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

Chen, Yuanyuan; Zhang, Liangyun

doi:https://doi.org/10.1155/2018/2912578

Advances in Mathematical Physics

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Research Article | Open Access

Volume 2018 | Article ID 2912578 | https://doi.org/10.1155/2018/2912578

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

Yuanyuan Chen¹and Liangyun Zhang¹

Academic Editor: Andrei D. Mironov

Received30 Apr 2018

Accepted03 Jul 2018

Published01 Aug 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.

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 -algebra automorphism of a Frobenius algebra which measures how far is from being a symmetric algebra, where is a fixed field. The automorphism is the identity if and only if 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 Section 3, we give some preliminary results about surgenerate bilinear forms and the element , where is an object in the Hom-category . 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 , such an element satisfies CHYBE if and only if there exists a surgenerate bilinear form on 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 is Frobenius if and only if the space of integrals in 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 over . The symbol denotes the transposition map.

Let be the Hom-category associated with the category of -modules (see [15] cited by [16]). Throughout, we use the conception of [15] for convenience.

Definition 1. A monoidal Hom-algebra ([15, Proposition 2.1]) is a vector space together with an element and linear maps such thatfor all .

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 . Let be a monoidal Hom-algebra in the rest of this section.

Right -Hom-modules have been introduced in [16, Definition 2.5]. In the following, we give the definition of a left -Hom-module similarly.

Definition 2. A left -Hom-module consists of in together with a morphism such that for all and .

A morphism of left -Hom-modules is a morphism of left -modules in the Hom-category . will be denoted the category of left -Hom-modules and left -linear morphisms between them.

If is both a left -Hom-module and a right -Hom-module such that the following compatibility conditionholds, then is called an -Hom-bimodule.

For any object , let be an morphism in . Consider the maps given by . We say that is a solution of the quantum Hom-Yang-Baxter equation (or the QHYBE for convenience) if

For elements , where is a monoidal Hom-algebra with unit . Then acts on via Hom-multiplication and is a solution of the quantum Hom-Yang-Baxter equation (or the QHYBE for convenience) ifwhere 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 element , where is an object in . 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. A Hom-Lie algebra in [9] is a triple consisting of vector space , bilinear map (called the “bracket”), and a linear endomorphism satisfying for any .

Now let be a Hom-Lie algebra. Element is called a solution of the classical Hom-Yang-Baxter equation (or the CHYBE for convenience) ifandHere where and .

Let be finite dimensional. A bilinear form in is called surgenerate if there exists an isomorphism in given by Here is a morphism in which means that , for all . And this makes sure that is also a morphism in .

A Hom-Lie algebra is said to be quasi-Frobenius if is finitely generated projective -module having a skew-symmetric surgenerate bilinear form satisfying the following equation:for all .

Lemma 4. Let be finite dimensional. Define a homomorphism by , for all , where is the twisting given by . Then is an isomorphism in .

Proof. First is well defined in , since . Define a morphism by , where is a dual basis for . We can easily check that is the inverse of . Indeed, for any , and

The endomorphism algebra 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 is given by , for .

Proposition 5. Let be finite dimensional with a surgenerate bilinear form in . Further let , be a dual basis of , and . We set , and define a homomorphism in by Further we define the mapping as follows: for any .
Then(1) for all .(2) is an isomorphism in .(3).(4)The element does not depend on the choice of the dual bases , of .(5)The mapping does not depend on the choice of the dual bases , of .

Proof. (1) Since , , for all . Given , set . Then for all . Since is surgenerate, .
(2) Define a homomorphism in by It is not difficult to show that is an isomorphism with an inverse . Since , is an isomorphism in .
(3) For all , we have by (1). On the other hand, , for all , so , and hence . Further, for all by (1). Therefore , which implies .
(4) follows directly from (3).
(5) For , define the mapping by in . Obviously, . It follows from (4) that does not depend on the choice of the dual bases , of .

Lemma 6. Let be finite dimensional with a surgenerate bilinear form in . Further let be a dual basis of . Choose such that for all . We set , and write , where . Next we set , , and .
Then(1), where is the identity of ;(2).

Proof. Since , the first assertion is true.
The second statement is obvious.

Theorem 7. Let be a Hom-Lie algebra with a skew-symmetric surgenerate bilinear form . Suppose that is finite dimensional and is a dual basis of . We set . Then(1).(2)The element does not depend on the choice of dual basis of .(3)The element satisfies CHYBE if and only if satisfies (16).

Proof. The first two statements follow from (3) and (4) of Proposition 5.
Next, if satisfies the CHYBE, then by (12) we have . Choose such that . Then from (4) of Proposition 5, so . Therefore we have which proves that is skew-symmetric.
Last, we only need to show that satisfies the equation if and only if satisfies (16). Since , we have and Then So if and only if which is equivalent to

Note that if is a quasi-Frobenius Hom-Lie algebra, then Lemma 6 implies that 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 , we can check that is a right -Hom-module via the action given by where and is the dual Hom-coalgebra structure on , for all .

Definition 8 (see [16, Definition 5.1]). A finite dimensional monoidal Hom-algebra is called Frobenius if is an isomorphism as right -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 , the following assertions are equivalent.(1) is Frobenius.(2) as left -Hom-modules, which is called Frobenius isomorphism.(3)There exists a Frobenius structure of . Equivalently, there exist elements (called Frobenius homomorphism), and such that and , for any .(4)There exists a Hom-associative, nondegenerate bilinear form for . That is, there exists a bilinear map in , such that , and if or for any , then .

Let be a finite dimensional monoidal Hom-algebra. And further we assume that is Frobenius with Frobenius isomorphism , Frobenius structure and , and the corresponding bilinear form . By the proof of the above lemma, we know that for all . Then there exists a dual basis and such that . Setting , we note that for all So for all . Thus we can also refer to and as dual basis of .

Note that there is an automorphism called the Nakayama automorphism of defined in [16, Section 5]. It is easy to check that is a Hom-algebra automorphism satisfying , that is, , for all .

If a Frobenius monoidal Hom-algebra has an augmentation , then the element in is called a left norm of if , for any .

Given a monoidal Hom-algebra , we denote We consider as an -Hom-bimodule under the actions given byandfor all .

Lemma 10. Let be a monoidal Hom-algebra and . Then(1);(2).

Proof. (1) (2) From (1), we have

Theorem 11. Let be a Frobenius monoidal Hom-algebra with Nakayama automorphism and let , be dual bases of . We set . Then(1).(2)The element does not depend on the choice of the dual bases of of .(3), for all .(4).(5).(6)The element satisfies the braid relation

Proof. The first two statements follow directly from (3) and (4) of Proposition 5.
(3) According to Proposition 5 (1), for any , we have which proves (3).
(4) For any given , define a map given by . is a homomorphism of -Hom-bimodules, where the actions on are given by (37) and (38). Then (3) implies that so , for any .
It is obvious that by setting in the equality of (3). The statements (5) and (6) follow from Lemma 10.

Theorem 12. Let be a finite dimensional monoidal Hom-algebra with two bases and , and . Then the following conditions are equivalent:(1);(2);(3);(4);(5) is a Frobenius algebra with Frobenius homomorphism such that , for all .

Proof. From Lemma 10, 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 and sofor all , which proves (4).
Due to Theorem 11, it is enough to prove that (4) implies (5). Define a bilinear map in as follows: , for all Since and are two bases of , it follows that is a well-defined surgenerate bilinear form. And we claim that is Hom-associative. Indeed, it is enough to show that , for all Since for any , , then equality (45) implies that and hence which implies that Thus , for all and , by the -invariance of . Setting , we conclude that is a Frobenius homomorphism and for all .

A Frobenius monoidal Hom-algebra with Frobenius homomorphism is called symmetric if , for all . In this case, the Nakayama automorphism is the identity map of .

Lemma 13. Let be a monoidal Hom-algebra with two bases and , and . Suppose that . Then(1) and ;(2).

Proof. The first assertion of (1) is proved in Theorem 12, and the second one can be showed similarly.
(2) From (1), we have

Theorem 14. Let be a symmetric monoidal Hom-algebra with Frobenius homomorphism , and , be dual bases of . We set . Then(1), for any ;(2) and ;(3)the element is a solution of QHYBE, and .

Proof. Since is symmetric, the corresponding Nakayama automorphism is the identity. It follows from Theorem 11 (3) that , for all . The last two statements are true from Lemma 13.

Next we consider the converse of the above theorem.

Theorem 15. Let be a finite dimensional monoidal Hom-algebra with two bases and . Further let . Then the following conditions are equivalent:(1) and ;(2) and ;(3) and ;(4), for all ;(5) is a symmetric monoidal Hom-algebra with Frobenius homomorphism such that , for all .

Proof. We have proved that (4) implies the first three equalities in Lemma 13. 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 in as follows: , for all We have seen that is a Hom-associative surgenerate bilinear form by Theorem 12, and is a Frobenius homomorphism. To show , we only need to show , for , which is equivalent to prove that , for . From (4) we see that , for all , so by Proposition 5 (1). Hence which implies for all . Thus , for all . Taking , we get so , for all .

5. Frobenius Monoidal Hom-Hopf Algebras

It is known that a finite dimensional monoidal Hom-Hopf algebra is Frobenius if and only if the space of integrals in 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]). A monoidal Hom-coalgebra is an object in the category together with linear maps and such thatfor all

Definition 17. A monoidal Hom-bialgebra [16, Definition 2.3] is a bialgebra in the category . This means that is a monoidal Hom-algebra and is a monoidal Hom-coalgebra such that and are Hom-algebra maps; that is, for any ,

Definition 18 (see [16, Definition 2.5]). A monoidal Hom-bialgebra is called a monoidal Hom-Hopf algebra if there exists a morphism (called antipode) in (i.e., ), such that, for any ,In fact, monoidal Hom-coalgebras, monoidal Hom-bialgebras, and monoidal Hom-Hopf algebras are coalgebras, bialgebras, and Hopf algebras in the category , 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 is given as follows: for all ,

Definition 19 (see [16, Definition 3.1]). A left integral in is an -invariant element (i.e., ) such that for all .

Lemma 20. Let be a monoidal Hom-Hopf algebra with antipode . Further let be an -invariant element in . Then for all , if and only if is a left integral in .

Proof. Let . Since for any , we have if and only if , which means that is a left integral in .

Lemma 21. Let be a monoidal Hom-bialgebra and Frobenius with Frobenius homomorphism such that and with left norm , for all . Then is a monoidal Hom-Hopf algebra.

Proof. We define in by for all . Then by the properties of , we see that for all .
It is not difficult to check that the endomorphism in forms a monoidal Hom-algebra with Hom-multiplication convolution , twisting map and unit . Indeed, for all And the Hom-associativity is true by the Hom-coassociativity of . The map is epimorphism, since . By Lemma 4 in [25], is an isomorphism with inverse map . So . Thus is an antipode of .

Theorem 22. Let be a monoidal Hom-Hopf algebra with antipode and also Frobenius with Frobenius homomorphism such that for all and with the Nakayama automorphism . Further let , be dual bases of . Then(1) is a left norm and a left integral in . Further , and ;(2) generates the Hom-submodule of left integral in and

Proof. (1) Setting , we have and by the -invariance of and Proposition 5 (1). Further, for all , Since , can also be considered as dual bases of , hence we see that which implies that is -invariant. Thus is a left integral in by Definition 19.
Next, , so is a left norm.
From Lemma 21, we know that the antipode is given by And is bijective when if finite dimensional by Theorem 5.3 of [16]. So forms a basis of . Therefore for some . Then we rewrite the expression of as for all . Applying we get for all . On the other hand . So . It follows that , so , and hence By the counitary property of we have . And , so by applying . Therefore . Similarly, , which implies that . According to the above, so we obtain (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 is a left integral in by Lemma 21, so generates the Hom-submodule of left integral in . Since by (1), we see that by applying and the definition of in Proposition 5. So .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the Fundamental Research Funds for the Central Universities (no. KJQN201550), the National Natural Science Foundation of China (no. 11401311, no. 11571173), and the Science Foundation of Jiangsu Province (no. BK20140676, no. BK20141358).

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Copyright © 2018 Yuanyuan Chen and Liangyun Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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