and by α and β which is given by 2n2 structure constants aij and bij, where
α(ei)=∑j=1najiej and β(ej)=∑k=1nbkjek. Requiring the algebra structure to be BiHom-associative and unital gives rise to sub-variety Hn (resp. Hun) of kn3+2n2. Base changes in A result in the natural transport of the structure action of GLn(k) on Hn. Thus the isomorphism classes of
n-dimensional BiHom-algebras are in one-to-one correspondence with the orbits of the action of GLn(k) on Hn
(rep. Hun).
Furthermore, we shall consider the class of BiHom-bialgebras which are
Hom-associative algebras equipped with a compatible BiHomCoalgebra structure,
in particular BiHomHopf algebras where the additional structures like
the comultiplication Δ and the counit ϵ can be expressed in a base in a similar way
as the multiplication above. We shall also give a classification
of these algebras up to isomorphism in low dimension n≤3.
The paper is organized as follows. In the first section we give the basics about BiHom-associative algebras and provide some new properties. Moreover, we discuss unital BiHom-associative algebras. In Section 2 is dedicated to describe algebraic varieties of BiHom-associative algebras and provide classifications, up to isomorphism, of 2-dimensional (resp. 3-dimensional) BiHom-associative algebras. Moreover, in Section 3
we shall recall the definitions of BiHom-bialgebras and BiHomHopf algebras
and present the classification up to dimension 3.
Acknowledgements. I’d like to thank my thesis
supervisor Abdenacer Makhlouf and Martin Bordomann for giving me the problem, for
many fruitful discussions and for their constant support.
1. Structure of BiHom-associative algebras
Let K be an algebraically closed field of characteristic [math], A be a linear space over K. We refer to a Hom-algebra by a
4-tuple (A,μ,α,β), where μ:A×A→A is a bilinear map (multiplication) and α and β be two homomorphisms of A (twist map).
1.1. Definitions
Definition 1.1**.**
[9].
A BiHom-associative algebra is a 4-tuple (A,μ,α,β) consisting of a linear space A, a bilinear map μ:A×A→A and
two linear space homomorphism α:A→A and β:A→A satisfying
[TABLE]
[TABLE]
[TABLE]
Usually such BiHom-associative algebras are called multiplicative. Since we are dealing only with multiplicative BiHom-associative algebras, we shall call them BiHom-associative algebras for simplicity. We denote the set of all BiHom-associative algebras by H.
In the language of Hopf algebras, the multiplication of a BiHom-associative algebra over A consists of a
linear map μ:A⊗A→A, and Condition (1.1) can be written as
[TABLE]
Definition 1.2**.**
Let (A,μA,αA,βA) and (B,μB,αB,βB) be two BiHom-associative algebras. A linear map φ:A→B is called a BiHom-associative algebras morphism if
[TABLE]
In particular, BiHom-associative algebras (A,μA,αA,βA) and (B,μB,αB,βB) are isomorphic if φ is also bijective.
Definition 1.3**.**
A unital BiHom-associative algebra (A,μ,α,β,u) is called unital if there exists an element uA∈A (called a unit) such that
α(uA)=uA,β(uA)=uA and auA=α(a) and uAa=β(a), ∀a∈A.
A morphism of unital BiHom-associative algebras ϕ:A⟶B is called unital if ϕ(uA)=uB.
1.2. Structure of BiHom-associative algebras
We state in this section some properties on the structure of BiHom-associative algebras which are not necessarily multiplicative.
Proposition 1.4** ([11]).**
Let (A,μ,α,β) be a BiHom-associative algebra and γ:A→A be a BiHom-associative algebra morphism.
Then (A,γμ,γα,γβ) is a BiHom-associative algebra.
Definition 1.5**.**
Let (A,μ,α,β) be a BiHom-associative algebra. If there is an associative algebra (A,μ′) such that
μ(x,y)=(α⊗β)μ′(x,y),∀x,y∈A, we say that (A,μ,α,β) is of associative type and (A,μ′) is its compatible associative algebra or the untwist of (A,μ,α,β).
Proposition 1.6**.**
Let (A,μ,α,β) be an n-dimensional BiHom-associative algebra and ϕ:A→A be an invertible linear map. Then there is an
isomorphism with an n-dimensional BiHom-associative algebra (A,μ′,ϕαϕ−1,ϕβϕ−1) where
μ′=ϕ∘μ∘(ϕ−1⊗ϕ−1). Furthermore, if {Cijk} are the structure constants of μ with
respect to the basis {e1,…,en}, then μ′ has the same structure constants with respect to the basis
{ϕ(e1),…,ϕ(en)}.
Proof.
We prove for any invertible linear map ϕ:A→A,(A,μ′,ϕαϕ−1,ϕβϕ−1) is a BiHom-associative algebra.
[TABLE]
So (A,μ′,ϕαϕ−1,ϕβϕ−1) is a BiHom-associative algebra.
It is also multiplicative. Indeed, for α
[TABLE]
We have also for β
[TABLE]
Therefore ϕ:(A,μ,α,β)→(A,μ′,ϕαϕ−1,ϕβϕ−1) is a BiHom-associative algebras morphism, since
ϕ∘μ=ϕ∘μ∘(ϕ−1⊗ϕ−1)∘(ϕ⊗ϕ)=μ′∘(ϕ⊗ϕ) and
(ϕαϕ−1)∘ϕ=ϕ∘α and (ϕβϕ−1)∘ϕ=ϕ∘β.
It is easy to see that
{ϕ(ei),⋯,ϕ(en)} is a basis of A. For i,j=1,⋯,n, we have
\begin{array}[]{ll}\mu_{2}(\phi(e_{i}),\phi(e_{j}))&=\phi\mu_{1}(\phi^{-1}(e_{i}),\phi^{-1}(e_{j}))=\phi\mu(e_{i},e_{j})=\sum_{k=1}^{n}\mathcal{C}^{k}_{ij}\phi(e_{k}).\end{array}
∎
Remark 1.7*.*
A BiHom-associative algebra (A,μ,α,β) is isomorphic to an associative algebra if and only if α=β=id. Indeed,
ϕ∘α∘ϕ−1=ϕ∘β∘ϕ−1=id is equivalent to α=β=id.
Remark 1.8*.*
Proposition 1.6 is useful to make a classification of BiHom-associative algebras. Indeed, we have to consider the class of morphisms which are conjugate. Representations of these classes are given by Jordan forms of the matrix.
Any n×n matrix over K is equivalent up to basis change to Jordan’s canonical form, then we choose ϕ such that the matrix of ϕαϕ−1=γ and λ=ϕβϕ−1, where γ and λ are Jordan canonical forms.
Hence, to obtain the classification, we consider only Jordan forms for the structure map of Hom-associative algebras.
Proposition 1.9**.**
Let (A,μ,α,β) be a Hom-associative algebra over K. Let (A,μ′,ϕαϕ−1,ϕβϕ−1) be
its isomorphic Hom-associative algebra described in Proposition 1.6. If ψ is an automorphism of (A,μ,α,β), then
ϕψϕ−1 is an automorphism of (A,μ,ϕαϕ−1,ϕβϕ−1).
Proof.
Note that γ=ϕαϕ−1. We have
[TABLE]
.
For β, we pose λ=ϕβϕ−1, we have
[TABLE]
For any x,y∈A,
[TABLE]
By Definition, ϕψϕ−1 is an automorphism of (A,μ′,ϕαϕ−1,ϕβϕ−1).
∎
The following characterization was given for Hom-Lie algebras in [12].
Proposition 1.10**.**
*Given two BiHom-associative algebras (A,μA,αA,βA) and (B,μB,αB,βB) over field K, there is a
BiHom-associative algebra (A⊕B,μA⊕B,αA+αB,βA+βB), where μA⊕B the usual multiplication
(a+b)(a′+b′)=(a,a′)+(b,b′).*
μA⊕B(.,.):(A⊕B)×(A⊕B)→(A⊕B)* is given by*
[TABLE]
and the linear map
(αA+αB,βA+βB):A⊕B→A⊕B is given by
[TABLE]
Proof.
For any a,a′,a′′∈A,b,b′,b′′∈B, by direct computation, we get
[TABLE]
This ends the proof.
∎
A morphism of BiHom-associative algebras
ϕ:(A,μA,αA,βA)→(B,μB,αB,βB) is a linear map ϕ:A→B such that
ϕ∘μA(a,b)=μB∘(ϕ(a),ϕ(b)),∀a,b∈A,αB∘ϕ=ϕ∘αA and βB∘ϕ=ϕ∘βB
Denote by ξϕ⊂A⊕B, the graph of the linear map ϕ:A→B.
Proposition 1.11**.**
A linear map ϕ:(A,μA,αA,βA)→(B,μB,αB,βB) is a BiHom-associative algebras morphism if and only if the graph ξϕ⊂A⊕B is a BiHom-associative sub-algebras of (A⊕B,μA⊕B,αA+αB,βA+βB).
Proof.
Let ϕ:(A,μA,αA,βA)→(B,μB,αB,βB) be a BiHom-associative algebra morphism. Then for any
a,b∈A, we have
[TABLE]
Thus the graph ξϕ is closed under the product μA⊕B. Furthermore, since αB∘ϕ=ϕ∘αA and
βB∘ϕ=ϕ∘βA
we have
[TABLE]
which implies that
(αA+αB,βA+βB)⊂ξϕ. Thus ξϕ is a BiHom-associative sub-algebra of
(A⊕B,μA⊕B,αA+αB,βA+βB).
Conversely, if the graph ξϕ⊂A⊕B is a Hom-associative sub-algebra of
(A⊕B,μA⊕B,αA+αB,βA+βB), then we have
[TABLE]
which implies that
μB(ϕ(a),ϕ(b))=ϕ∘μA(a,b). Furthermore, (αA+βA,αB+βB)(ξϕ)⊂ξϕ yields that
[TABLE]
which is equivalent to the condition
αB∘ϕ(a)=ϕ∘αA(a) and βB∘ϕ(a)=ϕ∘βA(a). Therefore, ϕ is a BiHom-associative algebra morphism.
∎
1.3. Unital BiHom-associative algebras
In this section we discuss unital BiHom-associative algebras. We denote by Hun the set of n-dimensional unital
BiHom-associative algebras.
Definition 1.12**.**
A BiHom-associative algebra (A,μ,α,β) is called unital if there exists an element u∈A such that
μ(x,u)=α(x) and μ(u,x)=β(x) for all x∈A.
Proposition 1.13**.**
Let (A,μ,α,β) be a BiHom-associative algebra. We set A~=span(A,u) the vector space generated by elements of
A and u. Assume μ(x,u)=α(x),andμ(u,x)=β(x)∀x∈A,α(u)=u and β(u)=u.
Then (A~,μ,α,β,u) is a unital
Hom-associative algebra.
Proof.
It is straightforward to check the Hom-associativity. For example
\begin{array}[]{ll}\mu(\mu(x,y),\beta(u))=\mu(\mu(x,y),u)=\alpha(\mu(x,y))=\mu(\alpha(x),\alpha(y))=\mu(\alpha(x),\mu(y,u)).\end{array}
∎
Remark 1.14*.*
Some unital BiHom-associative cannot be obtained as an extension of a non unital BiHom-associative algebra.
Remark 1.15*.*
Let (A,μ,α,β,u) be an n-dimensional unital BiHom-associative algebra and ϕ:A→A be an invertible linear map
such that ϕ(u)=u. Then it is isomorphic to a n-dimensional BiHom-associative algebra (A,μ′,ϕαϕ−1,ϕβϕ−1,u)
where μ′=ϕ∘μ∘(ϕ−1⊗ϕ−1). Moreover, if {Cijk} are the structure constants of μ with respect to the basis {e1,…,en} with e1=u being the unit, then μ′ has the same structure constants with respect to the basis
{ϕ(e1),…,ϕ(en)} with u the unit element.
Indeed, we use Proposition 1.6 and Definition 1.3. The unit is conserved since
[TABLE]
Proposition 1.16**.**
Let (A,μA,αA,βA,uA) and (B,μB,αB,βB,uB) be two unital BiHom-associative algebras with ϕ(uA)=uB. Suppose there exists a BiHom-associative algebra morphism ϕ:A→B. If (A,μA′,uA′) is an untwist of
(A,μA,αA,βA,uA) then there exists an untwist of (B,μB,αB,βB,uB) such that
ϕ:(A,μA′,uA′)→(B,μB′,uB′) is an algebra morphism.
Proof.
Because ϕ is a homomorphism from (A,μA,αA,βA,uA) to (B,μB,αB,βB,uB). Then
αBϕ=ϕαA, and βB∘ϕ=ϕ∘βA for all x∈A, we have
μB(ϕ(x),ϕ(uA))=μB(ϕ(x),uB)=αB∘ϕ(x) and μB(ϕ(uA),ϕ(x))=μB(uB,ϕ(x))=βB∘ϕ(x). We have also
ϕ∘μA(x,uA)=ϕ∘αA(x) and ϕ∘μA(uA,x)=ϕ∘βA(x). By Proposition 1.13, we can see that
(AB,μB,uB) is also an associative algebra.
Furthermore
\begin{array}[]{ll}\mu^{\prime}_{B}(\phi(x),\phi(u_{A}))=\mu^{\prime}_{B}(\phi(x),u_{B})&=\phi\circ\alpha^{\prime}_{A}\circ\phi(x)=\phi\circ\alpha_{A}\circ\mu_{A}(x,u_{A})\\
&=\alpha_{B}\circ\phi\circ\mu_{A}(x,u_{A})=\alpha_{B}\circ\mu_{B}(\phi(x),u_{B})\quad\text{and}\end{array}
\begin{array}[]{ll}\mu^{\prime}_{B}(\phi(u_{A}),\phi(x))=\mu^{\prime}_{B}(u_{B},\phi(x))&=\phi\circ\beta^{\prime}_{A}\circ\phi(x)=\phi\circ\alpha_{A}\circ\mu_{A}(u_{A},x)\\
&=\beta_{B}\circ\phi\circ\mu_{A}(u_{A},x)=\beta_{B}\circ\mu_{B}(u_{B},\phi(x)).\end{array}
∎
2. Algebraic varieties of BiHom-associative algebras and Classifications
In this section, we deal with Algebraic varieties of BiHom-associative algebras with a fixed dimension. A BiHom-associative algebra is identified with its structures constants with respect to a fixed basis. Their set corresponds to an algebraic variety where the ideal is generated by polynomials corresponding to the BiHom-associativity condition.
2.1. Algebraic varieties Hn and their action of linear group
.
Let A be a n-dimensional K-linear space and {e1,⋯,en} be a basis of A. A BiHom-algebra structure on A with product μ and a structure map α and β is determined by n3 structure constants Cijk where
μ(ei,ej)=∑k=1nCijkek and by 2n2 structure constants aji and bkj, where
α(ei)=∑j=1najiej and β(ej)=∑k=1nbkjek.
If we require this algebra structure to be BiHom-associative, then this limits the set of structure constants
(Cijk,aij,bjk) to a cubic sub-variety of the affine algebraic variety Kn3+2n2 defined by the following polynomial equations system :
[TABLE]
Moreover if μ is commutative,
we have Cijk=Cjiki,j,k=1,⋯,n.
The first set of equation correspond to the BiHom-associative condition
μ(α(ei),μ(ej,ek))=μ(μ(ei,ej),β(ek)) and the second set to multiplicativity condition
α∘μ(ei,ej)=μ(α(ei),α(ej)) and β∘μ(ej,ek)=μ(β(ej),β(ek)).
We denote by Hn the set of all n-dimensional multiplicative Hom-associative algebras.
The group GLn(K) acts on the algebraic varieties of BiHom-structures by the so-called transport of structure action defined as follows. Let A=(A,μ,α,β) be a n-dimensional BiHom-associative algebra defined by multiplication μ and a linear map
α and β. Given f∈GLn(K), the action f⋅A transports the structure,
[TABLE]
defined for x,y∈A by
[TABLE]
The conjugate class is given by Θ(f,(A,μ,α,β))=(A,f−1∘μ∘(f⊗f),f∘α∘f−1,f∘β∘f−1)) for f∈GLn(K).
The orbit of a Hom-associative algebra A of Hasn is given by
[TABLE]
The orbits are in 1-1 correspondence with the isomorphism classes of n-dimensional BiHom-associative algebras.
The stabilizer is
[TABLE]
We characterize –in terms of structure constants– the fact that two BiHom-associative algebras are in the same orbit (or isomorphic).
Let (A,μA,αA,βA) and (B,μB,αB,βB) be two n-dimensional BiHom-associative algebras. They are isomorphic if there exists
φ∈GLn(K) such that
[TABLE]
Remark 2.1*.*
Conditions (2.3) are equivalent to
μA=φ−1∘μB∘φ⊗φ, αA=φ−1∘αB∘φ and
βA=φ−1∘βB∘φ.
We set with respect to a basis {ei}i=1,⋯,n:
φ(ei)=∑p=1ndpiep,α(ei)=∑j=1najiej,β(ei)=∑k=1nbkiek,i∈{1,…,n}
μA(ei,ej)=∑k=1nCijkek,μB(ei,ej)=∑k=1nC~ijkek,i,j∈{1,…,n}.
Conditions (2.3) translate to the following
system
[TABLE]
We shall check whether the previous are isomorphic. In particular, we shall provide all 2-dimensional BiHom-associative algebras, corresponding to solutions of the system (3.6). To this end, we use a computer algebra system.
2.2. Classification in low dimensions
Theorem 2.2**.**
Every 2-dimensional multiplicative BiHom-associative algebra is isomorphic to one of the following pairwise non-isomorphic BiHom-associative algebras (A,∗,α,β) appearing in Table 1,
where ∗ is the multiplication and α and β the structure maps. We set {e1,e2} to be a basis of K2.
Table 1
- H12
: e1∗e1=e2,e1∗e2=e2,e2∗e1=−e1,e2∗e2=e2,α(e1)=e1,α(e2)=e2,β(e1)=−e1,β(e2)=e2;
- H22
: e1∗e1=e2,e1∗e2=e1,e2∗e1=−e1,e2∗e2=−e2,α(e1)=−e1,α(e2)=e2,β(e1)=e1,β(e2)=e2;
- H32
: e1∗e1=e1,e1∗e2=e2,e2∗e1=e2,e2∗e2=e2,α(e1)=e1,α(e2)=e2,β(e1)=e1,β(e2)=e2;
- H42
: e1∗e1=e1,e1∗e2=e2,e2∗e1=e2,e2∗e2=e1,α(e1)=e1,α(e2)=e2,β(e1)=e1,β(e2)=e2;
- H52
: e1∗e1=e1,e1∗e2=e2,e2∗e1=e1,e2∗e2=e2,α(e1)=e1,α(e2)=e2,β(e1)=e1,β(e2)=e2;
- H62
: e1∗e1=e1,e1∗e2=e1,e2∗e1=e1,e2∗e2=e2,α(e1)=e1,α(e2)=e2,β(e1)=e1,β(e2)=e2;
- H72
: e1∗e2=e1,e2∗e1=−e1,e2∗e2=−e2,α(e1)=−e1,α(e2)=e2,β(e1)=e1,β(e2)=e2;
- H82
: e1∗e1=−e1,e1∗e2=−e2,e2∗e2=e2,α(e1)=e1,α(e2)=−e2,β(e1)=e1,β(e2)=e2;
- H92
: e1∗e1=e1,e1∗e2=−e2,e2∗e1=e2,α(e1)=e1,α(e2)=e2,β(e1)=e2,β(e2)=−e2;
- H102
: e1∗e2=e1,e2∗e1=e1,e2∗e2=e1+e2,α(e1)=e1,α(e2)=e2,β(e1)=e1,β(e2)=e2;
- H112
: e1∗e1=e2,e1∗e2=e1,e2∗e1=e1,e2∗e2=e2,α(e1)=e1,α(e2)=e2,β(e1)=e1,β(e2)=e2;
- H122
: e1∗e2=e1,e2∗e1=e1,e2∗e2=e1,β(e2)=e1;
- H132
: e2∗e2=e1,α(e1)=e1,α(e2)=e1+e2,β(e1)=e1,β(e2)=e2.
Theorem 2.3**.**
Every 2-dimensional unital multiplicative BiHom-associative algebra is isomorphic to one of the following pairwise non-isomorphic BiHom-associative algebras (A,∗,α,β) appearing in Table 2,
where ∗ is the multiplication and α and β the structure maps. We set {e1,e2} to be a basis of
K2 where e1 is the unit :
Table 2
- Hu12
: e1∗e1=e1,e1∗e2=e2,e2∗e1=e2,e2∗e2=e1+e2,α(e1)=β(e1)=e1,α(e2)=β(e2)=e2;
- Hu22
: e1∗e1=e1,e1∗e2=e2,e2∗e1=e2,α(e1)=β(e1)=e1,α(e2)=β(e2)=e2;
- Hu32
: e1∗e1=e1,e1∗e2=−e2,e2∗e1=−e2,e2∗e2=e1,α(e1)=β(e1)=e1,α(e2)=β(e2)=−e2;
- Hu42
: e1∗e1=e1,e1∗e2=e2,e2∗e1=e2,e2∗e2=e2,α(e1)=β(e1)=e1,α(e2)=β(e2)=e2.
Theorem 2.4**.**
Every 3-dimensional BiHom-associative algebra is isomorphic to one of the following pairwise non-isomorphic BiHom-associative algebras (A,∗,α,β) where ∗ is the multiplication and α and β the structure maps. We set {e1,e2,e3} to be a basis of K3.
- H13
: e1∗e1=e1,e1∗e2=e1,e2∗e1=e2,e2∗e2=e2,e3∗e2=e3,e3∗e3=e3,α(e1)=e1,α(e2)=e2,β(e1)=e1,β(e2)=e1−e2;
- H23
: e1∗e1=e1,e1∗e2=e1,e1∗e3=e3,e2∗e1=e2,e2∗e2=e2,e2∗e3=e3,e3∗e1=e3,e3∗e2=e3,α(e1)=e1,α(e2)=e2,α(e3)=e3,β(e1)=e1,β(e2)=e1;
- H33
: e1∗e1=e1,e1∗e2=e1,e2∗e2=e2,e1∗e3=−e3,e3∗e2=e1−e2,α(e1)=e1,α(e2)=e2,α(e3)=−e3,β(e1)=e1,β(e2)=e1;
- H43
: e1∗e1=e1,e1∗e2=e1−e2,e2∗e1=e2,e2∗e2=−e1,α(e1)=e1,α(e2)=e2,β(e1)=e1;
- H53
: e1∗e1=e1,e1∗e2=e1,e2∗e1=e2,e2∗e2=e2,e3∗e1=e3,e3∗e2=e3,e3∗e3=e1−e3,α(e1)=e1,α(e2)=e2,α(e3)=e3,β(e1)=e1,β(e2)=e1;
- H63
: e1∗e1=e1,e1∗e2=e1,e1∗e3=e3,e2∗e1=e2,e2∗e2=e2,e2∗e3=e3,e3∗e1=e3,e3∗e2=e3,α(e1)=e1,α(e2)=e2,α(e3)=e3,β(e1)=e1,β(e2)=e1;
- H73
: e1∗e1=e1,e1∗e2=e2,e3∗e2=e3,e3∗e3=e3,α(e1)=e1,α(e2)=e2,β(e1)=e1;
- H83
: e1∗e3=e1−e2,e2∗e3=e1−e2,e3∗e3=e1−e2,α(e1)=e1,α(e2)=e1−e2,α(e3)=e2−e3,β(e1)=e1,β(e2)=e1−e2,β(e3)=e2−e3;
- H93
: e2∗e3=−e1,e3∗e2=e1,e3∗e3=e1,α(e1)=e1,α(e2)=e1+e2,α(e3)=e2+e3,β(e1)=e1,β(e2)=e1+e2,β(e3)=e2+e3;
- H103
: e1∗e3=e1+e2,e2∗e3=e1+e2,e3∗e1=e1−e2,e3∗e2=e1−e2,e3∗e3=e1−e2,α(e1)=e1,α(e2)=e1−e2,α(e3)=e2−e3,β(e1)=e1,β(e2)=e1−e2,β(e3)=e2−e3;
- H113
: e2∗e3=−e1,e3∗e2=e1,e3∗e3=e1,α(e1)=e1,α(e2)=e1+e2,α(e3)=e1+e2+e3,β(e1)=e1,β(e2)=e1+e2,β(e3)=e1+e2+e3;
- H123
: e1∗e3=e1−e2,e2∗e3=e1−e2,e3∗e3=e1−e2,α(e1)=e1,α(e2)=e1,α(e3)=e2,β(e1)=e1,β(e2)=e1,β(e3)=e2;
- H133
: e1∗e3=e1−e2,e2∗e3=e1−e2,e3∗e1=e1−e2,e3∗e2=e1−e2,e3∗e3=e1−e2,α(e1)=e1,α(e2)=e1,α(e3)=e2,β(e1)=e1,β(e2)=e1,β(e3)=e2.
Theorem 2.5**.**
Every 3-dimensional unital BiHom-associative algebra is isomorphic to one of the following pairwise non-isomorphic BiHom-associative algebras (A,∗,α,β) where ∗ is the multiplication and α and β the structure maps. We set {e1,e2,e3} to be a basis of K3.
- Hu13
: e1∗e1=e1,e1∗e2=e1−e2,e2∗e1=e2,e2∗e2=−e1,e3∗e3=−e3α(e1)=e1,α(e2)=e2,β(e1)=e1,β(e2)=e1−e2;
- Hu23
: e1∗e1=e1,e1∗e2=e1,e2∗e1=e2,e2∗e2=e2,e2∗e3=e1−e2,e3∗e1=e3,e3∗e2=e3,e3∗e3=e1−e2,α(e1)=e1,α(e2)=e2,α(e3)=e3,β(e1)=e1,β(e2)=e1;
- Hu33
: e1∗e1=e1,e1∗e2=e1,e2∗e1=e2,e2∗e2=e2,e3∗e1=e3,e3∗e2=e3,α(e1)=e1,α(e2)=e2,α(e3)=e3,β(e1)=e1,β(e2)=e1;
- Hu43
: e1∗e1=e1,e1∗e2=e1,e2∗e1=e2,e2∗e2=e2,e3∗e3=e3,α(e1)=e1,α(e2)=e2,β(e1)=e1,β(e2)=e1;
- Hu53
: e1∗e1=e1,e1∗e2=e1−e2,e2∗e1=e2,e3∗e1=e3,α(e1)=e1,α(e2)=e2,α(e3)=e3,β(e1)=e1,β(e2)=e1−e2;
- Hu63
: e1∗e1=e1,e1∗e2=e1,e2∗e1=e2,e2∗e2=e2,e2∗e3=e1−e2,e3∗e1=e3,e3∗e2=e3,α(e1)=e1,α(e2)=e2,α(e3)=e3,β(e1)=e1,β(e2)=e1;
- Hu73
: e1∗e1=e1,e1∗e2=e1,e2∗e1=e2,e2∗e2=e2+e3,e3∗e1=e3,e3∗e2=e3,α(e1)=e1,α(e2)=e2,α(e3)=e3,β(e1)=e1,β(e2)=e1;
- Hu83
: e1∗e1=e1,e1∗e2=e1−e2,e2∗e1=e2,e3∗e1=e3,e3∗e2=e3,α(e1)=e1,α(e2)=e2,α(e3)=e3,β(e1)=e1,β(e2)=e1−e2;
- H93
: e1∗e1=e1,e1∗e2=e1,e2∗e1=e2,e2∗e2=e2,e3∗e1=e3,e3∗e2=e3,α(e1)=e1,α(e2)=e2,α(e3)=e3,β(e1)=e1,β(e2)=e1;
- Hu103
: e1∗e1=e1,e1∗e2=e1,e1∗e3=e3,e2∗e1=e2,e2∗e2=e2,e2∗e3=e3,α(e1)=e1,α(e2)=e2,β(e1)=e1,β(e2)=e1,β(e3)=e3;
- Hu113
: e1∗e1=e1,e1∗e3=e3,e2∗e2=e2,e3∗e2=e2,α(e1)=e1,β(e1)=e1,β(e3)=e3;
- Hu123
: e1∗e1=e1,e1∗e2=e2,e2∗e3=e3,e3∗e3=e3,α(e1)=e1,β(e1)=e1,β(e2)=e2;
- Hu133
: e1∗e1=e1,e2∗e2=e2,e2∗e3=e2,e3∗e1=e3,α(e1)=e1,α(e3)=e3,β(e1)=e1.
3. BiHom-coassociative coalgebras and BiHom-bialgebras
In this Section, we show that for a fixed dimension n, the set of BiHom-bialgebras is endowed with a structure of an algebraic variety and a natural structure transport action which describes the set of isomorphic BiHom-algebras. Solving such systems of polynomial equations leads
to the classification of such structures. We shall now introduce the dual concept to BiHom-associative algebras:
Definition 3.1**.**
A BiHom-coassociative coalgebra is 4-tuple (A,Δ,ψ,ω) in which A is a linear space, ψ,ω:A⟶A
and Δ:A⟶A⊗A are linear maps, such that
[TABLE]
We call ψ and ω (in this order) the structure maps of A.
Let V be an n-dimensional vector space over K. Fixing a basis {ei}i={1,…,n} of V,
a multiplication μ (resp. linear maps α, β, ψ, ω and a comultiplication Δ) is identified with its n3,
2n2 structure constants Cijk∈K (resp. aji, bji, ξji, Dijk and γji)
where μ(ei⊗ej)=∑k=1nCijkek, α(ei)=∑j=1najiej, β(ei)=∑j=1nbjiej,
ψ(ei)=∑j=1nξjiej, ω(ei)=∑j=1nγjiej and Δ(ei)=∑j,k=1nDijkej⊗ek.
The counit ε is identified with its n structure constants ζi. We assume that e1 is the unit.
A family {(Cijk,aji,bji,ξji,γji,Dijk),…,i,j,k∈{1,…,n}}
represents a BiHom-coassociative coalgebra if the underlying family satisfies the appropriate conditions which translate to the following
polynomial equations:
[TABLE]
A morphism f:(A,ΔA,ψA,ωA)⟶(B,ΔB,ψB,ωB) of BiHom-coassociative coalgebras is a linear map
f:A⟶B such that ψB∘f=f∘ψA,ωB∘f=f∘ωA and (f⊗f)∘ΔA=ΔB∘f.
[TABLE]
A BiHom-coassociative coalgebra (A,Δ,ψ,ω) is called counital if there exists a linear map ε:A→K
(called a counit) such that
[TABLE]
We have ∑j=1nξjiζj=ζi,∑j=1nγjiζj=ζi,∑p=1n∑q=1nDipqεq=aij,∑p=1n∑q=1nDiqrεr=bij.
A morphism of counital BiHom-coassociative coalgebras f:A⟶B is called counital if f=εA, where εA and
εB are the counits of A and B, respectively.
Definition 3.2**.**
A BiHom-bialgebra is a 7-tuple (H,μ,Δ,α,β,ψ,ω), with the property that (H,μ,α,β) is a BiHom-associative
algebra, (H,Δ,ψ,ω) is a BiHom-coassociative coalgebra and moreover the following relations are satisfied, for h,h′∈H :
[TABLE]
[TABLE]
We say that H is a unital and counital BiHom-bialgebra if, in addition, it admits a unit uH and a counit εH such that
[TABLE]
We have Δ(e1)=e1⊗e1,ε(e1)=1,ψ(e1)=e1,ω(e1)=e1,∑j=1najiζj=ζi,and∑j=1nbjiηj=ηi.
Let us record the formula expressing the BiHom-coassociativity of Δ :
[TABLE]
3.1. Classification in Dimension 2 in Hi2
Thanks to computer Hom-algebra, we obtain the following Hom-coalgebras associated to the previons Hom algebras in order to obtain a
Hom-bialgebra structures. We denote the comultiplications by Δi,j2, where i indicates the item of the multiplication and j
the item of comultiplication.
Theorem 3.3**.**
The set of 2-dimensional BiHom-Bialgebras algebras yields two non-isomorphic algebras. Let {e1,e2} be a basis of
K2, then the BiHom-Bialgebras are given by the following non-trivial comultiplications.
- 1.
Δ(e1)=e1⊗e2+e2⊗e2,Δ(e2)=e1⊗e1−e2⊗e2,ψ(e1)=−e1,ψ(e2)=e2,ω(e1)=−e1,ω(e2)=e2;
2. 2.
Δ(e1)=e1⊗e1,Δ(e2)=e1⊗e1+e1⊗e2+e2⊗e1+e2⊗e2,ψ(e1)=e1,ω(e1)=e1,ω(e2)=e2;
3. 3.
Δ(e1)=e1⊗e1−e1⊗e2−e2⊗e1+e2⊗e2,Δ(e2)=e1⊗e1−e1⊗e2−e2⊗e1+e2⊗e2,ψ(e1)=e1,ψ(e2)=e2,ω(e1)=e1−e2,ω(e2)=e1−e2;
4. 4.
Δ(e1)=e1⊗e1−e1⊗e2−e2⊗e1+e2⊗e2,Δ(e2)=e1⊗e1−e1⊗e2−e2⊗e1+e2⊗e2,ψ(e1)=e1−e2,ψ(e2)=e1−e2,ω(e1)=e1,ω(e2)=e2;
5. 5.
Δ(e1)=e1⊗e1−e1⊗e2−e2⊗e1+e2⊗e2,Δ(e2)=e1⊗e1−e1⊗e2−e2⊗e1+e2⊗e2,ψ(e1)=e1−e2,ψ(e2)=e1−e2,ω(e1)=e1−e2,ω(e2)=e1−e2;
6. 6.
Δ(e1)=e1⊗e1−e1⊗e2−e2⊗e1+e2⊗e2,Δ(e2)=e1⊗e1−e1⊗e2−e2⊗e1+e2⊗e2,ψ(e1)=−e1+e2,ψ(e2)=−e1+e2,ω(e1)=e1,ω(e2)=e2;
7. 7.
Δ(e1)=e1⊗e1−e1⊗e2−e2⊗e1+e2⊗e2,Δ(e2)=e1⊗e1−e1⊗e2−e2⊗e1+e2⊗e2,ψ(e1)=e1,ψ(e2)=e2;
8. 8.
Δ(e1)=e1⊗e1+e1⊗e2+e2⊗e1+e2⊗e2,Δ(e2)=e1⊗e1+e1⊗e2+e2⊗e1+e2⊗e2,ψ(e1)=e1−e2,ψ(e2)=−e1+e2;
9. 9.
Δ(e1)=e1⊗e1+e1⊗e2+e2⊗e1+e2⊗e2,Δ(e2)=e1⊗e1+e1⊗e2+e2⊗e1+e2⊗e2,ψ(e1)=e1−e2,ψ(e2)=−e1+e2,ω(e1)=e1−e2,ω(e2)=−e1+e2;
10. 10.
Δ(e1)=e1⊗e1+e1⊗e2+e2⊗e1+e2⊗e2,Δ(e2)=e1⊗e1+e1⊗e2+e2⊗e1+e2⊗e2,ω(e1)=e1+e2,ω(e2)=e1+e2.
Remark 3.4*.*
There is no BiHom-Bialgabra whose underlying BiHom-associative algebra is given by H12.
Theorem 3.5**.**
The set of 2-dimensional unital BiHom-Bialgebras yields two non-isomorphic algebras. Let {e1,e2} be a basis of
K2, then the unital BiHom-Bialgebras are given by the following non-trivial comultiplications.
- 1.
Δ1,12(e1)=e1⊗e1,Δ1,12(e2)=−e1⊗e1+e1⊗e2+e2⊗e1+e2⊗e2,ψ(e1)=e1,ψ(e2)=e2,ω(e1)=e1,ω(e2)=e2,ε(e1)=1,ε(e2)=2;
2. 2.
Δ1,22(e1)=e1⊗e1,Δ1,22(e2)=−e1⊗e1+e1⊗e2+e2⊗e1+e2⊗e2,ψ(e1)=e1,ψ(e2)=e1,ω(e1)=e1,ω(e2)=e1,ε(e1)=1,ε(e2)=2;
3. 3.
Δ1,32(e1)=e1⊗e1,Δ1,32(e2)=e1⊗e1+e1⊗e2+e2⊗e1−e2⊗e2,ψ(e1)=e1,ψ(e2)=e1,ω(e1)=e1,ω(e2)=e1,ε(e1)=1,ε(e2)=−1;
4. 4.
Δ1,42(e1)=e1⊗e1,Δ1,42(e2)=e1⊗e1+e1⊗e2+e2⊗e1−e2⊗e2,ψ(e1)=e1,ψ(e2)=−e1,ω(e1)=e1,ω(e2)=−e1,ε(e1)=1,ε(e2)=−1;
5. 5.
Δ2,12(e1)=e1⊗e1,Δ2,22(e2)=−e1⊗e1+e1⊗e2+e2⊗e1+e2⊗e2,ψ(e1)=e1,ψ(e2)=e1,ω(e1)=e1,ω(e2)=e1,ε(e1)=1,ε(e2)=1;
6. 6.
Δ2,22(e1)=e1⊗e1,Δ2,32(e2)=e1⊗e1+e1⊗e2+e2⊗e1−e2⊗e2,ψ(e1)=e1,ψ(e2)=e2,ω(e1)=e1,ω(e2)=e2,ε(e1)=1,ε(e2)=1;
7. 7.
Δ4,12(e1)=e1⊗e1,Δ3,12(e2)=e2⊗e2,ψ(e1)=e1,ψ(e2)=e2,ω(e1)=e1,ω(e2)=e2,ε(e1)=1,ε(e2)=1;
8. 8.
Δ4,22(e1)=e1⊗e1,Δ4,22(e2)=e2⊗e2,ψ(e1)=e1,ψ(e2)=e1,ω(e1)=e1,ω(e2)=e1,ε(e1)=1,ε(e2)=1;
9. 9.
Δ4,32(e1)=e1⊗e1,Δ4,32(e2)=e1⊗e2+e2⊗e1−2e2⊗e2,ψ(e1)=e1,ψ(e2)=e2,ω(e1)=e1,ω(e2)=e2,ε(e1)=1,ε(e2)=1;
10. 10.
Δ4,42(e1)=e1⊗e1,Δ4,42(e2)=e1⊗e2+e2⊗e1−e2⊗e2,ψ(e1)=e1,ψ(e2)=e1,ω(e1)=e1,ω(e2)=e1,ε(e1)=1;
Remark 3.6*.*
There is no BiHom-Bialgabra whose underlying BiHom-associative algebra is given by Hu32.
3.2. Classification in Dimension 3 in Hi3
Thanks to computer Hom-algebra, we obtain the following Hom-coalgebras associated to the previons Hom algebras in order to obtain a
Hom-bialgebra structures. We denote the comultiplications by Δi,j3, where i indicates the item of the multiplication and j
the item of comultiplication.
Theorem 3.7**.**
The set of 3-dimensional BiHom-Bialgebras yields two non-isomorphic algebras. Let {e1,e2,e3} be a basis of
K3, then the BiHom-Bialgebras are given by the following non-trivial comultiplications.
- 1.
Δ(e1)=e1⊗e1,Δ(e2)=e1⊗e1+e1⊗e2−e1⊗e3+e2⊗e1+e2⊗e3−e3⊗e1−e3⊗e3,
Δ(e3)=e3⊗e3,ψ(e1)=e1,ψ(e2)=e1+e2,ψ(e3)=e3,ω(e1)=e1,ω(e2)=e1+e2,ω(e3)=e3;
2. 2.
Δ(e1)=e1⊗e1,Δ(e2)=e1⊗e1+e1⊗e2+e1⊗e3+e2⊗e1−e2⊗e2+e2⊗e3−e3⊗e1+e3⊗e2
Δ(e3)=e3⊗e3,ψ(e1)=e1,ψ(e2)=e1,ψ(e3)=e3,ω(e1)=e1,ω(e2)=e1,ω(e3)=e3;
3. 3.
Δ(e1)=e1⊗e1+e3⊗e3,Δ(e2)=e1⊗e2+e2⊗e1,Δ(e3)=e1⊗e3+e3⊗e1,ψ(e1)=e1,ψ(e3)=e3,ω(e1)=e1,ω(e3)=e3;
4. 4.
Δ(e1)=e1⊗e1+e3⊗e3,Δ(e2)=e1⊗e2+e2⊗e1,Δ(e3)=−e1⊗e3−e3⊗e1,ψ(e1)=e1,ψ(e3)=e3,ω(e1)=e1,ω(e3)=e3;
5. 5.
Δ(e1)=e1⊗e1,Δ(e2)=e1⊗e2+e2⊗e1+e2⊗e2,Δ(e3)=e1⊗e3+e2⊗e3+e3⊗e1+e3⊗e2−e3⊗e3,ψ(e1)=e1,ω(e1)=e1;
6. 6.
Δ(e1)=e1⊗e1,Δ(e2)=e1⊗e2+e2⊗e1+e2⊗e2,Δ(e3)=e1⊗e3+e2⊗e3+e3⊗e1+e3⊗e2−e3⊗e3,ψ(e1)=e1,ψ(e2)=e2,ω(e1)=e1,ω(e2)=e2;
7. 7.
Δ(e1)=0,Δ(e2)=−e1⊗e1+e1⊗e2+e2⊗e1,Δ(e3)=e1⊗e3+e3⊗e1,ψ(e1)=e1,ψ(e2)=e1−e2,ψ(e3)=−e3,ω(e1)=e1,ω(e2)=e1−e2,ω(e3)=−e3;
8. 8.
Δ(e1)=0,Δ(e2)=−e1⊗e1+e1⊗e2+e2⊗e1,Δ(e3)=e1⊗e3+e3⊗e1,ψ(e1)=e1,ψ(e2)=e2,ψ(e3)=e3,ω(e1)=e1,ω(e2)=e1−e2,ω(e3)=−e3;
9. 9.
Δ(e1)=0,Δ(e2)=−e1⊗e1+e1⊗e2+e2⊗e1,Δ(e3)=e1⊗e3+e3⊗e1,ψ(e1)=e1,ψ(e2)=e1−e2,ψ(e3)=−e3,ω(e1)=e1,ω(e2)=e2,ω(e3)=e3.
Remark 3.8*.*
There is no BiHom-Bialgabra whose underlying BiHom-associative algebra is given by H33,H53.
Theorem 3.9**.**
The set of 3-dimensional unital BiHom-Bialgebras yields two non-isomorphic algebras. Let {e1,e2,e3} be a basis of
K3, then the unital BiHom-Bialgebras are given by the following non-trivial comultiplications.
- 1.
Δ2,13(e1)=e1⊗e1,Δ(e2)=−e1⊗e1+e1⊗e2+e2⊗e1,Δ(e3)=e1⊗e1−e1⊗e2+2e1⊗e3−e2⊗e1+e2⊗e2−e2⊗e2+2e3⊗e1−e3⊗e2+e3⊗e3,ψ(e1)=e1,ψ(e2)=e1,ω(e1)=e1,ω(e2)=e1,ε(e1)=ε(e2)=1;
2. 2.
Δ3,13(e1)=e1⊗e1,Δ(e2)=e1⊗e1+e1⊗e2+e1⊗e3+e2⊗e1−e2⊗e2−e2⊗e3−e3⊗e1+e3⊗e2,Δ(e3)=−ae3⊗e3,ψ(e1)=e1,ψ(e2)=e2,ψ(e3)=e3,ω(e1)=e1,ω(e2)=e2,ω(e3)=e3,ε(e1)=ε(e2)=1;
3. 3.
Δ4,13(e1)=e1⊗e1,Δ(e2)=e2⊗e2,Δ(e3)=e1⊗e3+e3⊗e1+e3⊗e3,ψ(e1)=e1,ψ(e2)=e2,ψ(e3)=e3,ω(e1)=e1,ω(e2)=e2,ω(e3)=e3,ε(e1)=ε(e2)=1;
4. 4.
Δ12,13(e1)=e1⊗e1,Δ(e2)=e1⊗e2+e2⊗e1,Δ(e3)=e1⊗e3+e3⊗e1,ψ(e1)=e1,ω(e1)=e1,ε(e1)=1;
5. 5.
Δ15,13(e1)=e1⊗e1,Δ(e2)=e1⊗e1+e1⊗e2−e1⊗e3+e2⊗e1−e2⊗e3−e3⊗e1−e3⊗e2+2e3⊗e3,Δ(e3)=−e2⊗e2+e2⊗e3+e3⊗e2,ψ(e1)=e1,ψ(e3)=e1,ω(e1)=e1,ω(e3)=e1,ε(e1)=ε(e3)=1;
6. 6.
Δ15,23(e1)=e1⊗e1,Δ(e2)=e1⊗e2+e2⊗e1−ae2⊗e2−e2⊗e3−e3⊗e2,Δ(e3)=e1⊗e3+be2⊗e2−e3⊗e1−e3⊗e3,ψ(e1)=e1,ψ(e2)=ce2,ψ(e3)=de2+e3,ω(e1)=e1,ω(e2)=ce2,ω(e3)=de2+e3,ε(e1)=1;
7. 7.
Δ20,13(e1)=e1⊗e1,Δ(e2)=e2⊗e2,Δ(e3)=e1⊗e3+e3⊗e1+e3⊗e3,ψ(e1)=e1,,ψ(e2)=e2,ψ(e3)=e3,ω(e1)=e1,ω(e2)=e2,ω(e3)=e3,ε(e1)=ε(e2)=1;
8. 8.
Δ21,13(e1)=e1⊗e1,Δ(e2)=e1⊗e2+e2⊗e1+e2⊗e2,Δ(e3)=e3⊗e3,ψ(e1)=e1,ψ(e2)=e2,ω(e1)=e1,ω(e2)=e2,ε(e1)=1,ε(e2)=1;
9. 9.
Δ21,23(e1)=e1⊗e1,Δ(e2)=e1⊗e2+e2⊗e1,Δ(e3)=e1+⊗e3+e3⊗e1,ψ(e1)=e1,ω(e1)=e1,ε(e1)=1;
10. 10.
Δ22,13(e1)=e1⊗e1,Δ(e2)=e1⊗e2+e2⊗e1+e2⊗e2+e2⊗e3+e3⊗e2,Δ(e3)=e1+⊗e3+e3⊗e1+e3⊗e3,ψ(e1)=e1,ψ(e3)=e3,ω(e1)=e1,ω(e3)=e3,ε(e1)=1;
11. 11.
Δ22,23(e1)=e1⊗e1,Δ(e2)=e2⊗e2,Δ(e3)=e1+⊗e3+e3⊗e1+e3⊗e3,ψ(e1)=e1,ω(e1)=e1,ε(e1)=1;
12. 12.
Δ22,33(e1)=e1⊗e1,Δ(e2)=e2⊗e2,Δ(e3)=e1+⊗e3+e3⊗e1+e3⊗e3,ψ(e1)=e1,ψ(e3)=e3,ω(e1)=e1,ω(e3)=e3,ε(e1)=1;
13. 13.
Δ22,43(e1)=e1⊗e1,Δ(e2)=e2⊗e2,Δ(e3)=e1+⊗e3+e3⊗e1+e3⊗e3,ψ(e1)=e1,ψ(e2)=e2,ψ(e3)=e3,ω(e1)=e1,ω(e2)=e2,ω(e3)=e3,ε(e1)=ε(e2)=1.
Remark 3.10*.*
There is no unital BiHom-Bialgabra whose underlying unital BiHom-associative algebra is given by
Hu13,Hu53,Hu73,Hu83,Hu103.
We introduce the concept of BiHom-Hopf algebras.
Definition 3.11**.**
Let (H,μ,Δ,α,β) be a unital and counital BiHom-bialgebra with a unit 1H and a co-unit εH. 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 relation :
[TABLE]
A BiHom-Hopf algebra is a unital and counital BiHom-bialgebra with an antipode.
∑s=1n∑q,r=1n∑l,p=1n∑j,k=1nDijkSljgpkfqlarpbqrCsrt−ξi=0,∀i,t∈{1,n};
∑s=1n∑q,r=1n∑l,p=1n∑j,k=1nDijkfljSljbqlgrpasrCqrt−ξi=0,∀i,t∈{1,n}.
Proposition 3.12**.**
*The BiHom-bialgebra structures on K2 which are BiHom-Hopf algebras are given by the following pairs of multiplication and comultiplication
with the appropriate unit and conits :
(Hu12,Δ1,12),(Hu12,Δ1,22),(Hu12,Δ1,42),(Hu22,Δ2,12),(Hu22,Δ2,22),(Hu42,Δ4,12),(Hu42,Δ4,22),(Hu42,Δ4,32),(Hu42,Δ4,42).*
Proposition 3.13**.**
*The BiHom-bialgebra structures on K3 which are BiHom-Hopf algebras are given by the following pairs of multiplication and comultiplication
with the appropriate unit and conits :
(Hu33,Δ3,13),(Hu43,Δ4,13),(Hu123,Δ12,13),(Hu153,Δ15,13),(Hu153,Δ15,23),(Hu213,Δ21,13),(Hu213,Δ21,23),(Hu223,Δ22,13),
(Hu223,Δ22,23),
(Hu223,Δ22,33),(Hu223,Δ22,43).*
