The Factorization for a Class of Hom-Coalgebras

Dong, Lihong; Xue, Shuan; Zhang, Xiaohui

doi:https://doi.org/10.1155/2017/4653172

Advances in Mathematical Physics

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Research Article | Open Access

Volume 2017 | Article ID 4653172 | https://doi.org/10.1155/2017/4653172

The Factorization for a Class of Hom-Coalgebras

Lihong Dong,¹Shuan Xue,¹and Xiaohui Zhang²

Academic Editor: Yao-Zhong Zhang

Received15 May 2017

Revised10 Aug 2017

Accepted07 Sept 2017

Published29 Oct 2017

Abstract

Let be a Hom-Hopf algebra and a Hom-coalgebra. In this paper, we first introduce the notions of Hom-crossed coproduct and cleft coextension and then discuss the equivalence between them. Furthermore, we discuss the relation between cleft coextension and Hom-module coalgebra with the Hom-Hopf module structure and obtain a Hom-coalgebra factorization for Hom-module coalgebra with the Hom-Hopf module structure.

1. Introduction

The motivation to introduce Hom-type algebras comes for examples related to -deformations of Witt and Virasoro algebras, which play an important role in physics, mainly in conformal field theory. Hom-structures (Lie algebras, algebras, coalgebras, Hopf algebras) have been intensively investigated recently. Makhlouf and Silvestrov ([1, 2]) generalized the associativity to twisted associativity and naturally proposed the notion of Hom-associative algebras and were first to introduce Hom-coalgebras, Hom-bialgebras, Hom-Hopf algebras, and related objects.

The crossed products of algebras were independently introduced in [3, 4]. In [3], Blattner et al. showed the equivalence of crossed products and cleft extensions. In [5], Blattner and Montgomery gave several characterizations of crossed products. Lu and Wang [6] generalized the result in [3] to the case of Hom-Hopf algebras. The concept of crossed coproduct appeared as a dual version of the usual crossed product for Hopf algebras and it was studied in several papers; see [7–9]. In [8] the authors studied cleft coextension, a dual notion for that of cleft extension, and it was proved that a cleft coextension is isomorphic to a crossed coproduct. Naturally, what are the structure and the relation of crossed coproduct and cleft coextensions in the sense of Hom-structure?

In [10], Radford showed that a bialgebra with a projection had a factorization. In [11], Caenepeel et al. generalized Radford’s result. It is natural to ask under what conditions a Hom-coalgebra can be factorized into Hom-crossed coproduct. These two problems motivate us writing this paper.

This paper is organized as follows.

In Section 2, we recall some basic definitions and results, such as Hom-bialgebras, Hom-Hopf algebras, Hom-(co)module, Hom-Hopf module, and Hom-module coalgebras. In Section 3, let be a Hom-Hopf algebra and a Hom-coalgebra. We give the definition of Hom-Hopf algebra coacting weakly on Hom-coalgebra from the left, introduce the notion of Hom-crossed coproduct, and then discuss the necessary and sufficient conditions for to be Hom-crossed coproduct (see Theorem 4). In Section 4, we introduce the definition of the cleft coextension and discuss the equivalence between the Hom-crossed coproducts and cleft coextensions (see Theorem 10). Furthermore, we discuss the relation between cleft coextension and Hom-module coalgebra with the Hom-Hopf module structure and obtain which is a cleft coextension if and only if it is a right Hom-Hopf module (see Theorem 13). In Section 5, we obtain the Hom-coalgebra factorization for Hom-module coalgebra with the Hom-Hopf module structure (see Theorem 15).

2. Preliminaries

In this paper, all the vector spaces, tensor products, and homomorphisms are over a fixed field . For a coalgebra , we write for any (summation omitted).

We now recall from [2, 12–14] some definitions and results about the Hom-Hopf algebras, Hom-(co)modules, and so on.

2.1. Hom-Hopf Algebra

A Hom-algebra is a quadruple (abbr. ), where is a linear space, is a linear map, with notation , , and , such that, for any ,

A linear map is called a morphism of Hom-algebra if , , and .

A Hom-coalgebra is a quadruple (abbr. ), where is a linear space, , are linear maps, and , such that, for any ,

A linear map is called a morphism of Hom-coalgebra if , , and .

A Hom-bialgebra is a sextuple (abbr. ), where is a Hom-algebra and is a Hom-coalgebra, such that , are morphisms of Hom-algebra; that is, Furthermore, if there exists a linear map such that then we call (abbr. ) a Hom-Hopf algebra.

Let be a Hom-Hopf algebra; for we have the following properties for any :

2.2. Hom-Hopf Module

Let be a Hom-algebra; a right -Hom-module is a triple , where is a linear space, is a linear map, and is an automorphism of , such that, for any and ,

Let and be two right -Hom-modules. Then a linear morphism is called a morphism of right -Hom-modules if and .

Let be a Hom-coalgebra; a right -Hom-comodule is a triple , where is a linear space, is a linear map (write ), and is an automorphism of , such that, for any ,

Let and be two right -Hom-comodules. Then a linear morphism is called a morphism of right -Hom-comodules if and .

Let be a Hom-Hopf algebra; a right -Hom-Hopf module is a quadruple , where is a right -Hom-module and a right -Hom-comodule, such that, for all , ,

2.3. The Fundamental Theorem of Hom-Hopf Module

Let be a Hom-Hopf algebra and a right -Hom-Hopf module, and set , then is an isomorphism of right -Hom-Hopf module.

2.4. Hom-Module Coalgebra

Recall from [15], let be a Hom-Hopf algebra and a Hom-coalgebra, and if is a left -Hom-module, for all , , the following conditions hold: then is called an -Hom-module coalgebra.

3. Hom-Crossed Coproducts

Let be a Hom-Hopf algebra and a Hom-coalgebra. In this section, we give the definition of coacting weakly on from the left and introduce Hom-crossed coproduct. Then we discuss the necessary and sufficient conditions for to be Hom-crossed coproduct and get some properties about it.

Definition 1. Let be a Hom-Hopf algebra and a Hom-coalgebra. We say that coacts weakly on from the left if there is a linear map , such that for all the following conditions hold:(W1),(W2),(W3)

Definition 2. Let be a Hom-Hopf algebra and a Hom-coalgebra. Assume that coacts weakly on from the left. Let be a linear map; write . Define , whose underlying vector space is with the comultiplication given by We say that is a Hom-crossed coproduct if is coassociative and is the counit for all and .

Remark 3. If the cocycle is convolution invertible, we will denote its convolution inverse by .

Theorem 4. is a Hom-crossed coproduct if and only if the following conditions hold.
(CU) Normal Cocycle Condition(C) Cocycle Condition (TC) Twisted Comodule Condition

Proof. Directly computing, we can get that is the counit of if and only if holds.
Now, we prove that if is a Hom-crossed coproduct then the conditions and are satisfied. Because of coassociativity of , we can get Taking in the above equality, then applying to both sides, and using , we obtain ; and applying to both sides, is obtained too.
Conversely, suppose and hold, then This completes the proof.

Example 5. Consider the case when is trivial, that is, for all . Then the Hom-crossed coproduct is reduced to Hom-smash coproduct.

Proof. If , , then . Similarly we can get , so (CU) is satisfied.
For condition (C), the left hand side is and the right hand side is so is satisfied.
For condition , the left hand side is the right hand side is and then the Hom-crossed coproduct is Hom-smash coproduct. In this case, is a left -comodule coalgebra, and the condition is satisfied.

Let be 2-dimension Hopf group algebra with a basis . Then forms a Hom-Hopf algebra. Let be a vector space with a basis . Define the Hom-coalgebra structure on as follows.

The automorphism is given by the comultiplication and counit are given by It is easy to see that is a Hom-coalgebra.

Now consider the coaction defined by Then after a direct computation, we get that coacts weakly on from the left. Further, recall from (1) that if we define by , then is a Hom-crossed coproduct.

Proof. We can prove that the condition (CU) and the condition (C) hold for any the same as Example 5(1). Then we only prove that the condition (TC) holds. By the proof of Example 5, for , we can get that the left hand side of the condition (TC) is and the right hand side of the condition (TC) is so the condition (TC) holds for . Similarly, we can get the condition (TC) holds for .

(3) Let be a Hopf algebra and a coalgebra, and weakly coacts on from the left. Assume that is a Hopf automorphism of and is a coalgebra isomorphism of . Then we have Hom-Hopf algebra and Hom-coalgebra (see [2]). Furthermore assume that , and define the coaction , then weakly coacts on from the left. If is a crossed coproduct and , then is a Hom-crossed coproduct.

Proof. If is a crossed coproduct, then we can getFrom [2], we know that the multiplication and comultiplication of Hom-Hopf algebra are, respectively, given by and the comultiplication of Hom-coalgebra is given by First, we prove that weakly coacts on from the left. We only prove that (W1) of Definition 1 holds; the other two are easy to get. For any , so weakly coacts on from the left.
Then we prove that is a Hom-crossed coproduct. It is easy to see that the condition (CU) holds. For the condition , so the condition (CU) holds. Similarly we can prove that the condition holds by (28).

For a Hom-crossed coproduct, we can get the following properties, which are useful for the latter conclusions.

Lemma 6. Let be a Hom-crossed coproduct with invertible cocycle . Then the following equalities hold for any :
(i) (ii)

Proof. Applying to both sides of and then multiplying it (by convolution) to the right by the following map, we can get (i): Let us denote by , where the map is defined by We get from (i) that a right convolution inverse for is the map , which is given by Let be given by . Since . We get that is a left convolution inverse of , so . Therefore, we prove that (ii) holds.

Remark 7. Note that if is a Hom-crossed coproduct, then the map , defined by , is a Hom-coalgebra map, and we will also define the map , .

The following result is the generalization of Proposition 2.1 in [8].

Proposition 8. Let be a Hom-crossed coproduct. Then is convolution invertible in if and only if is convolution invertible in .

Proof. Assume first that is convolution invertible; for any , define , by Now we show that is a convolution inverse of . So is a left inverse for ; in order to show that is a right inverse we compute as follows: So is convolution invertible.
Next we prove that if is convolution invertible, then is also convolution invertible. Recalling we prove that is convolution invertible.
Firstly, we prove that Observe that if , then the equality becomes By applying we getAt the same time, we have So we get Secondly, let , which is given by and we prove that is the convolution inverse of .
On the one hand, So is a left convolution inverse for . On the other hand, Thirdly, let , then we have Since , we obtain that So has a right convolution inverse in . On the other hand, It shows that is a right convolution inverse for in ; thus .
Finally, we denote that , because we have that in and is a surjective Hom-coalgebra map, and we obtain in . So is the convolution inverse of .

4. Two Equivalent Characterizations of Cleft Coextensions

In this section, we introduce the notion of the cleft coextension and then discuss two equivalent characterizations of cleft coextensions.

Let be a Hom-Hopf algebra and a right -Hom-module coalgebra. Let denote the augmentation Hom-ideal ker which is a Hom-Hopf ideal, then is a Hom-coideal of , and is a Hom-coalgebra with a trivial right -Hom-module structure.

Definition 9. Let be a Hom-Hopf algebra and a right -Hom-module coalgebra. Denote by ; if there exists a right -Hom-module map which is convolution invertible, then is said to be a cleft coextension and write .

By Proposition 8, we get the main result of this section, which is the generalization of Theorem 2.10 in [8].

Theorem 10. The coextension is cleft if and only if is isomorphic to a Hom-crossed coproduct with invertible cocycle .

Proof. If is isomorphic to the Hom-crossed coproduct , we prove that the coextension is cleft. First, we show that is a right -Hom-module map. For any , we get On the other hand, and by Proposition 8, is an invertible cocycle, so is convolution invertible. Hence is cleft, so is cleft too.
Conversely, if the coextension is cleft, let us denote by the image in of the element . Define a weak coaction of on by The definition is well defined. The conditions , , and hold which can be directly verified.
Define , which is given by and is given by Similar to the proof of Proposition 8, we can obtain that , that is, is the convolution inverse of . With convolution inverse and weak coaction of on as the above, we can define a comultiplication and a counit as the Definition 2. The comultiplication is given by and the counit is given by for any , .
Recalling from [16], the isomorphism is given by with inverse Now, for any , , we have So we can obtain that and . Since is a Hom-coalgebra, is a Hom-crossed coproduct and is the required Hom-coalgebra isomorphism. Thus the proof is complete.

Now we give a lemma, which will be used in the sequel.

Lemma 11. Let be a Hom-Hopf algebra and a right -Hom-Hopf module; set and . Then
(1) as vector spaces, and the isomorphism is .
(2) If is a right -Hom-module coalgebra, then there exists a Hom-coalgebra structure on such that as Hom-coalgebras.

Proof. (1) It is easy to see that is an isomorphism of vector spaces.
(2) In order to prove that as Hom-coalgebras, we first give the Hom-coalgebra structure on . The comultiplication on is given by for any . With the above defined comultiplication, we can prove that is a Hom-coalgebra. Furthermore we can get that , which is as similar as defined in (1), is also a Hom-coalgebra morphism. So as Hom-coalgebras.

Then we discuss the relation between cleft coextension and Hom-module coalgebra with the Hom-Hopf module structure.

Proposition 12. Let be a cleft coextension. One assumes that coacts on to the right by then(1) is a right -Hom-comodule;(2) is a right -Hom-Hopf module.

Proof. Note that is convolution invertible right -Hom-module map, where the convolution inverse of is denoted by . We can getIn fact (1) Now we prove that is a right -Hom-comodule. For any , and ; thus is a right -Hom-comodule.
(2) For all and , we can calculate thus is a right -Hom-Hopf module.

Theorem 13. Let be a Hom-Hopf algebra and a right -Hom-module coalgebra, if , then is a cleft coextension if and only if is a right -Hom-Hopf module, and there exists a convolution invertible linear map satisfying .

Proof. Assume that is a cleft coextension, then there exists a convolution invertible Hom-module map such that , so we have . Define a map, clearly, . By Proposition 12, we see that is a right -Hom-Hopf module.
Conversely, it is sufficient to show that is a right -Hom-module map. Using the fact that is a right -Hom-Hopf module and , for any and , we get The proof is complete.

5. The Factorization of Hom-Module Coalgebra with Hopf Module Structure

In this section, we always assume that is a Hom-Hopf algebra and is a cleft coextension. Based on this assumption, can not only form a right -Hom-comodule, but also form a right -Hom-Hopf module. Furthermore, we obtain the factorization of Hom-coalgebra .

Now, we can obtain a Hom-coalgebra factorization for Hom-module coalgebra with the Hom-Hopf module structure.

Theorem 14. Let be a Hom-Hopf algebra and a right -Hom-Hopf module. One assumes that is a right -Hom-module coalgebra and set . The comultiplication on can be defined and the counit can be defined for any , , then is a Hom-coalgebra, and one writes it as . Moreover, as Hom-coalgebras.

Proof. Since is a right -Hom-Hopf module, is well defined and the map is an isomorphism of right -Hom-Hopf module.
We first verify that is a Hom-coalgebra. It is easy to see that . It is sufficient to show the coassociativity. For any , , we calculate as follows: In a similar way Thus, is a Hom-coalgebra.
Next we show that is a Hom-coalgebra morphism. Therefore, as Hom-coalgebras.

By the above results, we obtain the following result.

Theorem 15. Let be a Hom-Hopf algebra and a cleft coextension, then as Hom-coalgebras, where , and

Proof. By Theorems 10 and 14, we have and as Hom-coalgebras, respectively. Since is a cleft coextension, we get is a right -Hom-Hopf module by Proposition 12, where the comodule structure is defined by We know that is an isomorphism of Hom-coalgebras by Lemma 11. Set We only need to verify that is a Hom-coalgebra morphism. First we prove that In factOn one hand, we can calculate For all , , we define by and is defined by On the other hand, we have Therefore, as Hom-coalgebras.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the NNSF of China (no. 11601132), the Natural Science Foundation of Henan Province (no. 152300410086), the Research Fund of PhD of Henan Normal University (no. qd14151), the TianYuan Special Funds of the National Natural Science Foundation of China (no. 11626138), and the NSF of Shandong Province (no. ZR2016AQ03).

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Copyright © 2017 Lihong Dong et al. 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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