# Hom-Yang-Baxter Equations and Frobenius Monoidal Hom-Algebras

Authors: Yuanyuan Chen; Liangyun Zhang
Journal: Advances in Mathematical Physics
 (2018)

---

## Abstract

It is shown that quasi-Frobenius Hom-Lie algebras are connected with a class of solutions of the classical Hom-Yang-Baxter equations. Moreover, a similar relation is discussed on Frobenius (symmetric) monoidal Hom-algebras and solutions of quantum Hom-Yang-Baxter equations. Monoidal Hom-Hopf algebras with Frobenius structures are studied at last.


---

## Body

## 1. Introduction

Hom-type algebras appeared first in physical contexts, in connection with twisted, discretized, or deformed derivatives and corresponding generalizations, discretizations, and deformations of vector fields and differential calculus. The notion of Hom-Lie algebras was introduced by Hartwig, Larsson, and Silvestrov in [1–3] as part of a study of deformations of Witt algebras and Virasoro algebras. Because of close relation to discrete and deformed vector fields and differential calculus, Hom-Lie algebras are widely studied recently; see [4–9].Hom-associative algebras generalize the concept of associative algebras. They were introduced by Makhlouf and Silvestrov in [10]. Hom-associative algebras and their related structures have recently become rather popular, due to the prospect of having a general framework in which one can produce many types of natural deformations of algebras. Among them, there are such structures as Hom-coassociative coalgebras, Hom-Hopf algebras, Hom-alternative algebras, Hom-Jordan algebras, Hom-Poisson algebras, Hom-Leibniz algebras, infinitesimal Hom-bialgebras, Hom-power associative algebras, and quasi-triangular Hom-bialgebras (see [11–16]). Furthermore, some categories of Hom-modules on Hom-Hopf algebras are studied, such as the category of Hom-Hopf modules and the category of Yetter-Drinfel’d Hom-modules in [17].The concept of Frobenius algebras is very important because of the connections to such diverse areas as group representations, homology of a compact oriented manifold, topological quantum field theories, quantum cohomology, Gorenstein rings in commutative algebras, Hopf algebras, coding theory, Lie quasi-Frobenius algebras, and the classical (quantum) Yang-Baxter equation (see [18–21]). In addition, there is a “quantum version” of the classical result that any finite dimensional Hopf algebra is Frobenius. The main properties of Frobenius algebras were developed by Nakayama in [22]. Nakayama automorphism is a distinguished k-algebra automorphism of a Frobenius algebra A which measures how far A is from being a symmetric algebra, where k is a fixed field. The automorphism is the identity if and only if A is symmetric. A connection between solutions of the classical Yang-Baxter equation and quasi-Frobenius Lie algebras was studied by Drinfel’d [23]. The authors in [19] gave the relation between Frobenius algebras and solutions of the quantum Yang-Baxter equation.Naturally, we would like to do some research in Frobenius Hom-algebras. Some preliminary results about monoidal Hom-algebras with Frobenius property have been studied, such as when finite dimensional monoidal Hom-Hopf algebras are Frobenius associated with integral spaces [16], the structures of separable and Frobenius monoidal Hom-algebras related to the Hom-Frobenius-separability equation, as well as the Nakayama automorphism of Frobenius monoidal Hom-algebras [24], and so on.The main purpose of this paper is to do further research in Frobenius Hom-algebras connected with solutions of the quantum Hom-Yang-Baxter equation. This paper is organized as follows. In Section3, we give some preliminary results about surgenerate bilinear forms and the element Q∈(M⊗M,μ⊗μ), where (M,μ) is an object in the Hom-category H~(Mk). In addition, a connection between solutions of the classical Hom-Yang-Baxter equation (CHYBE) and quasi-Frobenius Hom-Lie algebras is also discussed. Specifically, for Hom-Lie algebra (L,α), such an element Q∈L⊗L satisfies CHYBE if and only if there exists a surgenerate bilinear form on (L,α) such that it is quasi-Frobenius. In Section 4, a similar relationship between Frobenius (symmetric) monoidal Hom-algebras and the quantum Hom-Yang-Baxter equation (QHYBE) is discussed. For any Frobenius monoidal Hom-algebra, there exists a class of solutions of the QHYBE. And a converse result when monoidal Hom-algebras are Frobenius is also presented. It is known that a finite dimensional monoidal Hom-Hopf algebra (H,α) is Frobenius if and only if the space of integrals in H is not zero. Last, the properties of monoidal Hom-Hopf algebras with Frobenius structures are studied in Section 5.


## 2. Preliminaries

Throughout the paper, all vector spaces, tensor products, and homomorphisms are overk. The symbol τ denotes the transposition map.LetH~(Mk) be the Hom-category associated with the category Mk of k-modules (see [15] cited by [16]). Throughout, we use the conception of [15] for convenience.Definition 1.
Amonoidal Hom-algebra ([15, Proposition 2.1]) is a vector space A together with an element 1A∈A and linear maps (1)m:A⊗A→A;a⊗b↦ab,α∈AutkA such that(2)αabc=abαc,(3)αab=αaαb,(4)a1A=1Aa=αa,(5)α1A=1A,for all a,b,c∈A.The definition of monoidal Hom-algebras is the same to the unital monoidal Hom-associative algebras defined in [16, Definition 2.1]. A monoidal Hom-algebra is indeed a unital and associative algebra in the Hom-category H~(Mk). Let (A,α) be a monoidal Hom-algebra in the rest of this section.Right(A,α)-Hom-modules have been introduced in [16, Definition 2.5]. In the following, we give the definition of a left (A,α)-Hom-module similarly.Definition 2.
Aleft(A,α)-Hom-module consists of (M,μ) in H~(Mk) together with a morphism ψ:A⊗M→M,ψ(a⊗m)=a·m such that (6)αa·b·m=ab·μm,μa·m=αa·μm,1A·m=μm,for all a,b∈A and m∈M.Amorphism of left(A,α)-Hom-modules is a morphism of left A-modules in the Hom-category H~(Mk). H~(AM) will be denoted the category of left (A,α)-Hom-modules and left (A,α)-linear morphisms between them.If(M,μ) is both a left (A,α)-Hom-module and a right (A,α)-Hom-module such that the following compatibility condition(7)αa·m·b=a·m·αbholds, then (M,μ) is called an (A,α)-Hom-bimodule.For any object(M,μ)∈H~(Mk), let r:(M⊗M,μ⊗μ)→(M⊗M,μ⊗μ) be an morphism in H~(Mk). Consider the maps (8)r12,r13,r23:M⊗M⊗M,μ⊗μ⊗μ→M⊗M⊗M,μ⊗μ⊗μ given by r12=r⊗μ,r13=(id⊗τ)(r⊗μ)(id⊗τ),r23=r⊗μ. We say that r is asolution of the quantum Hom-Yang-Baxter equation (or the QHYBE for convenience) if(9)r12r13r23=r23r13r12.For elementsR=∑i=1nai⊗bi∈A⊗A, where (A,α) is a monoidal Hom-algebra with unit 1A. Then R acts on A⊗A via Hom-multiplication and is asolution of the quantum Hom-Yang-Baxter equation (or the QHYBE for convenience) if(10)α⊗3R12R13R23=R23R13α⊗3R12.where R12=∑i=1n‍ai⊗bi⊗1A,R13=∑i=1n‍ai⊗1A⊗bi,R23=∑i=1n‍1A⊗ai⊗bi. In what follows, we often omit the summation symbols for convenience.


## 3. Hom-Lie Algebras and Classical Hom-Yang-Baxter Equation

In this section, we give some preliminary results about surgenerate bilinear forms and a specific elementQ∈(M⊗M,μ⊗μ), where (M,μ) is an object in H~(Mk). In addition, an equivalent relation between solutions of the classical Hom-Yang-Baxter equation (CHYBE) and quasi-Frobenius Hom-Lie algebras is also discussed.Definition 3.
AHom-Lie algebra in [9] is a triple (L,[-,-],α) consisting of vector space L, bilinear map [-,-]:L⊗2→L (called the “bracket”), and a linear endomorphism α:L→L satisfying (11)αx,y=αx,αy,multiplicativityx,y+y,x=0,anti-symmetric↺x,y,αz=0,Hom-Jacobiidentity for any x,y,z∈L.Now let(L,[-,-],α) be a Hom-Lie algebra. Element R=∑i=1n‍ai⊗bi∈L⊗L is called a solution ofthe classical Hom-Yang-Baxter equation (or the CHYBE for convenience) if(12)∑i=1nai⊗bi=-∑i=1nbi⊗ai,and(13)R12,R13+R12,R23+R13,R23=0.Here (14)R12,R13=ai,aj⊗αbi⊗αbj,R12,R23=αai⊗bi,ak⊗αbk,R13,R23=αaj⊗αak⊗bj,bk, where R12=R⊗1=ai⊗bi⊗1,R13=(τ⊗id)(1⊗R)=aj⊗1⊗bj and R23=1⊗R=1⊗ak⊗bk.Let(M,μ)∈H~(Mk) be finite dimensional. A bilinear form B:(M⊗M,μ⊗μ)→(k,id) in H~(Mk) is calledsurgenerate if there exists an isomorphism ϕ:(M,μ)→(M∗,(μ∗)-1) in H~(Mk) given by (15)ϕxy=By,x. Here B is a morphism in H~(Mk) which means that B(μ(x),μ(y))=B(x,y), for all x,y∈M. And this makes sure that ϕ is also a morphism in H~(Mk).A Hom-Lie algebra(L,[-,-],α) is said to bequasi-Frobenius if L is finitely generated projective k-module having a skew-symmetric surgenerate bilinear form B:L⊗L→k satisfying the following equation:(16)Bx,y,αz+By,z,αx+Bz,x,αy=0,for all x,y,z∈L.Lemma 4.
Let(M,μ)∈H~(Mk) be finite dimensional. Define a homomorphism ρM:(M⊗M,μ⊗μ)→(Hom(M∗,M),ι) by ρM(a⊗b)(f)=f(a)b, for all a,b∈M,f∈M∗, where ι:Hom(M∗,M)→Hom(M∗,M) is the twisting given by ι(ψ)=μψμ∗. Then ρM is an isomorphism in H~(Mk).Proof.
Firstρm is well defined in H~(Mk), since ρM(μ(a)⊗μ(b))(f)=fμ(a)μ(b)=μ(ρM(a⊗b)μ∗(f)). Define a morphism λM:Hom(M∗,M)→M⊗M by ψ↦ei⊗ψ(fi), where (ei,fi) is a dual basis for M. We can easily check that λM is the inverse of ρM. Indeed, for any a,b∈M,f∈M∗, (17)λMρMa⊗b=ei⊗ρMa⊗bfi=ei⊗fiab=a⊗b, and (18)ρMλMψf=ρMei⊗ψfif=feiψfi=ψf.The endomorphism algebraEndk(M,μ) can be considered as a monoidal Hom-algebra, where the Hom-multiplication is the composite of morphisms, the unit is the identity homomorphism, and the twisting map ι:Endk(M,μ)→Endk(M,μ) is given by ι(φ)=μφμ-1, for φ∈Endk(M,μ).Proposition 5.
Let(M,μ)∈H~(Mk) be finite dimensional with a surgenerate bilinear form B:(M⊗M,μ⊗μ)→(k,id) in H~(Mk). Further let ei∈M,fi∈M∗,i=1,2,3,…,n, be a dual basis of M, and ei=ϕ-1(fi),i=1,2,3,…,n. We set Q=ei⊗ei∈M⊗M, and define a homomorphism in H~(Mk)(19)χM:M⊗M,μ⊗μ→EndkM,μ,ι by (20)χMa⊗bx=ϕb,μ-1xμa=Bx,μbμa. Further we define the mapping Tr:Endk(M,μ)→k as follows: (21)Trφ=fi,φei=ϕei,φei=Bφei,ei, for any φ∈Endk(M,μ).
Then(1)
x=B(x,ei)ei=B(ei,x)ei for all x∈M.(2)
χM is an isomorphism in H~(Mk).(3)
Q=ρM-1(ϕ-1)=χM-1(idM).(4)
The elementQ does not depend on the choice of the dual bases ei,ei,i=1,2,…,n, of M.(5)
The mappingTr does not depend on the choice of the dual bases ei,ei,i=1,2,…,n, of M.Proof.
(1) Sincefi(x)=ϕ(ei)(x)=B(x,ei), x=fi(x)ei=B(x,ei)ei, for all x∈M. Given x∈M, set y=B(ei,x)ei. Then (22)Bz,y=Bz,Bei,xei=Bei,xBz,ei=BBz,eiei,x=Bz,x, for all z∈M. Since B is surgenerate, y=x.
(2) Define a homomorphismς:(M⊗M∗,μ⊗(μ∗)-1)→(Endk(M,μ),ι) in H~(Mk) by (23)ςa⊗fx=fμ-1xμa,a,x∈M,f∈M∗. It is not difficult to show that ς is an isomorphism with an inverse ς-1:f↦f(ei)⊗fi. Since χM=ς(id⊗ϕ), χM is an isomorphism in H~(Mk).
(3) For allk=1,2,…,n, we have (24)ρMQfk=fkeiei=Bei,ekei=Bei,ekei=ek by (1). On the other hand, ϕ-1(fk)=ek, for all k=1,2,…,n, so ρM(Q)=ϕ-1, and hence Q=ρM-1(ϕ-1). Further, χM(Q)(x)=B(μ-1(x),ei)μ(ei)=x for all x∈M by (1). Therefore χM(Q)=idM, which implies Q=χM-1(idM).
(4) follows directly from (3).
(5) Forφ∈End(M,μ), define the mapping Λ:(M⊗M,μ⊗μ)→(k,idk) by Λ(a⊗b)=B(φ(a),b) in H~(Mk). Obviously, Tr(φ)=Λ(Q)=B(φ(ei),ei). It follows from (4) that Tr(φ) does not depend on the choice of the dual bases ei,ei,i=1,2,…,n, of M.Lemma 6.
Let(M,μ)∈H~(Mk) be finite dimensional with a surgenerate bilinear form B:(M⊗M,μ⊗μ)→(k,id) in H~(Mk). Further let ei∈M,fi∈M∗,i=1,2,3,…,n be a dual basis of M. Choose e1,e2,…,en∈M such that B(ei,ej)=δij for all i,j=1,2,3,…,n. We set Q=ei⊗ei∈M⊗M, and write ei=∑j=1n‍ljiej, where lji∈k. Next we set nij=ϕ(ej)(ei), N=(nij)i,j=1n∈Mn(k), and L=(lji)i,j=1n.
Then(1)
LN=E=NL, where E is the identity of Mn(k);(2)
Q=∑i=1n‍ljiei⊗ej.Proof.
Sinceδji=B(ej,ei)=B(ej,∑k=1n‍lkiek)=∑k=1n‍B(ej,ek)lki=∑k=1n‍bjkrki, the first assertion is true.
The second statement is obvious.Theorem 7.
Let(L,[-,-],α) be a Hom-Lie algebra with a skew-symmetric surgenerate bilinear form B:L⊗L→k. Suppose that L is finite dimensional and {ei,ei},i=1,2,…,n is a dual basis of L. We set Q=ei⊗ei∈L⊗L. Then(1)
Q=ρL-1(M)(ϕ-1)=χL-1(idL).(2)
The elementQ does not depend on the choice of dual basis of L.(3)
The elementQ satisfies CHYBE if and only if B satisfies (16).Proof.
The first two statements follow from (3) and (4) of Proposition5.
Next, ifQ satisfies the CHYBE, then by (12) we have ej⊗ej=τ(ej⊗ej)=-ej⊗ej. Choose wi∈L such that B(ej,wi)=δij. Then from (4) of Proposition 5, (25)ei⊗wi=ei⊗ei=-ei⊗ei, so wi=-ei. Therefore we have (26)Bei,ej=Bej,wi=Bej,-ei=-Bej,ei, which proves that B is skew-symmetric.
Last, we only need to show thatQ satisfies the equation [Q12,Q13]+[Q12,Q23]+[Q13,Q23]=0 if and only if B satisfies (16). Since ei⊗ei=-ei⊗ei, we have (27)Q12,Q13=ei,ej⊗αei⊗αej=Bei,ej,ekek⊗αei⊗αej=Bα-1ei,ej,ekαekαei⊗αejQ12,Q23=αek⊗ek,ej⊗αej=αek⊗Bei,ek,ejei⊗αej=Bei,ek,ejαek⊗ei⊗αej=Bej,ek,eiαek⊗ei⊗αej=Bα-1ej,ek,eiαek⊗αei⊗αejand (28)Q13,Q23=-αek⊗αei⊗ek,ei=-αek⊗αei⊗Bej,ek,eiej=-Bej,α-1ek,eiαek⊗αei⊗αej=Bα-1ek,ei,ejαek⊗αei⊗αej.
Then(29)Q12,Q13+Q12,Q23+Q13,Q23=Bα-1ei,ej,ek+Bα-1ej,ek,ei+Bα-1ek,ei,ejαek⊗αei⊗αej.So (30)Q12,Q13+Q12,Q23+Q13,Q23=0 if and only if (31)Bα-1ei,ej,ek+Bα-1ej,ek,ei+Bα-1ek,ei,ej=0which is equivalent to (32)Bei,ej,αek+Bej,ek,αei+Bek,ei,αej=0.Note that if(L,[-,-],α) is a quasi-Frobenius Hom-Lie algebra, then Lemma 6 implies that rijei⊗ej=ej⊗ej=-Q is skew-symmetric.


## 4. Frobenius Monoidal Hom-Algebras and Quantum Hom-Yang-Baxter Equations

A relationship between Frobenius (symmetric) monoidal Hom-algebras and QHYBE is mainly discussed in this section.For a finite dimensional monoidal Hom-algebra(A,α), we can check that (A∗,(α∗)-1) is a right (A,α)-Hom-module via the action ↼ given by (33)θ↼a=θα-2αa-=θ1α-1aα∗-2θ2, where Δ(θ)=θ1⊗θ2 and Δ is the dual Hom-coalgebra structure on (A∗,(α∗)-1), for all a∈A,θ∈A∗.Definition 8 (see [16, Definition 5.1]).
A finite dimensional monoidal Hom-algebra(A,α) is calledFrobenius if (34)ϕ:A,α→A∗,α∗-1is an isomorphism as right (A,α)-Hom-modules.Recall from [16, Proposition 5.2] that a finite dimensional monoidal Hom-algebra being Frobenius has several equivalent characterization as follows.Lemma 9.
For a finite dimensional monoidal Hom-algebra(A,α), the following assertions are equivalent.(1)
(A,α) is Frobenius.(2)
(A,α)≅(A∗,(α∗)-1) as left (A,α)-Hom-modules, which is called Frobenius isomorphism.(3)
There exists a Frobenius structure of(A,α). Equivalently, there exist elements φ∈A∗ (called Frobenius homomorphism), and ai⊗bi∈A⊗A such that φα=φ,α(ai)⊗α(bi)=ai⊗bi and φ(xai)bi=aiφ(bix)=x, for any x∈A.(4)
There exists a Hom-associative, nondegenerate bilinear form for(A,α). That is, there exists a bilinear map B:A×A→k in H~(Mk), such that B(α(a),bc)=B(ab,α(c)), and if B(x,a)=0 or B(a,x)=0 for any x∈A, then a=0.Let(A,α) be a finite dimensional monoidal Hom-algebra. And further we assume that (A,α) is Frobenius with Frobenius isomorphism ϕ:(A,α)→(A∗,(α∗)-1), Frobenius structure φ∈A∗ and ai⊗bi∈A⊗A, and the corresponding bilinear form B:A⊗A→k. By the proof of the above lemma, we know that (35)Bx,y=φxy=ϕyx, for all x,y∈A. Then there exists a dual basis e1,e2,…,en∈A and f1,f2,…,fn∈A∗ such that x=fi(x)ei=∑i=1n‍fi,xei. Setting ei=ϕ-1(fi),i=1,2,…,n, we note that fi(x)=ϕ(ei)(x)=φ(xei) for all x∈A,i=1,2,…,n. So x=φ(xei)ei for all x∈A. Thus we can also refer to e1,e2,…,en and e1,e2,…,en as dual basis of A.Note that there is an automorphismσ:(A,α)→(A,α) calledthe Nakayama automorphism of(A,α) defined in [16, Section 5]. It is easy to check that σ is a Hom-algebra automorphism satisfying B(x,y)=B(y,σ(x)), that is, φ(xy)=φ(yσ(x)), for all x,y∈A.If a Frobenius monoidal Hom-algebra(A,α) has an augmentation ε, then the element N in A is calleda left norm of(A,α) if φ(aN)=ε(a), for any a∈A.Given a monoidal Hom-algebra(A,α), we denote (36)QA=Q∈A⊗A∣1⊗aQ=Qa⊗1,∀a∈A=Q=ai⊗bi∈A⊗A∣αai⊗abi=aia⊗αbi,∀a∈A. We consider (A⊗A,α⊗α) as an (A,α)-Hom-bimodule under the actions given by(37)a·x⊗y=α-1ax⊗αyand(38)x⊗y·b=αx⊗yα-1b,for all x,y,a,b∈A.Lemma 10.
Let(A,α) be a monoidal Hom-algebra and Q=ai⊗bi∈Q(A). Then(1)
Q12Q13=Q23Q12=Q13Q23;(2)
α⊗3(Q12)(Q23Q12)=(Q23Q12)α⊗3(Q23).Proof.
(1)(39)Q12Q13=aiaj⊗αbi⊗αbj=αai⊗ajbi⊗αbj=Q23Q12=αai⊗αaj⊗bibj=Q13Q23.
(2) From (1), we have(40)α⊗3Q12Q23Q12=α⊗3Q12Q13Q23=Q12Q13α⊗3Q23=Q23Q12α⊗3Q23.Theorem 11.
Let(A,α) be a Frobenius monoidal Hom-algebra with Nakayama automorphism σ and let ei,ei,i=1,2,…,n, be dual bases of A. We set Q=ei⊗ei∈A⊗A. Then(1)
Q=ρA-1(ϕ-1)=χA-1(idA).(2)
The elementQ does not depend on the choice of the dual bases of ei,ei of A.(3)
(a⊗b)Q=Q(b⊗σ(a)), for all a,b∈A.(4)
α(x)((eiz)ei)=((eiz)ei)σ(x).(5)
Q12Q13=Q23Q12=Q13Q23.(6)
The elementQ satisfies the braid relation (41)α⊗3Q12Q23Q12=Q23Q12α⊗3Q23.Proof.
The first two statements follow directly from (3) and (4) of Proposition5.
(3) According to Proposition5 (1), for any a,b∈A, we have (42)a⊗bQ=aei⊗bei=aei⊗Bej,beiej=aei⊗φejbeiej=aφα-1ejbαeiei⊗ej=aφα-2ejα-1beiei⊗ej=aα-2ejα-1b⊗ej=α-1aα-1ejb⊗ej=φα-1aα-1ejeieib⊗ej=eib⊗φaα-1α-1ejeiej=eib⊗φα-1α-1ejeiσaej=eib⊗φα-1ejα-1eiσaej=eib⊗φejeiσaej=eib⊗eiσa=Qb⊗σa, which proves (3).
(4) For any givenz∈A, define a map λz:A⊗A→A given by λz(a⊗b)=(az)b. λz is a homomorphism of (A,α)-Hom-bimodules, where the actions on A⊗A are given by (37) and (38). Then (3) implies that (43)α2xeiα-1zei=αxeiα-1zαei=xeizαei=λzxei⊗αei=λzx⊗1ei⊗ei=λzei⊗ei1⊗σx=λzαei⊗eiσx=αeizeiσx=eiα-1zeiσαx, so α(x)((eiz)ei)=((eiz)ei)σ(x), for any x,z∈A.
It is obvious thatQ∈Q(A) by setting a=1 in the equality of (3). The statements (5) and (6) follow from Lemma 10.Theorem 12.
Let(A,α) be a finite dimensional monoidal Hom-algebra with two bases e1,e2,…,en and e1,e2,…,en, and Q=ei⊗ei∈A⊗A. Then the following conditions are equivalent:(1)
Q12Q13=Q23Q12;(2)
Q12Q13=Q13Q23;(3)
Q23Q12=Q13Q23;(4)
(1⊗a)Q=Q(a⊗1);(5)
A is a Frobenius algebra with Frobenius homomorphism φ:A→k such that φ(eiej)=δij, for all i,j=1,2,…,n.Proof.
From Lemma10, it is enough to show that one of the conditions (1), (2), and (3) implies (4). We may as well show that (3) implies (4), and the others can be proved analogously. From (3), it follows that (44)Q23Q12=αei⊗ejei⊗αej=αei⊗αej⊗eiej=Q13Q23 and so(45)ejei⊗αej=αej⊗eiej,for all i=1,2,…,n, which proves (4).
Due to Theorem11, it is enough to prove that (4) implies (5). Define a bilinear map B:(A⊗A,α⊗α)→(k,id) in H~(Mk) as follows: B(ei,ej)=B(α(ei),α(ej))=δij, for all i,j=1,2,…,n. Since e1,e2,…,en and e1,e2,…,en are two bases of A, it follows that B is a well-defined surgenerate bilinear form. And we claim that B is Hom-associative. Indeed, it is enough to show that B(ejx,α(ei))=B(α(ej),xei), for all i,j=1,2,…,n. Since for any a∈A, a=B(ei,a)ei=B(a,ei)ei, then equality (45) implies that (46)αei⊗xei=ejx⊗αej and hence (47)αei⊗Bej,α-1xeiαej=Bα-1ejx,eiαei⊗αej, which implies that (48)Bej,α-1xei=Bα-1ejx,ei.Thus B(α(ej),xei)=B(ejx,α(ei)), for all i,j=1,2,…,n and x∈A, by the α-invariance of B. Setting φ(x)=B(1,α-1(x)), we conclude that φ is a Frobenius homomorphism and (49)φeiej=B1,α-1eiej=Bei,ej=δij, for all i,j=1,2,…,n.A Frobenius monoidal Hom-algebra with Frobenius homomorphismφ:A→k is calledsymmetric if φ(xy)=φ(yx), for all x,y∈A. In this case, the Nakayama automorphism σ is the identity map of A.Lemma 13.
Let(A,α) be a monoidal Hom-algebra with two bases e1,e2,…,en and e1,e2,…,en, and Q=ei⊗ei∈A⊗A. Suppose that (a⊗b)Q=Q(b⊗a). Then(1)
Q12Q13=Q23Q12=Q13Q23 and Q12Q23=Q23Q13=Q13Q12;(2)
α⊗3(Q12)(Q13Q23)=(Q23Q13)α⊗3(Q12).Proof.
The first assertion of (1) is proved in Theorem12, and the second one can be showed similarly.
(2) From (1), we have(50)α⊗3Q12Q13Q23=Q12Q13α⊗3Q23=Q23Q12α⊗3Q23=α⊗3Q23Q12Q23=α⊗3Q23Q13Q12=Q23Q13α⊗3Q12.Theorem 14.
Let(A,α) be a symmetric monoidal Hom-algebra with Frobenius homomorphism φ:A→k, and {ei},{ej},i,j=1,2,…,n, be dual bases of A. We set Q=ei⊗ei∈A⊗A. Then(1)
(a⊗b)Q=Q(b⊗a), for any a,b∈A;(2)
Q12Q13=Q23Q12=Q13Q23 and Q12Q23=Q23Q13=Q13Q12;(3)
the elementQ is a solution of QHYBE, and τ(Q)=Q.Proof.
Since(A,α) is symmetric, the corresponding Nakayama automorphism is the identity. It follows from Theorem 11 (3) that (a⊗b)Q=Q(b⊗a), for all a,b∈A. The last two statements are true from Lemma 13.Next we consider the converse of the above theorem.Theorem 15.
Let(A,α) be a finite dimensional monoidal Hom-algebra with two bases e1,e2,…,en and e1,e2,…,en. Further let Q=ei⊗ei∈A⊗A. Then the following conditions are equivalent:(1)
Q12Q13=Q23Q12 and Q12Q23=Q23Q13;(2)
Q12Q13=Q13Q23 and Q12Q23=Q13Q12;(3)
Q23Q12=Q13Q23 and Q23Q13=Q13Q12;(4)
(a⊗b)Q=Q(b⊗a), for all a∈A;(5)
A is a symmetric monoidal Hom-algebra with Frobenius homomorphism φ:A→k such that φ(eiej)=δij, for all i,j=1,2,…,n.Proof.
We have proved that (4) implies the first three equalities in Lemma13. And each of the first three conditions implies the fourth one in the same way as it was done in Theorem 12. Taking into account Theorem 14, we have only to prove that (4) implies (5). Set B:(A⊗A,α⊗α)→(k,id) in H~(Mk) as follows: B(ei,ej)=B(α(ei),α(ej))=δij, for all i,j=1,2,…,n. We have seen that B is a Hom-associative surgenerate bilinear form by Theorem 12, and φ(x)=B(1,α-1(x)) is a Frobenius homomorphism. To show φ(xy)=φ(yx), we only need to show B(1,xy)=B(1,yx), for x,y∈A, which is equivalent to prove that B(x,y)=B(y,x), for x,y∈A. From (4) we see that aei⊗α(ei)=α(ej)⊗eja, for all a∈A, so (51)Baei,ejej⊗αei=αej⊗Bei,ejaeiby Proposition 5 (1). Hence (52)Bα-1aei,ejαej⊗αei=αej⊗Bei,α-1ejaαei, which implies (53)Baei,αej=Bαei,eja, for all i,j=1,2,…,n,a∈A. Thus B(ax,α(y))=B(α(x),ya), for all x,y∈A. Taking x=1A, we get (54)Bαa,αy=B1,ya=Bαy,αa, so B(a,y)=B(y,a), for all a,y∈A.


## 5. Frobenius Monoidal Hom-Hopf Algebras

It is known that a finite dimensional monoidal Hom-Hopf algebra(H,α) is Frobenius if and only if the space of integrals in H is not zero. In this section, we do further research on monoidal Hom-Hopf algebras with Frobenius structures.Definition 16 (see [16, Definition 2.2]).
Amonoidal Hom-coalgebra is an object (C,γ) in the category H~(Mk) together with linear maps Δ:C→C⊗C,Δ(c)=c1⊗c2 and ε:C→k such that(55)γ-1c1⊗Δc2=Δc1⊗γ-1c2,c1εc2=γ-1c=εc1c2,(56)Δγc=γc1⊗γc2,εγc=εc,for all c∈C.Definition 17.
Amonoidal Hom-bialgebra [16, Definition 2.3] H=(H,α,m,η,Δ,ε) is a bialgebra in the category H~(Mk). This means that (H,α,m,η) is a monoidal Hom-algebra and (H,α,Δ,ε) is a monoidal Hom-coalgebra such that Δ and ε are Hom-algebra maps; that is, for any h,g∈H, (57)Δhg=ΔhΔg,Δ1H=1H⊗1H,εhg=εhεg,ε1H=1k.Definition 18 (see [16, Definition 2.5]).
A monoidal Hom-bialgebra(H,α) is called amonoidal Hom-Hopf algebra if there exists a morphism (called antipode) S:H→H in H~(Mk) (i.e., S∘α=α∘S), such that, for any h∈H,(58)Sh1h2=εh1H=h1Sh2.
In fact, monoidal Hom-coalgebras, monoidal Hom-bialgebras, and monoidal Hom-Hopf algebras are coalgebras, bialgebras, and Hopf algebras in the categoryH~(Mk), respectively. Further, the antipodes of monoidal Hom-Hopf algebras have similar properties (see [16]) of those of Hopf algebras. The self-duality of finite dimensional monoidal Hom-Hopf algebras also holds from [16], and the dual structure of monoidal Hom-Hopf algebra (H,α,m,η,Δ,ε,S) is given as follows: for all g,h∈H,ϕ,φ∈H∗, (59)ϕφ,h=ϕ,h1φ,h2,1H∗=ε;Δϕ,g⊗h=ϕ,gh,εH∗=η;α∗-1ϕ=ϕ∘α-1,S∗ϕ=ϕ∘S-1.Definition 19 (see [16, Definition 3.1]).
Aleft integral in (H,α) is an α-invariant element t∈H (i.e., α(t)=t) such that (60)ht=εht, for all h∈H.Lemma 20.
Let(H,α) be a monoidal Hom-Hopf algebra with antipode S. Further let φ be an α∗-invariant element in H∗. Then for all h∈H, (61)h1φh2=φh1H if and only if φ is a left integral in H∗.Proof.
Letψ∈H∗. Since for any h∈H, (62)εH∗ψφh=φhψ1H=ψφh1H,ψφh=ψh1φh2=ψh1φh2, we have h1φ(h2)=φ(h)1H if and only if ψφ=εH∗(ψ)φ, which means that φ is a left integral in H∗.Lemma 21.
Let(H,α) be a monoidal Hom-bialgebra and Frobenius with Frobenius homomorphism φ:(H,α)→k such that h1φ(h2)=φ(h)1H and with left norm N, for all h∈H. Then (H,α) is a monoidal Hom-Hopf algebra.Proof.
We defineS:(H,α)→(H,α) in H~(Mk) by (63)Sh=αN1φα-1hN2, for all h∈H. Then by the properties of φ, we see that (64)h1Sh2=h1αN1φα-1h2N2=h1αN1φh2αN2=φhαN1H=φα-1hN1H=εα-1h1H=ηεh, for all h∈H.
It is not difficult to check that the endomorphismEndk(H,α) in H~(Mk) forms a monoidal Hom-algebra with Hom-multiplication convolution ∗, twisting map id and unit ηε. Indeed, for all h∈H,f∈Endk(H,α)(65)f∗ηεh=fh1ηεh2=fh. And the Hom-associativity is true by the Hom-coassociativity of (H,α). The map (66)-∗S:EndkH,α→EndkH,α;f↦f∗S is epimorphism, since f=f∗ηε=f∗id∗S=f∗id∗S. By Lemma 4 in [25], -∗S is an isomorphism with inverse map -∗id. So S∗id=(ηε∗S)∗id=ηε. Thus S is an antipode of (H,α).Theorem 22.
Let(H,α) be a monoidal Hom-Hopf algebra with antipode S and also Frobenius with Frobenius homomorphism φ:(H,α)→k such that (67)h1φh2=φh1H,for all h∈H and with the Nakayama automorphism σ. Further let {ei},{ei},i=1,2,…,n, be dual bases of H. Then(1)
N=ε(ei)ei=S(ei)ε(ei) is a left norm and a left integral in H. Further Δ(N)=Sα-1(ei)⊗α-1(ei), and ε(N)1H=eiei=S2(ei)ei=ε(ei)ε(ei)1H;(2)
φ generates the Hom-submodule of left integral in H∗ and (68)TrS2=εNεH∗φ.Proof.
(1) SettingN=ε(ei)ei, we have ε(N)=ε(ei)ε(ei) and (69)φN=εeiφei=εφeiei=εφ1Heiei=ε1H=1 by the α-invariance of φ and Proposition 5 (1). Further, for all x∈H, (70)xN=εeixei=εeiφejxeiej=εφejxeieiej=εφα-1ejxαeieiej=εφα-1α-1ejxeieiej=εα-1α-1ejxej=εα-1ejxej=εejεxej=εxN. Since {α(ei)},{α-1(ei)},i=1,2,…,n, can also be considered as dual bases of H, hence we see that (71)α-1N=εeiα-1ei=εαeiα-1ei=εeiei=N,which implies that N is α-invariant. Thus N is a left integral in H by Definition 19.
Next,φ(xN)=φ(ε(x)N)=ε(x)φ(N)=ε(x), so N is a left norm.
From Lemma21, we know that the antipode S:(H,α)→(H,α) is given by (72)Sx=αN1φα-1xN2. And S is bijective when H if finite dimensional by Theorem 5.3 of [16]. So {Sα-1(ei)} forms a basis of H. Therefore Δ(N)=Sα-1(ei)⊗ai for some ai∈H. Then we rewrite the expression of S as (73)Sx=Seiφα-1xai, for all x∈H. Applying S-1 we get (74)x=eiφα-1xai, for all x∈H. On the other hand x=eiφ(xei). So φ(xα(ai))=φ(α-1(x)ai)=φ(xei). It follows that α(ai)=ei, so ai=α-1(ei), and hence (75)ΔN=Sα-1ei⊗α-1ei. By the counitary property of (H,α) we have N=αSα-1eiεα-1(ei)=S(ei)ε(ei). And N1S(N2)=Sα-1(ei)Sα-1(ei)=ε(N)1H, so α-1(eiei)=ε(N)S-1(1H)=ε(N)1H by applying S-1. Therefore eiei=ε(N)1H. Similarly, S2α-1(ei)α-1(ei)=ε(N)1H, which implies that S2(ei)ei=ε(N)1H. According to the above, so we obtain (76)eiei=S2eiei=εN1H=εeiεei1H.
(2) From Proposition 3.4 of [16], we know that the integral in finite dimensional monoidal Hom-Hopf algebras does not necessarily exist. But if it exists, it is unique up to a scalar multiplication. And φ is a left integral in H∗ by Lemma 21, so φ generates the Hom-submodule of left integral in H∗. Since ε(N)1H=S2(ei)ei by (1), we see that (77)εNφ1H=φεN1H=φS2eiei=TrS2 by applying φ and the definition of Tr in Proposition 5. So Tr(S2)=ε(N)φ(1H)=ε(N)εH∗(φ).



---

*Source: 2912578-2018-08-01.xml*