The Factorization for a Class of Hom-Coalgebras
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.
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 , Blattner et al. showed the equivalence of crossed products and cleft extensions. In , Blattner and Montgomery gave several characterizations of crossed products. Lu and Wang  generalized the result in  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  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 , Radford showed that a bialgebra with a projection had a factorization. In , 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).
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).
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 , 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 ). 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 , 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 :
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 .
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