Research Article | Open Access

Lihong Dong, Shuan Xue, Xiaohui Zhang, "The Factorization for a Class of Hom-Coalgebras", *Advances in Mathematical Physics*, vol. 2017, Article ID 4653172, 14 pages, 2017. https://doi.org/10.1155/2017/4653172

# The Factorization for a Class of Hom-Coalgebras

**Academic Editor:**Yao-Zhong Zhang

#### 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 [7â€“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 Radfordâ€™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, 12â€“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.

*Exampleâ€‰â€‰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 Exampleâ€‰â€‰5(1). Then we only prove that the condition (TC) holds. By the proof of Exampleâ€‰â€‰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 Propositionâ€‰â€‰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: