# The Factorization for a Class of Hom-Coalgebras

Authors: Lihong Dong; Shuan Xue; Xiaohui Zhang
Journal: Advances in Mathematical Physics
 (2017)

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## Abstract

Let(H,α) be a Hom-Hopf algebra and (C,β) a Hom-coalgebra. In this paper, we first introduce the notions of Hom-crossed coproduct C⋊λH 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.


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## Body

## 1. Introduction

The motivation to introduce Hom-type algebras comes for examples related toq-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 Section2, 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 (H,α) be a Hom-Hopf algebra and (C,β) a Hom-coalgebra. We give the definition of Hom-Hopf algebra H coacting weakly on Hom-coalgebra C from the left, introduce the notion of Hom-crossed coproduct, and then discuss the necessary and sufficient conditions for C⋊λH 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 (B,·,ϕ,β) 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 fieldk. For a coalgebra C, we write Δ(c)=c1⊗c2 for any c∈C (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(A,μ,1A,α) (abbr. (A,α)), where A is a linear space, μ:A⊗A→A is a linear map, with notation μa⊗a′=aa′, 1A∈A, and α∈Autk(A), such that, for any a,a′,a′′∈A, (1)αaa′=αaαa′,a1A=1Aa=αa,αaa′a′′=aa′αa′′,α1A=1A.A linear mapf:(A,μA,1A,αA)→(B,μB,1B,αB) is called a morphism of Hom-algebra if αBf=fαA, f(1A)=1B, and fμA=μB(f⊗f).A Hom-coalgebra is a quadruple(C,Δ,ε,β) (abbr. (C,β)), where C is a linear space, Δ:C→C⊗C, ε:C→k are linear maps, and β∈Autk(C), such that, for any c∈C, (2)βc1⊗βc2=βc1⊗βc2,εc1c2=c1εc2=βc,βc1⊗c21⊗c22=c11⊗c12⊗βc2,εβ=ε.A linear mapf:(C,ΔC,εC,βC)→(D,ΔD,εD,βD) is called a morphism of Hom-coalgebra if βDf=fβC, εDf=εC, and ΔDf=(f⊗f)ΔC.A Hom-bialgebra is a sextuple(H,μ,1H,Δ,ε,γ) (abbr. (H,γ)), where (H,μ,1H,γ) is a Hom-algebra and (H,Δ,ε,γ) is a Hom-coalgebra, such that Δ, ε are morphisms of Hom-algebra; that is, (3)Δhh′=ΔhΔh′,Δ1H=1H⊗1H,εhh′=εhεh′,ε1H=1.Furthermore, if there exists a linear map S:H→H such that (4)Sh1h2=h1Sh2=εh1H,Sγh=γSh,then we call (H,μ,1H,Δ,ε,S,γ) (abbr. (H,S,γ)) a Hom-Hopf algebra.Let(H,S,γ) be a Hom-Hopf algebra; for S we have the following properties for any h,g∈H: (5)Sh1⊗Sh2=Sh2⊗Sh1,Shg=SgSh,εS=ε.


### 2.2. Hom-Hopf Module

Let(A,β) be a Hom-algebra; a right (A,β)-Hom-module is a triple (M,·,α), where M is a linear space, ·:M⊗A→M is a linear map, and α is an automorphism of M, such that, for any a,a′∈A and m∈M, (6)αm·aa′=m·a·βa′,m·1A=αm,αm·a=αm·βa.Let(M,·M,αM) and (N,·N,αN) be two right (A,β)-Hom-modules. Then a linear morphism f:M→N is called a morphism of right (A,β)-Hom-modules if f(m·Ma)=f(m)·Na and αMf=fαN.Let(C,β) be a Hom-coalgebra; a right (C,β)-Hom-comodule is a triple (M,ρ,α), where M is a linear space, ρ:M→M⊗C is a linear map (write ρ(m)=m(0)⊗m(1)), and α is an automorphism of M, such that, for any m∈M, (7)αm0⊗m11⊗m12=m00⊗m01⊗βm1,m0εm1=αm,αm0⊗αm1=αm0⊗βm0.Let(M,ρM,αM) and (N,ρN,αN) be two right (C,β)-Hom-comodules. Then a linear morphism f:M→N is called a morphism of right (C,β)-Hom-comodules if f(m)(0)⊗f(m)(1)=f(m(0))⊗m(1) and αMf=fαN.Let(H,α) be a Hom-Hopf algebra; a right H-Hom-Hopf module is a quadruple (M,·,ρ,β), where M is a right H-Hom-module and a right H-Hom-comodule, such that, for all m∈M, h∈H, (8)ρm·h=m0·h1⊗m1h2.


### 2.3. The Fundamental Theorem of Hom-Hopf Module

Let(H,α) be a Hom-Hopf algebra and (B,·,ρ,β) a right H-Hom-Hopf module, and set A=BcoH=b∈B∣ρb=βb⊗1, then (9)φ:A⊗H⟶B,b⊗h⟼b·his an isomorphism of right H-Hom-Hopf module.


### 2.4. Hom-Module Coalgebra

Recall from [15], let (H,β) be a Hom-Hopf algebra and (C,α) a Hom-coalgebra, and if (C,·,α) is a left (H,β)-Hom-module, for all c∈C, h∈H, the following conditions hold: (10)h·c1⊗h·c2=h1·c1⊗h2·c2,εh·c=εhεc,then (C,·,α) is called an (H,β)-Hom-module coalgebra.


## 2.1. Hom-Hopf Algebra

A Hom-algebra is a quadruple(A,μ,1A,α) (abbr. (A,α)), where A is a linear space, μ:A⊗A→A is a linear map, with notation μa⊗a′=aa′, 1A∈A, and α∈Autk(A), such that, for any a,a′,a′′∈A, (1)αaa′=αaαa′,a1A=1Aa=αa,αaa′a′′=aa′αa′′,α1A=1A.A linear mapf:(A,μA,1A,αA)→(B,μB,1B,αB) is called a morphism of Hom-algebra if αBf=fαA, f(1A)=1B, and fμA=μB(f⊗f).A Hom-coalgebra is a quadruple(C,Δ,ε,β) (abbr. (C,β)), where C is a linear space, Δ:C→C⊗C, ε:C→k are linear maps, and β∈Autk(C), such that, for any c∈C, (2)βc1⊗βc2=βc1⊗βc2,εc1c2=c1εc2=βc,βc1⊗c21⊗c22=c11⊗c12⊗βc2,εβ=ε.A linear mapf:(C,ΔC,εC,βC)→(D,ΔD,εD,βD) is called a morphism of Hom-coalgebra if βDf=fβC, εDf=εC, and ΔDf=(f⊗f)ΔC.A Hom-bialgebra is a sextuple(H,μ,1H,Δ,ε,γ) (abbr. (H,γ)), where (H,μ,1H,γ) is a Hom-algebra and (H,Δ,ε,γ) is a Hom-coalgebra, such that Δ, ε are morphisms of Hom-algebra; that is, (3)Δhh′=ΔhΔh′,Δ1H=1H⊗1H,εhh′=εhεh′,ε1H=1.Furthermore, if there exists a linear map S:H→H such that (4)Sh1h2=h1Sh2=εh1H,Sγh=γSh,then we call (H,μ,1H,Δ,ε,S,γ) (abbr. (H,S,γ)) a Hom-Hopf algebra.Let(H,S,γ) be a Hom-Hopf algebra; for S we have the following properties for any h,g∈H: (5)Sh1⊗Sh2=Sh2⊗Sh1,Shg=SgSh,εS=ε.


## 2.2. Hom-Hopf Module

Let(A,β) be a Hom-algebra; a right (A,β)-Hom-module is a triple (M,·,α), where M is a linear space, ·:M⊗A→M is a linear map, and α is an automorphism of M, such that, for any a,a′∈A and m∈M, (6)αm·aa′=m·a·βa′,m·1A=αm,αm·a=αm·βa.Let(M,·M,αM) and (N,·N,αN) be two right (A,β)-Hom-modules. Then a linear morphism f:M→N is called a morphism of right (A,β)-Hom-modules if f(m·Ma)=f(m)·Na and αMf=fαN.Let(C,β) be a Hom-coalgebra; a right (C,β)-Hom-comodule is a triple (M,ρ,α), where M is a linear space, ρ:M→M⊗C is a linear map (write ρ(m)=m(0)⊗m(1)), and α is an automorphism of M, such that, for any m∈M, (7)αm0⊗m11⊗m12=m00⊗m01⊗βm1,m0εm1=αm,αm0⊗αm1=αm0⊗βm0.Let(M,ρM,αM) and (N,ρN,αN) be two right (C,β)-Hom-comodules. Then a linear morphism f:M→N is called a morphism of right (C,β)-Hom-comodules if f(m)(0)⊗f(m)(1)=f(m(0))⊗m(1) and αMf=fαN.Let(H,α) be a Hom-Hopf algebra; a right H-Hom-Hopf module is a quadruple (M,·,ρ,β), where M is a right H-Hom-module and a right H-Hom-comodule, such that, for all m∈M, h∈H, (8)ρm·h=m0·h1⊗m1h2.


## 2.3. The Fundamental Theorem of Hom-Hopf Module

Let(H,α) be a Hom-Hopf algebra and (B,·,ρ,β) a right H-Hom-Hopf module, and set A=BcoH=b∈B∣ρb=βb⊗1, then (9)φ:A⊗H⟶B,b⊗h⟼b·his an isomorphism of right H-Hom-Hopf module.


## 2.4. Hom-Module Coalgebra

Recall from [15], let (H,β) be a Hom-Hopf algebra and (C,α) a Hom-coalgebra, and if (C,·,α) is a left (H,β)-Hom-module, for all c∈C, h∈H, the following conditions hold: (10)h·c1⊗h·c2=h1·c1⊗h2·c2,εh·c=εhεc,then (C,·,α) is called an (H,β)-Hom-module coalgebra.


## 3. Hom-Crossed Coproducts

Let(H,α) be a Hom-Hopf algebra and (C,β) a Hom-coalgebra. In this section, we give the definition of H coacting weakly on C from the left and introduce Hom-crossed coproduct. Then we discuss the necessary and sufficient conditions for C⋊λH to be Hom-crossed coproduct and get some properties about it.Definition 1.
Let(H,α) be a Hom-Hopf algebra and (C,β) a Hom-coalgebra. We say that H coacts weakly on C from the left if there is a linear map C→H⊗C, c↦c-1⊗c0 such that for all c∈C the following conditions hold:(W1)
α2(c-1)⊗c01⊗c02=c1-1c2-1⊗c10⊗c20,(W2)
ε(c0)c-1=ε(c)1,(W3)
ε(c-1)c0=β(c).Definition 2.
Let(H,α) be a Hom-Hopf algebra and (C,β) a Hom-coalgebra. Assume that H coacts weakly on C from the left. Let λ∈Hom(C,H⊗H) be a linear map; write λ(c)=λ1(c)⊗λ2(c). Define C⋊λH, whose underlying vector space is C⊗H with the comultiplication given by (11)Δλc⊗h=c1⊗α-4c21-1α-3λ1c22α-1h1⊗β-2c210⊗α-2λ2c22α-1h2.We say that (C⋊λH,β⊗α) is a Hom-crossed coproduct if Δλ is coassociative and ε⊗ε is the counit for all c∈C and h∈H.Remark 3.
If the cocycleλ is convolution invertible, we will denote its convolution inverse by λ-1(c)=λ1-1(c)⊗λ2-1(c).Theorem 4.
(
C
⋊
λ
H
,
β
⊗
α
) is a Hom-crossed coproduct if and only if the following conditions hold.
(CU) Normal Cocycle Condition
(12)
i
d
⊗
ε
λ
=
ε
⊗
i
d
λ
=
1
H
ε
.
(C) Cocycle Condition
(13)
λ
1
c
1
α
-
1
λ
1
c
2
1
⊗
λ
2
c
1
α
-
1
λ
1
c
2
2
⊗
α
λ
2
c
2
=
α
-
1
c
1
-
1
λ
1
c
2
⊗
α
-
1
λ
1
c
10
α
-
1
λ
2
c
2
1
⊗
α
-
1
λ
2
c
10
α
-
1
λ
2
c
2
2
.
(TC) Twisted Comodule Condition
(14)
λ
1
c
1
α
-
2
c
2
-
11
⊗
λ
2
c
1
α
-
2
c
2
-
12
⊗
c
20
=
α
-
1
c
1
-
1
λ
1
c
2
⊗
α
-
2
c
10
-
1
λ
2
c
2
⊗
β
-
1
c
100
.Proof.
Directly computing, we can get thatε⊗ε is the counit of (C⋊λH,β⊗α) if and only if (CU) holds.
Now, we prove that if(C⋊λH,β⊗α) is a Hom-crossed coproduct then the conditions (C) and (TC) are satisfied. Because of coassociativity of (C⋊λH,β⊗α), we can get (15)Δλ⊗β⊗αΔλc⊗h=c11⊗α-4c121-1α-3λ1c122α-5c21-11α-4λ1c221α-2h11⊗β-2c1210⊗α-2λ2c122α-5c21-12α-4λ1c222α-2h12⊗β-1c210⊗α-1λ2c22h2=αc1⊗α-3c21-1α-2λ1c22h1⊗β-2c2101⊗α-6c21021-1α-5λ1c21022α-3λ2c221α-2h21⊗β-4c210210⊗α-4λ2c21022α-3λ2c222α-2h22=β⊗α⊗ΔλΔλc⊗h.Taking h=1H in the above equality, then applying (ε⊗id)2⊗id⊗ε to both sides, and using (CU), we obtain (TC); and applying (ε⊗id)3 to both sides, (C) is obtained too.
Conversely, suppose(C) and (TC) hold, then (16)β⊗α⊗ΔλΔλc⊗h=W1αc1⊗α-5c211-1α-7c2121-1c2122-1α-2λ1c22h1⊗β-2c2110⊗α-6c21210-1α-5λ1c21220·α-3λ2c221α-2h21⊗β-4c212100⊗α-4λ2c21220α-3λ2c222α-2h22=αc1⊗α-5c211-1α-5c212-1α-5c221-1α-4λ1c222h1⊗β-2c2110⊗α-4c2120-1α-5λ1c2210α-5λ2c2221α-2h21⊗β-3c21200⊗α-4λ2c2210α-4λ2c2222α-1h22=Cαc1⊗α-5c211-1α-5c212-1α-4λ1c221α-5λ1c2221h1⊗β-2c2110⊗α-4c2120-1α-4λ2c221α-5λ1c2222α-2h21⊗β-3c21200⊗α-2λ2c222α-1h22=αc1⊗α-3c21-1α-7c2211-1α-6λ1c2212α-5λ1c2221h1⊗β-1c210⊗α-6c22110-1α-4λ2c2212α-4λ1c2222α-2h21⊗β-4c221100⊗α-2λ2c222α-1h22=TCαc1⊗α-3c21-1α-6λ1c2211α-8c2212-11α-5λ1c2221h1⊗β-1c210⊗α-4λ2c2211α-6c2212-12α-4λ1c2222α-2h21⊗β-3c22120⊗α-2λ2c222α-1h22=αc1⊗α-3c21-1α-4λ1c2211α-7c2212-11α-5λ1c2221α-2h11⊗β-1c210⊗α-3λ2c2211α-7c2212-12α-5λ1c2222α-2h12⊗β-3c22120⊗α-2λ2c222h2=Δλ⊗β⊗αΔλc⊗h.This completes the proof.Example  5. (1) Consider the case when λ is trivial, that is, λ(c)=ε(c)1H⊗1H for all c∈C. Then the Hom-crossed coproduct is reduced to Hom-smash coproduct.Proof.
Ifλ(c)=ε(c)1H⊗1H, ∀c∈C, then (id⊗ε)λ(c)=ε(c)(id⊗ε)(1H⊗1H)=1Hε(c). Similarly we can get (ε⊗id)λ=1Hε, so (CU) is satisfied.
For condition (C), the left hand side is(17)λ1c1α-1λ1c21⊗λ2c1α-1λ1c22⊗αλ2c2=εc1εc21H⊗1H⊗1H=εc1H⊗1H⊗1H,and the right hand side is (18)α-1c1-1λ1c2⊗α-1λ1c10α-1λ2c21⊗α-1λ2c10α-1λ2c22=α-1c1-1εc21H⊗α-1λ1c101H⊗α-1λ2c101H=α-1c1-1εc21H⊗εc101H⊗1H=εc1H⊗1H⊗1H,so (C) is satisfied.
For condition(TC), the left hand side is (19)λ1c1α-2c2-11⊗λ2c1α-2c2-12⊗c20=εc1α-1c2-11⊗α-1c2-12⊗c20=c-11⊗c-12⊗βc0,the right hand side is (20)α-1c1-1λ1c2⊗α-2c10-1λ2c2⊗β-1c100=c1-1εc2⊗α-1c10-1⊗β-1c100=αc-1⊗c0-1⊗c00,and then the Hom-crossed coproduct is Hom-smash coproduct. In this case, C is a left H-comodule coalgebra, and the condition (TC) is satisfied.(
2
) Let H2 be 2-dimension Hopf group algebra with a basis {1,g}. Then (H2,id) forms a Hom-Hopf algebra. Let H4 be a vector space with a basis {e,c,x,y}. Define the Hom-coalgebra structure on H4 as follows.The automorphismβ:H4→H4 is given by (21)βe=e,βc=c,βx=-x,βy=-y;the comultiplication and counit are given by (22)Δe=e⊗e,Δc=c⊗c,Δx=-x⊗c-e⊗x,Δy=-y⊗e-c⊗y,εe=1,εc=1,εx=0,εy=0.It is easy to see that H4 is a Hom-coalgebra.Now consider the coactionρH4:H4→H2⊗H4 defined by (23)e-1⊗e0≔1⊗e,c-1⊗c0≔1⊗c,x-1⊗x0≔-g⊗x,y-1⊗y0≔-g⊗y.Then after a direct computation, we get that H2 coacts weakly on H4 from the left. Further, recall from (1) that if we define λ:H4→H2⊗H2 by λ(h)=ε(h)1⊗1, then (H4⋊λH2,β⊗id) is a Hom-crossed coproduct.Proof.
We can prove that the condition (CU) and the condition (C) hold for anyc∈H4 the same as Example  5(1). Then we only prove that the condition (TC) holds. By the proof of Example  5(1), for x∈H4, we can get that the left hand side of the condition (TC) is (24)λ1x1x2-11⊗λ2x1x2-12⊗x20=x-11⊗x-12⊗βx0=g⊗x⊗x, and the right hand side of the condition (TC) is (25)x1-1λ1x2⊗x10-1λ2x2⊗β-1x100=x-1⊗x0-1⊗x00=g⊗x⊗x, so the condition (TC) holds for x∈H4. Similarly, we can get the condition (TC) holds for e,c,y∈H4.(3) Let(H,Δ,μ) be a Hopf algebra and (C,Δ) a coalgebra, and H weakly coacts on C from the left. Assume that α is a Hopf automorphism of H and β is a coalgebra isomorphism of C. Then we have Hom-Hopf algebra (Hα,Δα,μα,α) and Hom-coalgebra (Cβ,Δβ,β) (see [2]). Furthermore assume that ρ(β(c))=(α⊗β)ρ(c), and define the coaction ρ(c)=(α⊗β)ρ(c), then Hα weakly coacts on Cβ from the left. If C⋊λH is a crossed coproduct and λβ=(α⊗α)λ, then Cβ⋊λHα is a Hom-crossed coproduct.Proof.
IfC⋊λH is a crossed coproduct, then we can get(26)id⊗ελ=ε⊗idλ=1Hε;(27)λ1c1λ1c21⊗λ2c1λ1c22⊗λ2c2=c1-1λ1c2⊗λ1c10λ2c21⊗λ2c10λ2c22;(28)λ1c1c2-11⊗λ2c1c2-12⊗c20=c1-1λ1c2⊗c10-1λ2c2⊗c100.
From [2], we know that the multiplication and comultiplication of Hom-Hopf algebra (Hα,Δα,μα,α) are, respectively, given by (29)μαh⊗g=αhαg,Δαh=αh1⊗αh2,∀h,g∈Hα,and the comultiplication of Hom-coalgebra (Cβ,Δβ,β) is given by (30)Δβc=βc1⊗βc2,∀c∈Cβ.
First, we prove thatHα weakly coacts on Cβ from the left. We only prove that (W1) of Definition 1 holds; the other two are easy to get. For any c∈Cβ, (31)α2c-1⊗c01⊗c02inHα⊗Cβ⊗Cβ=α3c-1⊗β2c01⊗β2c02inH⊗C⊗C=α3c1-1c2-1⊗β2c10⊗β2c20inH⊗C⊗C=c1-1c2-1⊗c10⊗c20,inHα⊗Cβ⊗Cβ, so Hα weakly coacts on Cβ from the left.
Then we prove thatCβ⋊λHα is a Hom-crossed coproduct. It is easy to see that the condition (CU) holds. For the condition (C), (32)λ1c1α-1λ1c21⊗λ2c1α-1λ1c22⊗αλ2c2inHα⊗Hα⊗Hα=α2λ1c1λ1c21⊗α2λ2c1λ1c22⊗α2λ2c2inH⊗H⊗H=27α2c1-1λ1c2⊗α2λ1c10λ2c21⊗α2λ2c10λ2c22inH⊗H⊗H=α-1c1-1λ1c2⊗α-1λ1λ1c10α-1λ2c21⊗α-1λ2c10α-1λ2c22,inHα⊗Hα⊗Hα so the condition (CU) holds. Similarly we can prove that the condition (TC) holds by (28).For a Hom-crossed coproduct, we can get the following properties, which are useful for the latter conclusions.Lemma 6.
LetC⋊λH be a Hom-crossed coproduct with invertible cocycle λ. Then the following equalities hold for any c∈C:
(i)(33)c-1⊗λ1c0⊗λ2c0=α-3λ1c11α-4λ1c121α-1λ1-1c2⊗α-3λ2c11α-4λ1c122α-2λ2-1c21⊗α-2λ2c12α-2λ2-1c22;
(ii)(34)c-1⊗λ1-1c0⊗λ2-1c0=α-1λ1c1α-4λ1-1c211α-3λ1-1c22⊗α-2λ2c11α-4λ1-1c212α-3λ2-1c22⊗α-2λ2c12α-2λ2-1c21.Proof.
Applyingα-3⊗α-3⊗α-3 to both sides of (C) and then multiplying it (by convolution) to the right by the following map, we can get (i): (35)c⟼α-1λ1-1c⊗α-2λ2-1c1⊗α-2λ2-1c2. Let us denote by F1,F2,F3:C→H⊗H⊗H, where the map is defined by (36)F1c=c-1⊗λ1c0⊗λ2c0.We get from (i) that a right convolution inverse for F1 is the map F2, which is given by (37)F2c=α-1λ1c1α-4λ1-1c211α-3λ1-1c22⊗α-2λ2c11α-4λ1-1c212α-3λ2-1c22⊗α-2λ2c12α-2λ2-1c21.Let F3 be given by F3(c)=c-1⊗λ1-1(c0)⊗λ2-1(c0). Since (F3∗F1)(c)=ε(c)1⊗1⊗1. We get that F3 is a left convolution inverse of F1, so F3=F2. Therefore, we prove that (ii) holds.Remark 7.
Note that if(C⋊λH,β⊗α) is a Hom-crossed coproduct, then the map π:C⋊λH→C, defined by π(c⊗h)=ε(h)β-1(c), is a Hom-coalgebra map, and we will also define the map θ:C⋊λH→H, θ(c⊗h)=ε(c)α-1(h).The following result is the generalization of Proposition  2.1 in [8].Proposition 8.
LetB=C⋊λH be a Hom-crossed coproduct. Then θ is convolution invertible in Hom(B,H) if and only if λ is convolution invertible in Hom(C,H⊗H).Proof.
Assume first thatλ is convolution invertible; for any c⋊h∈B, define μ:B→H, by (38)μc⋊h=Sα-3λ1-1cα-3hα-2λ2-1c.Now we show that μ is a convolution inverse of θ. (39)μ∗θc⋊h=μc1⊗α-4c21-1α-3λ1c22α-1h1θβ-2c210⊗α-2λ2c22α-1h2=Sα-3λ1-1c1α-3α-4c21-1α-3λ1c22α-1h1α-2λ2-1c1εc210α-3λ2c22α-2h2=Sα-3λ1-1c1α-4λ1c2α-4h1α-2λ2-1c1α-2λ2c2α-2h2=Sα-3λ1-1c1λ1c2α-2h1α-3λ2-1c1λ2c2α-2h2=εcεh. So μ is a left inverse for θ; in order to show that μ is a right inverse we compute as follows: (40)θ∗μc⋊h=θc1⊗α-4c21-1α-3λ1c22α-1h1μβ-2c210⊗α-2λ2c22α-1h2=εc1α-5c21-1α-4λ1c22α-2h1Sα-5λ1-1c210α-5λ2c22α-4h2α-4λ2-1c210=α-4c1-1α-3λ1c2α-2h1Sα-5λ1-1c10α-4λ2c2α-3h2α-3λ2-1c10=α-3c1-1α-2λ1c2α-3h1Sα-3h2Sα-5λ1-1c10α-4λ2c2α-4λ2-1c10=α-3c1-1α-2λ1c2εhSα-4λ1-1c10α-3λ2c2α-3λ2-1c10=34α-4λ1c11α-7λ1-1c1211α-6λ1-1c122α-2λ1c2εhSα-6λ2c111α-8λ1-1c1212α-7λ2-1c122α-3λ2c2α-5λ2c112α-5λ2-1c121=α-3λ1c11α-5λ1-1c121εc21εhSα-5λ2c111α-6λ1-1c1221α-5λ2c112α-4λ2-1c12=α-2λ1c1α-3λ1-1c21εhSα-4λ2c11α-4λ1-1c22α-4λ2c12α-3λ2-1c2=εcεh.So θ is convolution invertible.
Next we prove that ifθ is convolution invertible, then λ is also convolution invertible. Recalling (41)π:B⟶C,c⊗h⟼εhβ-1c,we prove that λπ is convolution invertible.
Firstly, we prove that(42)λπb=α-3θb11α-3θ-1b21⊗α-3θb12α-3θ-1b22. Observe that if b=c⊗h, then the equality θ(b1)θ-1(b2)=ε(b)1 becomes (43)α-4c1-1α-3λ1c2α-2h1θ-1β-1c10⊗α-1λ2c2α-1h2=εcεh1. By applying ΔH we get(44)α-4c1-11α-3λ1c21α-2h11θ-1β-1c10⊗α-1λ2c2α-1h21⊗α-4c1-12α-3λ1c22α-2h12θ-1β-1c10⊗α-1λ2c2α-1h22=εcεh1⊗1.At the same time, we have (45)b11⊗b12⊗b2=c11⊗α-4c121-1α-3λ1c122α-5c21-11α-4λ1c221α-2h11⊗β-2c1210⊗α-2λ2c122α-5c21-12α-4λ1c222α-2h12⊗β-2c210⊗α-2λ2c22α-1h2.So we get (46)α-3θb11α-3θ-1b21⊗α-3θb12α-3θ-1b22=α-3λ1c1α-9c21-11α-8λ1c221α-6h11_θ-1β-6c210⊗α-6λ2c22α-5h21_⊗α-3λ2c1α-9c21-12α-8λ1c222α-6h12_θ-1β-6c210⊗α-6λ2c22α-5h22_=44α-2λ1c1⊗α-2λ2c1εc2εh=α-1λ1c⊗α-1λ2cεh=α-1λcεh=λπb.
Secondly, letω:B→H⊗H, which is given by (47)ωb=α-4θb111α-2θ-1b2⊗α-4θb112α-3θ-1b12,and we prove that ω is the convolution inverse of λπ.
On the one hand,(48)λπ∗ωb=λπb1ωb2=α-3θb111α-3θ-1b121α-4θb2111α-2θ-1b22⊗α-3θb112α-3θ-1b122α-4θb2112α-3θ-1b212=α-1θb11α-5θ-1b2111α-5θb2121α-3θ-1b222⊗α-1θb12α-5θ-1b2112α-5θb2122α-3θ-1b221=α-1θb11α-2θ-1b222εb21⊗α-1θb12α-2θ-1b221=θb1α-1θ-1b22⊗α-2θb211α-2θ-1b222=θb1α-1θ-1b22⊗1εb21=θb1θ-1b2⊗1=1⊗1εb.So ω is a left convolution inverse for λπ. On the other hand, (49)ω∗λπb=ωb1λπb2=α-4θb1111α-2θ-1b12α-3θb211α-3θ-1b221⊗α-4θb1112α-3θ-1b112α-3θb212α-3θ-1b222=α-2θb111α-4θ-1b211α-4θb212α-4θ-1b2221⊗α-2θb112α-3θ-1b12α-4θb221α-4θ-1b2222=α-2θb111α-3θ-1b2221εb21⊗α-2θb112α-3θ-1b12α-4θb221α-4θ-1b2222=α-2θb111α-2θ-1b221⊗α-2θb112α-3θ-1b12α-3θb21α-3θ-1b222=α-1θb11α-2θ-1b221⊗α-1θb12α-4θ-1b211α-4θb212α-3θ-1b222=α-1θb11α-2θ-1b221⊗α-1θb12α-2θ-1b222εb21=α-1θb11α-1θ-1b21⊗α-1θb12α-1θ-1b22=1⊗1εd.Thirdly, let j:C→B,j(c)=c⊗1, then we have (50)π⊗idΔjc=π⊗idΔβc⊗1=πβc1⊗α-2c21-1α-1λ1c22⊗β-1c210⊗λ2c22=c1εc21-1ελ1c22⊗β-1c210⊗λ2c22=c1⊗c21⊗εc221=c1⊗βc2⊗1=c1⊗jc2. Since (λπ)∗ω=1, we obtain that (51)λπ∗ωjc=λπjc1ωjc2=λc1ωjc2=εc1.So λ has a right convolution inverse ωj in Hom(C,H⊗H). On the other hand, (52)λπ∗ωjπb=λπb1ωjπb2=λπb1ωjπb2=επb1=εb1.It shows that ωjπ is a right convolution inverse for λπ in Hom(B,H⊗H); thus ωjπ=ω.
Finally, we denote thatλ′=ωj, because we have that (λπ)∗(λ′π)=(λ′π)∗(λπ)=1 in Hom(B,H⊗H) and π is a surjective Hom-coalgebra map, and we obtain λ∗λ′=λ′∗λ=1 in Hom(C,H⊗H). So λ′ is the convolution inverse of λ.


## 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(H,α) be a Hom-Hopf algebra and (B,β) a right H-Hom-module coalgebra. Let H+ denote the augmentation Hom-ideal ker ε which is a Hom-Hopf ideal, then BH+ is a Hom-coideal of B, and B/BH+ is a Hom-coalgebra with a trivial right H-Hom-module structure.Definition 9.
Let(H,α) be a Hom-Hopf algebra and (B,β) a right H-Hom-module coalgebra. Denote by C=B/BH+; if there exists a right H-Hom-module map ϕ:B→H which is convolution invertible, then B/C is said to be a cleft coextension and write (B,·,ϕ,β).By Proposition8, we get the main result of this section, which is the generalization of Theorem  2.10 in [8].Theorem 10.
The coextensionB/C is cleft if and only if B is isomorphic to a Hom-crossed coproduct C⋊λH with invertible cocycle λ.Proof.
IfB is isomorphic to the Hom-crossed coproduct C⋊λH, we prove that the coextension B/C is cleft. First, we show that ϕ:C⋊λH→H is a right H-Hom-module map. For any h,g∈H, we get (53)ϕc⊗h·g=ϕc⊗hg=εcα-1hg. On the other hand, (54)ϕc⊗h·g=ϕc·g1⊗hg2=εc·g1α-1hg2=εcα-1hg,and by Proposition 8, λ is an invertible cocycle, so ϕ is convolution invertible. Hence C⋊λH/C is cleft, so B/C is cleft too.
Conversely, if the coextensionB/C is cleft, let us denote by c¯ the image in C of the element c∈B. Define a weak coaction of H on C by (55)ρ:C⟶H⊗C,ρc¯=α-2ϕc11α-1ϕ-1c2⊗β-1c¯12.The definition is well defined. The conditions (W1), (W2), and (W3) hold which can be directly verified.
Defineλ:C→H⊗H, which is given by (56)λc¯=α-3ϕc11α-3ϕ-1c21⊗α-3ϕc12α-3ϕ-1c22, and ω:C→H⊗H is given by (57)ωc¯=α-4ϕc111α-2ϕ-1c2⊗α-4ϕc112α-3ϕ-1c12. Similar to the proof of Proposition 8, we can obtain that λ∗ω=ω∗λ=1, that is, ω is the convolution inverse of λ. With convolution inverse λ and weak coaction of H on C as the above, we can define a comultiplication Δ and a counit ε as the Definition 2. The comultiplication Δ is given by (58)Δc¯⊗h=c¯1⊗α-4c¯21-1α-3λ1c¯22α-1h1⊗β-2c¯210⊗α-2λ2c¯22α-1h2=c¯1⊗α-6ϕc2111α-5ϕ-1c212α-6ϕc2211α-6ϕ-1c2221α-1h1⊗β-3c¯2112⊗α-5ϕc2212α-5ϕ-1c2222α-1h2, and the counit ε is given by (59)εc¯⊗h=εcεh, for any c∈B, h∈H.
Recalling from [16], the isomorphism is given by (60)F:B⟶C⊗H,Fc=c¯1⊗ϕc2,with inverse (61)F-1c¯⊗h=c1·ϕ-1c2h.Now, for any c∈B, h∈H, we have (62)ΔFc=Δc¯1⊗ϕc2=c¯11⊗α-6ϕc12111α-5ϕ-1c1212α-6ϕc12211α-6ϕ-1c12221α-1ϕc21⊗β-3c¯12112⊗α-5ϕc12212α-5ϕ-1c12222α-1ϕc22=c¯11⊗α-4ϕc1211εc1221α-6ϕ-1c122221α-1ϕc21⊗β-2c¯1212⊗α-5ϕc12221α-6ϕ-1c122222α-1ϕc22=c¯11⊗α-4ϕc1211α-5ϕ-1c12221α-1ϕc21⊗β-2c¯1212⊗α-4ϕc1221α-5ϕ-1c12222α-1ϕc22=α-1c¯111⊗α-4ϕc1121α-3ϕ-1c211α-2ϕc221⊗β-2c¯1122⊗α-2ϕc12α-3ϕ-1c¯212α-2ϕc222=α-1c¯111⊗α-2ϕc1121εc2⊗β-2c¯1122⊗ϕc12=c¯11⊗α-1ϕc121⊗β-1c¯122⊗αϕc2=c¯11⊗ϕc12⊗c¯21⊗ϕc22=F⊗FΔc.So we can obtain that (F⊗F)Δ=ΔF and ε=εF. Since B is a Hom-coalgebra, C⊗H is a Hom-crossed coproduct and F 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(H,α) be a Hom-Hopf algebra and (B,·,ρ,β) a right H-Hom-Hopf module; set A=BcoH=b∈B∣ρb=βb⊗1 and C=B/BH+. Then
(1)C≃A as vector spaces, and the isomorphism is ψ:C→A,ψ(b¯)=b(0)·S(b(1)).
(2) If(B,·) is a right H-Hom-module coalgebra, then there exists a Hom-coalgebra structure on A such that C≃A as Hom-coalgebras.Proof.
(1) It is easy to see thatψ is an isomorphism of vector spaces.
(2) In order to prove thatC≃A as Hom-coalgebras, we first give the Hom-coalgebra structure on A. The comultiplication on A is given by (63)Δa=β-3a100·Sα-3a101⊗β-3a200·Sα-3a201,for any a∈A. With the above defined comultiplication, we can prove that A is a Hom-coalgebra. Furthermore we can get that ψ, which is as similar as defined in (1), is also a Hom-coalgebra morphism. So C≃A as Hom-coalgebras.Then we discuss the relation between cleft coextension and Hom-module coalgebra with the Hom-Hopf module structure.Proposition 12.
Let(B,·,ϕ,β) be a cleft coextension. One assumes that H coacts on B to the right by (64)ρb=β-2b11·α-3ϕ-1b12ϕb21⊗α-1ϕb22,then(1)
(B,ρ,β) is a right H-Hom-comodule;(2)
(B,·,ρ,β) is a right H-Hom-Hopf module.Proof.
Note thatϕ is convolution invertible right H-Hom-module map, where the convolution inverse of ϕ is denoted by ϕ-1. We can get(65)h1α-1ϕ-1b·h2=εhαϕ-1b.In fact (66)εhϕ-1β5b=ϕ-1β4b1εb2εh=ϕ-1β3b1ϕ∗ϕ-1b2·h=ϕ-1β3b1ϕb21·h1ϕ-1b22·h2=ϕ-1β2b1ϕb21h1αϕ-1b22·h2=ϕ-1βb1ϕb21αh1αϕ-1b22·h2=ϕ-1b11ϕb12αh1αϕ-1βb2·h2=εb1α2h1αϕ-1βb2·h2=α2h1αϕ-1β2b·h2.
(1) Now we prove that(B,ρ,β) is a right H-Hom-comodule. For any b∈B, (67)ρ⊗αρb=ρ⊗αβ-2b11·α-3ϕ-1b12ϕb21⊗α-1ϕb22=β-4b1111·α-5ϕ-1b12ϕb2111·α-3ϕ-1β-2b1112·α-3ϕ-1b12ϕb2112α-3ϕβ-2b112·α-3ϕ-1b12ϕb2121⊗α-1ϕβ-2b112·α-3ϕ-1b12ϕb2122⊗ϕb22=β-3b1111·α-6ϕ-1b12ϕb2111α-4ϕ-1β-2b1112·α-3ϕ-1b12ϕb2112α-3ϕβ-2b112·α-3ϕ-1b12ϕb2121⊗α-1ϕβ-2b112·α-3ϕ-1b12ϕb2122⊗ϕb22=65β-3b1111·εϕ-1b12ϕb211ϕ-1β-4b1112α-3ϕβ-2b112·α-3ϕ-1b12ϕb2121⊗α-1ϕβ-2b112·α-3ϕ-1b12ϕb2122⊗ϕb22=β-3b1111·ϕ-1β-4b1112α-3ϕβ-2b112·α-2ϕ-1b12ϕb211⊗α-1ϕβ-2b112·α-2ϕ-1b12ϕb212⊗ϕb22=β-3b1111·ϕ-1β-4b1112α-5ϕb112α-5ϕ-1b12ϕb211⊗α-3ϕb112α-3ϕ-1b12ϕb212⊗ϕb22=β-3b1111·ϕ-1β-4b1112α-6ϕb112α-5ϕ-1b12α-4ϕb211⊗α-4ϕb112α-3ϕ-1b12α-2ϕb212⊗ϕb22=β-1b11·ϕ-1β-2b12εb21α-4ϕb2211⊗α-2ϕb2212⊗ϕb22=β-1b11·ϕ-1β-2b12α-3ϕb211⊗α-1ϕb212⊗ϕb22=β-1b11·ϕ-1β-2b12α-2ϕb21⊗α-1ϕb221⊗α-1ϕb222=α⊗Δρb,and (β⊗ε)ρ(b)=b; thus (B,ρ,β) is a right H-Hom-comodule.
(2) For allb∈B and h∈H, we can calculate (68)ρb·h=β-2b·h11·α-3ϕ-1b·h12ϕb·h21⊗α-1ϕb·h22=β-2b11·h11·α-3ϕ-1b12·h12ϕb2·h21⊗α-1ϕb2·h22=β-1b11·α-3h11α-4ϕ-1b12·h12α-3ϕb2·h21⊗α-1ϕb2·h22=65β-1b11·εh1α-2ϕ-1b12α-3ϕb2·h21⊗α-1ϕb2·h22=β-1b11·α-2ϕ-1b12α-3ϕb21α-2h1⊗α-1ϕb22h2=β-2b11·α-3ϕ-1b12ϕb21·h1⊗α-1ϕb22h2=b0h1⊗b1h1;thus (B,·,ρ,β) is a right H-Hom-Hopf module.Theorem 13.
Let(H,α) be a Hom-Hopf algebra and (B,·,β) a right H-Hom-module coalgebra, if C=B/BH+, then B/C is a cleft coextension if and only if (B,·,ρ,β) is a right H-Hom-Hopf module, and there exists a convolution invertible linear map ϕ:B→H satisfying ϕ(b)=ε(b(0))β-1(b(1)).Proof.
Assume thatB is a cleft coextension, then there exists a convolution invertible Hom-module map ϕ:B→H such that εϕ=ε, so we have εϕ-1=ε. Define a map, (69)ρ:B⟶B⊗H,ρb=β-2b11·α-3ϕ-1b12ϕb21⊗α-1ϕb22,∀b∈B;clearly, ϕ(b)=ε(b(0))β-1(b(1)). By Proposition 12, we see that (B,·,ρ,β) is a right H-Hom-Hopf module.
Conversely, it is sufficient to show thatϕ is a right H-Hom-module map. Using the fact that (B,·,ρ,β) is a right H-Hom-Hopf module and ϕ(b)=ε(b(0))β-1(b(1)), for any b∈B and h∈H, we get (70)ϕb·h=εb·h0β-1b·h1=εb0εh1β-1b1α-1h2=ϕbh.The proof is complete.


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

In this section, we always assume that(H,α) is a Hom-Hopf algebra and (B,·,ϕ,β) is a cleft coextension. Based on this assumption, B can not only form a right H-Hom-comodule, but also form a right H-Hom-Hopf module. Furthermore, we obtain the factorization of Hom-coalgebra B.Now, we can obtain a Hom-coalgebra factorization for Hom-module coalgebra with the Hom-Hopf module structure.Theorem 14.
Let(H,α) be a Hom-Hopf algebra and (B,·,ρ,β) a right H-Hom-Hopf module. One assumes that (B,·,β) is a right H-Hom-module coalgebra and set A=BcoH=a∈B∣ρa=βa⊗1. The comultiplication on A⊗H can be defined (71)Δa⊗h=β-3a100·Sα-3a101⊗α-2a11α-1h1⊗β-3a200·Sα-3a201⊗α-2a21α-1h2,and the counit can be defined (72)εa⊗h=εaεh,for any a∈A, h∈H, then (A⊗H,α⊗β) is a Hom-coalgebra, and one writes it as A□·H. Moreover, B≃A□·H as Hom-coalgebras.Proof.
Since(B,·,ρ) is a right H-Hom-Hopf module, Δ is well defined and the map (73)φ:A⊗H⟶B,a⊗h⟼a·his an isomorphism of right H-Hom-Hopf module.
We first verify that(A⊗H,Δ,ε,β⊗α) is a Hom-coalgebra. It is easy to see that (ε⊗(β⊗α))Δ=((β⊗α)⊗ε)Δ. It is sufficient to show the coassociativity. For any a∈A, h∈H, we calculate as follows: (74)Δ⊗β⊗αΔa⊗h=β-6a100100·Sα-6a101222·Sα-6a100101Sα-6a101221⊗α-5a10011Sα-5a10121α-3a111α-2h11⊗β-6a100200·Sα-6a101122·Sα-6a100201Sα-6a101121⊗α-5a10021Sα-5a10111α-3a112α-2h12⊗β-2a200·Sα-2a201⊗α-1a21h2=β-5a100100·Sα-6a100101εa101221H⊗α-5a10011Sα-5a10121α-3a111α-2h11⊗β-5a100200·Sα-6a100201εa101121H⊗α-5a10021Sα-5a10111α-3a112α-2h12⊗β-2a200·Sα-2a201⊗α-1a21h2=β-5a100100·Sα-5a100101⊗α-5a10011Sα-4a1012α-3a111α-2h11⊗β-5a100200·Sα-5a100201⊗α-5a10021Sα-4a1011α-3a112α-2h12⊗β-2a200·Sα-2a201⊗α-1a21h2=β-4a10100·Sα-4a10101⊗α-4a1011Sα-4a1112α-4a1121α-2h11⊗β-4a10200·Sα-4a10201⊗α-4a1021Sα-4a1111α-4a1122α-2h12⊗β-2a200·Sα-2a201⊗α-1a21h2=β-4a10100·Sα-4a10101⊗α-3a1011Sα-5a1112α-5a1121α-2h11⊗β-4a10200·Sα-4a10201⊗α-3a1021Sα-5a1111α-5a1122α-2h12⊗β-2a200·Sα-2a201⊗α-1a21h2=β-4a10100·Sα-4a10101⊗α-3a1011εa1121α-1h11⊗β-4a10200·Sα-4a10201⊗α-3a1021Sα-4a111α-5a1122α-2h12⊗β-2a200·Sα-2a201⊗α-1a21h2=β-4a10100·Sα-4a10101⊗α-3a1011α-1h11⊗β-4a10200·Sα-4a10201⊗α-3a1021Sα-4a111α-4a112α-2h12⊗β-2a200·Sα-2a201⊗α-1a21h2=β-4a10100·Sα-4a10101⊗α-3a1011α-1h11⊗β-4a10200·Sα-4a10201⊗α-3a1021εa11α-1h12⊗β-2a200·Sα-2a201⊗α-1a21h2=β-3a1100·Sα-3a1101⊗α-2a111α-1h11⊗β-3a1200·Sα-3a1201⊗α-2a121α-1h12⊗β-2a200·Sα-2a201⊗α-1a21h2. In a similar way (75)β⊗α⊗ΔΔa⊗h=β-2a100·Sα-2a100⊗α-1a11h1⊗β-3a2100·Sα-3a2101⊗α-2a211α-1h21⊗β-3a2200·Sα-3a2201⊗α-2a221α-1h22. Thus, (A⊗H,Δ,ε,β⊗α) is a Hom-coalgebra.
Next we show thatφ is a Hom-coalgebra morphism. (76)φa⊗h1⊗φa⊗h2=β-3a100·Sα-3a101·α-2a11α-1h1⊗β-3a200·Sα-3a201·α-2a21α-1h2=β-1a10·Sα-4a111α-4a112α-1h1⊗β-1a20·Sα-4a211α-4a212α-1h2=a1·h1⊗a2·h2=φa1⊗h1⊗φa2⊗h2.Therefore, B≃A□·H as Hom-coalgebras.By the above results, we obtain the following result.Theorem 15.
Let(H,α) be a Hom-Hopf algebra and (B,·,ϕ,β) a cleft coextension, then (77)B≃C⋊λH≃A□·Has Hom-coalgebras, where C=B/BH+, and (78)A=b∈B∣β-2b11·α-3ϕ-1b12ϕb21⊗α-1ϕb22=βb⊗1.Proof.
By Theorems10 and 14, we have B≃C⋊λH and B≃A□·H as Hom-coalgebras, respectively. Since (B,·,ϕ,β) is a cleft coextension, we get (B,·,ρ,β) is a right H-Hom-Hopf module by Proposition 12, where the comodule structure is defined by (79)ρb=β-2b11·α-3ϕ-1b12ϕb21⊗α-1ϕb22.We know that (80)ψ:C⟶BcoH,b¯⟼b0·Sb1is an isomorphism of Hom-coalgebras by Lemma 11. Set (81)ψ¯=ψ⊗id,ψ¯:C⋊λH⟶A□·H.We only need to verify that ψ¯ is a Hom-coalgebra morphism. First we prove that (82)ψb¯=b1·ϕ-1b2.In fact(83)ψb¯=b0·Sb1=β-2b11·α-3ϕ-1b12ϕb21·Sα-1ϕb22=β-1b11·α-1ϕ-1b12εϕb2=b1·ϕ-1b2.On one hand, we can calculate (84)Δψ¯b¯⊗h=83Δb1·ϕ-1b2⊗h=β-3b1·ϕ-1b2100·Sα-3b1·ϕ-1b2101⊗α-2b1·ϕ-1b211α-1h1⊗β-3b1·ϕ-1b2200·Sα-3b1·ϕ-1b2201⊗α-2b1·ϕ-1b221α-1h2=β-3b1100·ϕ-1b2111·Sα-3b1101ϕ-1b2112⊗α-2b111ϕ-1b212α-1h1⊗β-3b1200·ϕ-1b2211·Sα-3b1201ϕ-1b2212⊗α-2b121ϕ-1b222α-1h2=β-2b1100·α-4ϕ-1b2111Sα-4ϕ-1b2112Sα-3b1101⊗α-2b111ϕ-1b212α-1h1⊗β-2b1200·α-4ϕ-1b2211Sα-4ϕ-1b2212Sα-3b1201⊗α-2b121ϕ-1b222α-1h2=β-2b1100·Sα-2b1101⊗α-2b111α-1ϕ-1b21α-1h1⊗β-2b1200·Sα-2b1201⊗α-2b121α-1ϕ-1b22α-1h2=β-1b110·Sα-2b1111⊗α-3b1112α-1ϕ-1b21α-1h1⊗β-1b120·Sα-2b1211⊗α-3b1212α-1ϕ-1b22α-1h2=β-3b1111·α-4ϕ-1b1112ϕb1121·Sα-3ϕb11221⊗α-4ϕb11222α-1ϕ-1b21α-1h1⊗β-3b1211·α-4ϕ-1b1212ϕb1221·Sα-3ϕb12221⊗α-4ϕb12222α-1ϕ-1b22α-1h2=β-2b1111·α-3ϕ-1b1112α-4ϕb1121Sα-5ϕb11221⊗α-4ϕb11222α-1ϕ-1b21α-1h1⊗β-2b1211·α-3ϕ-1b1212α-4ϕb1221Sα-5ϕb12221⊗α-4ϕb12222α-1ϕ-1b22α-1h2=β-2b1111·α-2ϕ-1b1112⊗α-2ϕb112α-1ϕ-1b21α-1h1⊗β-2b1211·α-2ϕ-1b1212⊗α-2ϕb122α-1ϕ-1b22α-1h2. For all b∈B, h∈H, we define ρ′:C→H⊗C by (85)ρ′b¯=b¯-1⊗b¯0=ϕb11αϕ-1b2⊗β-1b¯12,and λ:C→H⊗H is defined by (86)λb¯=λ1b¯⊗λ2b¯=ϕb1α-2ϕ-1b221⊗α-1ϕb21α-2ϕ-1b222.On the other hand, we have (87)ψ¯⊗ψ¯Δλb¯⊗h=ψ¯⊗ψ¯b¯1⊗α-4b¯21-1α-3λ1b¯22α-1h1⊗β-2b¯210⊗α-2λ2b¯22α-1h2=ψ¯⊗ψ¯b¯1⊗α-4ϕb2111αϕ-1b212α-3ϕb221α-2ϕ-1b22221α-1h1⊗β-3b¯2112⊗α-2α-1ϕb2221α-2ϕ-1b22222α-1h2=ψ¯⊗ψ¯b¯1⊗α-3ϕb2111α-4ϕ-1b212α-4ϕb221α-5ϕ-1b22221α-1h1⊗β-3b¯2112⊗α-2α-1ϕb2221α-2ϕ-1b22222α-1h2=ψ¯⊗ψ¯b¯1⊗α-2ϕb211εb221α-4ϕ-1b22221α-1h1⊗β-2b¯212⊗α-2α-1ϕb2221α-2ϕ-1b22222α-1h2=ψ¯⊗ψ¯b¯1⊗α-2ϕb211α-3ϕ-1b2221α-1h1⊗β-2b¯212⊗α-2ϕb221α-1ϕ-1b2222α-1h2=83b11·ϕ-1b12⊗α-2ϕb211α-3ϕ-1b2221α-1h1⊗β-2b2121·ϕ-1b2122⊗α-2ϕb221α-1ϕ-1b2222α-1h2=β-2b1111·α-2ϕ-1b1112⊗α-2ϕb112α-1ϕ-1b21α-1h1⊗β-2b1211·α-2ϕ-1b1212⊗α-2ϕb122α-1ϕ-1b22α-1h2.Therefore, B≃C⋊λH≃A□·H as Hom-coalgebras.



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*Source: 4653172-2017-10-29.xml*