# New BraidedT-Categories over Weak Crossed Hopf Group Coalgebras

Authors: Xuan Zhou; Tao Yang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

LetH be a weak crossed Hopf group coalgebra over group π; we first introduce a kind of new α-Yetter-Drinfel’d module categories 𝒲𝒴𝒟α(H) for α∈π and use it to construct a braided T-category 𝒲𝒴𝒟(H). As an application, we give the concept of a Long dimodule category H𝒲ℒH for a weak crossed Hopf group coalgebra H with quasitriangular and coquasitriangular structures and obtain that H𝒲ℒH is a braided T-category by translating it into a weak Yetter-Drinfel'd module subcategory 𝒲𝒴𝒟(H⊗H).


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## Body

## 1. Introduction

Braided crossed categories over a groupπ (i.e., braided T-categories), introduced by Turaev [1] in the study of 3-dimensionalhomotopy quantum field theories, are braided monoidal categories in Freyd-Yetter categories of crossed π-sets [2]. Such categories play an important role in the construction of homotopy invariants. By using braided T-categories,Virelizier [3, 4] constructed Hennings-type invariants of flat group bundles over complements of links in the 3-sphere.  Braided T-categories  also provide suitable mathematical formalism to describe the orbifold models of rational conformal field theory (see [5]).The methods of constructing braidedT-categories can be found in [5–8]. Especially, in [8], Zunino gave the definition of α-Yetter-Drinfel’d modules over Hopf group coalgebras and constructed a braided T-category, then proved that both the category of Yetter-Drinfel’d modules 𝒴𝒟(H) and the center of the category of representations of H as well as the category of representations of the quantum double of H are isomorphic as braided T-categories.Furthermore, in [6], Wang considered the dual setting of Zunino’s partial results, formed the category of Long dimodules over Hopf group algebras, and proved that the category is a braided T-subcategory of Yetter-Drinfel’d category 𝒴𝒟(H⊗B).Weak multiplier Hopf algebras, as a further development of the notion of the well-known multiplier Hopf algebras [9], were introduced by Van Daele and Wang [10]. Examples of such weak multiplier Hopf algebras can be constructed from weak Hopf group coalgebras [10, 11]. Furthermore, the concepts of weak Hopf group coalgebras are also regard as a natural generalization of weak Hopf algebras [12, 13] and Hopf group coalgebras [14].In this paper, we mainly generalize the above constructions shown in [6, 8], replacing their Hopf group coalgebras (or Hopf group algebras) by weak crossed Hopf group coalgebras [11] and provide new examples of braided T-categories.This paper is organized as follows. In Section1, we recall definitions and properties related to braided T-categories and weak crossed Hopf group coalgebras.In Section2, let H be a weak crossed Hopf group coalgebra over group π; α is a fixed element in π. We first introduce the concept of a (left-right) weak α-Yetter-Drinfel’d module and define the category 𝒲𝒴𝒟(H)=∐α∈π‍𝒲𝒴𝒟α(H), where 𝒲𝒴𝒟α(H) is the category of (left-right) weak α-Yetter-Drinfel’d modules. Then, we show that the category 𝒲𝒴𝒟(H) is a braided T-category.In Section3, we introduce a (left-right) weak α-Long dimodule category 𝒲ℒαHH for a weak crossed Hopf group coalgebra H. Then, we obtain a new category 𝒲ℒHH=∐α∈π𝒲ℒαHH‍ and show that as H is a quasitriangular and coquasitriangular weak crossed Hopf group coalgebra, then 𝒲ℒHH is a braided T-subcategory of Yetter-Drinfel’d category 𝒲𝒴𝒟(H⊗H).


## 2. Preliminary

Throughout the paper, letπ be a group with the unit 1 and let k be a field. All algebras, vector spaces, and so forth are supposed to be over k. We use the Sweedler-type notation [15] for the comultiplication and coaction, t for the flip map, and id for the identity map. In the section, we will recall some basic definitions and results related to our paper.


### 2.1. Weak Crossed Hopf Group Coalgebras

Recall from Turaev and Virelizier (see [1, 14]) that a group coalgebra over π is a family of k-spaces C={Cα}α∈π together with a family of k-linear maps Δ={Δα,β:Cαβ→Cα⊗Cβ}α,β∈π (called a comultiplication) and a k-linear map ε:C1→k (called a counit), such that Δ is coassociative in the sense that
(1)(Δα,β⊗idCγ)Δαβ,γ=(idCα⊗Δβ,γ)Δα,βγ,∀α,β,γ∈π.(idCα⊗ε)Δα,1=idCα=(ε⊗idCα)Δ1,α,∀α∈π.We use the Sweedler-type notation (see [14]) for a comultiplication; that is, we write
(2)Δα,β(c)=c(1,α)⊗c(2,β),foranyα,β∈π,c∈Cαβ.Recall from Van Daele and Wang (see [11]) that a weak semi-Hopf group coalgebra H={Hα,mα,1α,Δ,ε}α∈π is a family of algebras {Hα,mα,1α}α∈π and at the same time a group coalgebra {Hα,Δ={Δα,β},ε}α,β∈π, such that the following conditions hold.(i)
The comultiplicationΔα,β:Cαβ→Cα⊗Cβ is a homomorphism of algebras (not necessary unit preserving) such that
(3)(Δα,β⊗idHγ)Δαβ,γ(1αβγ)=(Δα,β(1αβ)⊗1γ)(1α⊗Δβ,γ(1βγ)),(Δα,β⊗idHγ)Δαβ,γ(1αβγ)=(1α⊗Δβ,γ(1βγ))(Δα,β(1αβ)⊗1γ),    for all α,β,γ∈π.(ii)
The counitε:H1→k is a k-linear map satisfying the identity
(4)ε(gxh)=ε(gx(2,1))ε(x(1,1)h)=ε(gx(1,1))ε(x(2,1)h),   for all g,h,x∈H1.A weak Hopf group coalgebra overπ is a weak semi-Hopf group coalgebra H={Hα,mα,1α,Δ,ε}α∈π endowed with a family of k-linear maps S={Sα:Hα→Hα-1}α∈π (called an antipode) satisfying the following equations:
(5)mα(Sα-1⊗idHα)Δα-1,α(h)=1(1,α)ε(h1(2,1)),mα(idHα⊗Sα-1)Δα,α-1(h)=ε(1(1,1)h)1(2,α),Sα(g(1,α))g(2,α-1)Sα(g(3,α))=Sα(g),
for all h∈H1,g∈Hα,  and α∈π.LetH be a weak Hopf group coalgebra. Define a family of linear maps εt={εαt:H1→Hα}α∈π and εs={εαs:H1→Hα}α∈π by the formulae
(6)εαt(h)=ε(1(1,1)h)1(2,α)=mα(idHα⊗Sα-1)Δα,α-1(h),εαs(h)=1(1,α)ε(h1(2,1))=mα(Sα-1⊗idHα)Δα-1,α(h),
for any h∈H1, where εt and εs are called the π-target and π-source counital maps.By Van Daele and Wang (see [11]), let H be a weak semi-Hopf group coalgebra. Then, we have the following equations:(1)
ε(gh)=ε(gε1t(h)), ε(gh)=ε(ε1s(g)h),  for all g,h∈H1,(2)
x(1,α)⊗εβt(x(2,1))=1(1,α)x⊗1(2,β),  for  all x∈Hα,α,β∈π,(3)
εβs(x(1,1))⊗x(2,α)=1(1,β)⊗x1(2,α), for  all x∈Hα,α,β∈π,(4)
εαt(ε1t(x)y)=εαt(x)εαt(y), εαs(xε1s(y))=εαs(x)εαs(y),  for all x,y∈H1.Similarly, for anyα∈π and h∈H1, define ε~αt(h)=ε(h1(1,1))1(2,α), ε~αs(h)=1(1,α)ε(1(2,1)h). Then, we have(1)
ε~αs(h(1,1))⊗h(2,β)=1(1,a)⊗1(2,β)h, for all h∈Hβ,α,β∈π,(2)
x(1,α)⊗ε~αt(x(2,1))=x1(1,α)⊗1(2,β),  for  all x∈Hα,α,β∈π.A weak Hopf group coalgebraH={Hα,mα,1α,Δ,ε,S}α∈π is called a weak crossed Hopf group coalgebra if it is endowed with a family of algebra isomorphisms φ={φα:Hβ→Hαβα-1}α,β∈π (called a crossing) such that (φα⊗φα)∘Δβ,γ=Δαβα-1,αγα-1∘φα, ε∘φα=ε, and φαβ=φα∘φβ for all α,β,γ∈π.IfH is crossed with the crossing φ={φα}α∈π, then we have
(7)φβ∘εαs=εβαβ-1s∘φβ,φβ∘εαt=εβαβ-1t∘φβ,∀α,β∈π.A quasitriangular weak crossed Hopf group coalgebra overπ is a pair (H,R) where H is a weak crossed Hopf group coalgebra together with a family of maps R={Rα,β∈Δ¯β-1,α-1cop(1αβ)(Hα⊗Hβ)Δα,β(1αβ)} satisfying the following conditions:(1)
Rα,βΔα,β(h)=Δ¯β-1,α-1cop(h)Rα,β, for all h∈Hαβ,α,β∈π,(2)
(idHα⊗Δβ,γ)(Rα,βγ)=(Rα,γ)1β3(Rα,β)12γ,  for all α,β,γ∈π,(3)
(Δ¯α,β⊗idHγ)(Rβ-1α-1,γ)=(Rα-1,γ)1β-13(Rβ-1,γ)α-123,  for all α,β,γ∈π, where Δ¯α,β=(φβ⊗idHβ-1)∘Δβ-1α-1β,β-1, Δ¯α,βcop=tHα-1,Hβ-1∘(φβ⊗idHβ-1)∘Δβ-1α-1β,β-1 for all α,β∈π, and such that there exists a family of R¯={R¯α,β∈Δα,β(1αβ)(Hα⊗Hβ)Δ¯β-1,α-1cop(1αβ)} with
(8)Rα,βR¯α,β=Δβ-1,α-1cop(1αβ),R¯α,βRα,β=Δα,β(1αβ),(φβ⊗φβ)(Rα,γ)=Rβαβ-1,βγβ-1,
for all α,β,γ∈π. In this paper, we denote Rα,β=aα⊗bβ.Recall from [16] that a coquasitriangular weak Hopf group coalgebra (H,σ) consists of a weak Hopf group coalgebra H={Hα,mα,1α,Δ,ε,S}α∈π and a map σ:H1⊗H1→k satisfying
(9)σ(h(1,1),g(1,1))h(2,α)g(2,α)=g(1,α)h(1,α)σ(h(2,1),g(2,1)),σ(a,bc)=σ(a(1,1),c)σ(a(2,1),b),σ(ab,c)=σ(a,c(1,1))σ(b,c(2,1)),ε(a(1,1)b(1,1))σ(b(2,1),a(2,1))ε(b(3,1)a(3,1))=σ(b,a),
and there exists σ-1:H1⊗H1→k such that
(10)σ(a(1,1),b(1,1))σ-1(a(2,1),b(2,1))=ε(ba),σ-1(a(1,1),b(1,1))σ(a(2,1),b(2,1))=ε(ab),ε(a(1,1)b(1,1))σ-1(a(2,1),b(2,1))ε(b(3,1)a(3,1))=σ-1(a,b),
for all h,g∈Hα, a,b,c∈H1, where σ-1 is called a weak inverse of σ.


### 2.2.   BraidedT-Categories

We recall that a monoidal category𝒞 is called a crossed category over group π if it consists of the following data.(1)
A family of subcategories{𝒞α}α∈π such that 𝒞 is a disjoint union of this family and such that for any U∈𝒞α and V∈𝒞β, U⊗V∈𝒞αβ. Here, the subcategory 𝒞α is called the αth component of 𝒞.(2)
A group homomorphismψ:π→aut(𝒞):β↦ψβ, the conjugation, (where aut(𝒞) is the group of invertible strict tensor functors from 𝒞 to itself) such that ψβ(𝒞α)=𝒞βαβ-1 for any α,β∈π. Here, the functors ψβ are called conjugation isomorphisms.We will use the Turaev’s left index notation in [1]: for any object U∈𝒞α, V,W∈𝒞β and any morphism f:V→W in 𝒞β, we set
(11)VU=ψα(V)∈𝒞αβα-1,fU=ψα(f):VU⟶WU.Recall form [1] that a braided T-category is a crossed category 𝒞 endowed with braiding, that is a family of isomorphisms,
(12)c={cU,V∈𝒞(U⊗V,(VU)⊗U)}U,V∈𝒞
satisfying the following conditions:(1)
for any morphismf∈𝒞α(U,U′) with α∈π, g∈𝒞(V,V′), we have
(13)((αg)⊗f)∘cU,V=cU′,V′∘(f⊗g);(2)
for allU,V,W∈𝒞, we have
(14)cU⊗V,W=aU⊗VW,U,V∘(cU,VW⊗idV)∘aU,VW,V-1∘(idU⊗cV,W)∘aU,V,W,cU,V⊗W=aUV,UW,U-1∘(idVU⊗cU,W)∘aUV,U,W∘(cU,V⊗idW)∘aU,V,W-1;(3)
for anyU,V∈𝒞, α∈π,
(15)ψα(cU,V)=cψα(U),ψα(V).


## 2.1. Weak Crossed Hopf Group Coalgebras

Recall from Turaev and Virelizier (see [1, 14]) that a group coalgebra over π is a family of k-spaces C={Cα}α∈π together with a family of k-linear maps Δ={Δα,β:Cαβ→Cα⊗Cβ}α,β∈π (called a comultiplication) and a k-linear map ε:C1→k (called a counit), such that Δ is coassociative in the sense that
(1)(Δα,β⊗idCγ)Δαβ,γ=(idCα⊗Δβ,γ)Δα,βγ,∀α,β,γ∈π.(idCα⊗ε)Δα,1=idCα=(ε⊗idCα)Δ1,α,∀α∈π.We use the Sweedler-type notation (see [14]) for a comultiplication; that is, we write
(2)Δα,β(c)=c(1,α)⊗c(2,β),foranyα,β∈π,c∈Cαβ.Recall from Van Daele and Wang (see [11]) that a weak semi-Hopf group coalgebra H={Hα,mα,1α,Δ,ε}α∈π is a family of algebras {Hα,mα,1α}α∈π and at the same time a group coalgebra {Hα,Δ={Δα,β},ε}α,β∈π, such that the following conditions hold.(i)
The comultiplicationΔα,β:Cαβ→Cα⊗Cβ is a homomorphism of algebras (not necessary unit preserving) such that
(3)(Δα,β⊗idHγ)Δαβ,γ(1αβγ)=(Δα,β(1αβ)⊗1γ)(1α⊗Δβ,γ(1βγ)),(Δα,β⊗idHγ)Δαβ,γ(1αβγ)=(1α⊗Δβ,γ(1βγ))(Δα,β(1αβ)⊗1γ),    for all α,β,γ∈π.(ii)
The counitε:H1→k is a k-linear map satisfying the identity
(4)ε(gxh)=ε(gx(2,1))ε(x(1,1)h)=ε(gx(1,1))ε(x(2,1)h),   for all g,h,x∈H1.A weak Hopf group coalgebra overπ is a weak semi-Hopf group coalgebra H={Hα,mα,1α,Δ,ε}α∈π endowed with a family of k-linear maps S={Sα:Hα→Hα-1}α∈π (called an antipode) satisfying the following equations:
(5)mα(Sα-1⊗idHα)Δα-1,α(h)=1(1,α)ε(h1(2,1)),mα(idHα⊗Sα-1)Δα,α-1(h)=ε(1(1,1)h)1(2,α),Sα(g(1,α))g(2,α-1)Sα(g(3,α))=Sα(g),
for all h∈H1,g∈Hα,  and α∈π.LetH be a weak Hopf group coalgebra. Define a family of linear maps εt={εαt:H1→Hα}α∈π and εs={εαs:H1→Hα}α∈π by the formulae
(6)εαt(h)=ε(1(1,1)h)1(2,α)=mα(idHα⊗Sα-1)Δα,α-1(h),εαs(h)=1(1,α)ε(h1(2,1))=mα(Sα-1⊗idHα)Δα-1,α(h),
for any h∈H1, where εt and εs are called the π-target and π-source counital maps.By Van Daele and Wang (see [11]), let H be a weak semi-Hopf group coalgebra. Then, we have the following equations:(1)
ε(gh)=ε(gε1t(h)), ε(gh)=ε(ε1s(g)h),  for all g,h∈H1,(2)
x(1,α)⊗εβt(x(2,1))=1(1,α)x⊗1(2,β),  for  all x∈Hα,α,β∈π,(3)
εβs(x(1,1))⊗x(2,α)=1(1,β)⊗x1(2,α), for  all x∈Hα,α,β∈π,(4)
εαt(ε1t(x)y)=εαt(x)εαt(y), εαs(xε1s(y))=εαs(x)εαs(y),  for all x,y∈H1.Similarly, for anyα∈π and h∈H1, define ε~αt(h)=ε(h1(1,1))1(2,α), ε~αs(h)=1(1,α)ε(1(2,1)h). Then, we have(1)
ε~αs(h(1,1))⊗h(2,β)=1(1,a)⊗1(2,β)h, for all h∈Hβ,α,β∈π,(2)
x(1,α)⊗ε~αt(x(2,1))=x1(1,α)⊗1(2,β),  for  all x∈Hα,α,β∈π.A weak Hopf group coalgebraH={Hα,mα,1α,Δ,ε,S}α∈π is called a weak crossed Hopf group coalgebra if it is endowed with a family of algebra isomorphisms φ={φα:Hβ→Hαβα-1}α,β∈π (called a crossing) such that (φα⊗φα)∘Δβ,γ=Δαβα-1,αγα-1∘φα, ε∘φα=ε, and φαβ=φα∘φβ for all α,β,γ∈π.IfH is crossed with the crossing φ={φα}α∈π, then we have
(7)φβ∘εαs=εβαβ-1s∘φβ,φβ∘εαt=εβαβ-1t∘φβ,∀α,β∈π.A quasitriangular weak crossed Hopf group coalgebra overπ is a pair (H,R) where H is a weak crossed Hopf group coalgebra together with a family of maps R={Rα,β∈Δ¯β-1,α-1cop(1αβ)(Hα⊗Hβ)Δα,β(1αβ)} satisfying the following conditions:(1)
Rα,βΔα,β(h)=Δ¯β-1,α-1cop(h)Rα,β, for all h∈Hαβ,α,β∈π,(2)
(idHα⊗Δβ,γ)(Rα,βγ)=(Rα,γ)1β3(Rα,β)12γ,  for all α,β,γ∈π,(3)
(Δ¯α,β⊗idHγ)(Rβ-1α-1,γ)=(Rα-1,γ)1β-13(Rβ-1,γ)α-123,  for all α,β,γ∈π, where Δ¯α,β=(φβ⊗idHβ-1)∘Δβ-1α-1β,β-1, Δ¯α,βcop=tHα-1,Hβ-1∘(φβ⊗idHβ-1)∘Δβ-1α-1β,β-1 for all α,β∈π, and such that there exists a family of R¯={R¯α,β∈Δα,β(1αβ)(Hα⊗Hβ)Δ¯β-1,α-1cop(1αβ)} with
(8)Rα,βR¯α,β=Δβ-1,α-1cop(1αβ),R¯α,βRα,β=Δα,β(1αβ),(φβ⊗φβ)(Rα,γ)=Rβαβ-1,βγβ-1,
for all α,β,γ∈π. In this paper, we denote Rα,β=aα⊗bβ.Recall from [16] that a coquasitriangular weak Hopf group coalgebra (H,σ) consists of a weak Hopf group coalgebra H={Hα,mα,1α,Δ,ε,S}α∈π and a map σ:H1⊗H1→k satisfying
(9)σ(h(1,1),g(1,1))h(2,α)g(2,α)=g(1,α)h(1,α)σ(h(2,1),g(2,1)),σ(a,bc)=σ(a(1,1),c)σ(a(2,1),b),σ(ab,c)=σ(a,c(1,1))σ(b,c(2,1)),ε(a(1,1)b(1,1))σ(b(2,1),a(2,1))ε(b(3,1)a(3,1))=σ(b,a),
and there exists σ-1:H1⊗H1→k such that
(10)σ(a(1,1),b(1,1))σ-1(a(2,1),b(2,1))=ε(ba),σ-1(a(1,1),b(1,1))σ(a(2,1),b(2,1))=ε(ab),ε(a(1,1)b(1,1))σ-1(a(2,1),b(2,1))ε(b(3,1)a(3,1))=σ-1(a,b),
for all h,g∈Hα, a,b,c∈H1, where σ-1 is called a weak inverse of σ.


## 2.2.   BraidedT-Categories

We recall that a monoidal category𝒞 is called a crossed category over group π if it consists of the following data.(1)
A family of subcategories{𝒞α}α∈π such that 𝒞 is a disjoint union of this family and such that for any U∈𝒞α and V∈𝒞β, U⊗V∈𝒞αβ. Here, the subcategory 𝒞α is called the αth component of 𝒞.(2)
A group homomorphismψ:π→aut(𝒞):β↦ψβ, the conjugation, (where aut(𝒞) is the group of invertible strict tensor functors from 𝒞 to itself) such that ψβ(𝒞α)=𝒞βαβ-1 for any α,β∈π. Here, the functors ψβ are called conjugation isomorphisms.We will use the Turaev’s left index notation in [1]: for any object U∈𝒞α, V,W∈𝒞β and any morphism f:V→W in 𝒞β, we set
(11)VU=ψα(V)∈𝒞αβα-1,fU=ψα(f):VU⟶WU.Recall form [1] that a braided T-category is a crossed category 𝒞 endowed with braiding, that is a family of isomorphisms,
(12)c={cU,V∈𝒞(U⊗V,(VU)⊗U)}U,V∈𝒞
satisfying the following conditions:(1)
for any morphismf∈𝒞α(U,U′) with α∈π, g∈𝒞(V,V′), we have
(13)((αg)⊗f)∘cU,V=cU′,V′∘(f⊗g);(2)
for allU,V,W∈𝒞, we have
(14)cU⊗V,W=aU⊗VW,U,V∘(cU,VW⊗idV)∘aU,VW,V-1∘(idU⊗cV,W)∘aU,V,W,cU,V⊗W=aUV,UW,U-1∘(idVU⊗cU,W)∘aUV,U,W∘(cU,V⊗idW)∘aU,V,W-1;(3)
for anyU,V∈𝒞, α∈π,
(15)ψα(cU,V)=cψα(U),ψα(V).


## 3. Yetter-Drinfel’d Categories for Weak Crossed Hopf Group Coalgebras

In this section, we first introduce the definition of weakα-Yetter-Drinfel’d modules over a weak crossed Hopf group coalgebra H and then use it to construct a class of braided T-categories.Definition 1.
LetH be a weak crossed Hopf group coalgebra over group π and let α be a fixed element in π. A (left-right) weak α-Yetter-Drinfel’d module, or simply a 𝒲𝒴𝒟α-module, is a couple V=(V,ρV={ρλV}λ∈π), where V is a left Hα-module and, for any λ∈π, ρλV:V→V⊗Hλ is a k-linear morphism, such that(1)
V is coassociative in the sense that, for any λ1,λ2∈π, we have
(16)(idV⊗Δλ1,λ2)∘ρλ1λ2V=(ρλ1V⊗idHλ2)∘ρλ2V;(2)
V is counitary in the sense that
(17)(idV⊗ε)∘ρ1V=idV;(3)
V is crossed in the sense that, for any λ∈π, h∈Hα,
(18)ρλV(h·v)=h(2,α)·v(0)⊗h(3,λ)v(1,λ)S-1φα-1(h(1,αλ-1α-1)), where ρλV(v)=v(0)⊗v(1,λ).Given two𝒲𝒴𝒟α-modules (V,ρV) and (W,ρW), a morphism f:(V,ρV)→(W,ρW) of this two 𝒲𝒴𝒟α-modulesis an Hα-linear map f:V⇀W, such that, for any λ∈π,
(19)ρλW∘f=(f⊗idHλ)∘ρλV.Then, we can form the category𝒲𝒴𝒟α(H) of 𝒲𝒴𝒟α-modules where the composition of morphisms of 𝒲𝒴𝒟α-modules is the standard composition of the underlying linear maps.Proposition 2.
Equation (18) is equivalent to the following equations:
(20)h(1,α)·v(0)⊗h(2,λ)v(1,λ)=(h(2,α)·v)(0)⊗(h(2,α)·v)(1,λ)φα-1(h(1,αλα-1)),(21)ρλV(v)=v(0)⊗v(1,λ)∈V⊗tαλHλ:=Δα,λ(1αλ)·(V⊗Hλ),
for any v∈V, h∈Hαλ.Proof.
Assume that (20) and (21) hold for all h∈Hαλ, v∈V. We compute
(22)h(2,α)·v(0)⊗h(3,λ)v(1,λ)S-1φα-1(h(1,αλ-1α-1))=(h(3,α)·v)(0)⊗(h(3,α)·v)(1,λ)×φα-1(h(2,αλα-1))S-1φα-1(h(1,αλ-1α-1))=(h(3,α)·v)(0)⊗(h(3,α)·v)(1,λ)×φα-1(h(2,αλα-1)S-1(h(1,αλ-1α-1)))=(h(2,α)·v)(0)⊗(h(2,α)·v)(1,λ)×φα-1S-1(εαλ-1α-1t(h(1,1)))=(1(2,α)′h(2,α)·v)(0)⊗(1(2,α)′h(2,α)·v)(1,λ)×φα-1(1(1,αλα-1))×ε(1(2,1)1(1,1)′h(1,1))=(1(2,α)h·v)(0)⊗(1(2,α)h·v)(1,λ)φα-1(1(1,αλα-1))=1(1,α)·(h·v)(0)⊗1(2,λ)(h·v)(1,λ)=(h·v)(0)⊗(h·v)(1,λ)
as required.
Conversely, suppose thatV is crossed in the sense of (18). We first note that
(23)v(0)⊗v(1,λ)=1(2,α)·v(0)⊗1(3,λ)v(1,λ)S-1φα-1(1(1,αλ-1α-1))=1(1,α)′1(2,α)·v(0)⊗1(2,λ)′v(1,λ)S-1×φα-1(1(1,αλ-1α-1))=1(1,α)′·(1(2,α)·v(0))⊗1(2,λ)′[v(1,λ)S-1φα-1(1(1,αλ-1α-1))]∈V⊗tαλHλ.
To show that (21) is satisfied, for all h∈Hαλ, we do the following calculations:
(24)(h(2,α)·v)(0)⊗(h(2,α)·v)(1,λ)φα-1(h(1,αλα-1))=h(3,α)·v(0)⊗h(4,λ)v(1,λ)S-1φα-1(h(2,αλ-1α-1))×φα-1(h(1,αλα-1))=h(2,α)·v(0)⊗h(3,λ)v(1,λ)φα-1S-1(εαλ-1α-1s(h(1,1)))=h(2,α)1(2,α)′·v(0)⊗h(3,λ)v(1,λ)φα-1S-1(1(1,αλ-1α-1))×ε(h(1,1)1(1,1)′1(2,1))=h(1,α)1(2,α)·v(0)⊗h(2,λ)v(1,λ)φα-1S-1(1(1,αλ-1α-1))=h(1,α)1(1,α)′1(2,α)·v(0)⊗h(2,λ)1(2,λ)′v(1,λ)S-1×φα-1(1(1,αλ-1α-1))=h(1,α)1(2,α)·v(0)⊗h(2,λ)1(3,λ)v(1,λ)S-1×φα-1(1(1,αλ-1α-1))=h(1,α)·v(0)⊗h(2,λ)v(1,λ).
This completes the proof.Proposition 3.
If(V,ρV)∈𝒲𝒴𝒟α(H), (W,ρW)∈𝒲𝒴𝒟β(H), then V⊗tαβW = Δα,β(1αβ)·(V⊗W)∈𝒲𝒴𝒟αβ(H) with the action and coaction structures as follows:
(25)h·(v⊗w)=h(1,α)·v⊗h(2,β)·w,ρλV⊗tαβW(v⊗w)=v(0)⊗w(0)⊗w(1,λ)φβ-1(v(1,βλβ-1)),
for all  h∈Hαβ, λ∈π, v⊗w∈V⊗tαβW.Proof.
It is easy to prove thatV⊗tαβW is a left Hαβ-module, and the proof of coassociativity of V⊗tαβW is similar to the Hopf group coalgebra case. For all v⊗w∈V⊗tαβW, we have
(26)(idV⊗tαβW⊗ε)∘ρ1V⊗tαβW(v⊗w)=ε(w(1,1)φβ-1(1′)(2,1))×ε(φβ-1(1′)(1,1)φβ-1(v(1,1)))v(0)⊗w(0)=ε((1(2,β)·w)(1,1)φβ-1(1(1,1))φβ-1(1(2,1)′))×ε(φβ-1(1(1,1)′)φβ-1(v(1,1)))v(0)⊗(1(2,β)·w)(0)=ε(1(5,1)w(1,1)S-1φβ-1(1(3,1))φβ-1(1(2,1)))×ε(1(1,1)v(1,1))v(0)⊗1(4,β)·w(0)=ε(1(4,1)w(1,1)φβ-1S-1(ε1s(1(2,1))))ε(1(1,1)v(1,1))v(0)⊗1(3,β)·w(0)=ε(1(4,1)w(1,1)ε1tS-1φβ-1(1(2,1)))ε(1(1,1)v(1,1))v(0)⊗1(3,β)·w(0)=ε((1(2,β)·w)(1,1))ε(1(1,1)v(1,1))v(0)⊗(1(2,β)·w)(0)=ε((1(2,α)·v)(1,1)φα-1(1(1,1)))(1(2,α)·v)(0)⊗1(3,β)·w=ε(1(3,1)v(1,1)φα-1S-1ε1s(1(1,1)))1(2,α)·v(0)⊗1(4,β)·w=ε((1(1,α)·v)(1,1))(1(1,α)·v)(0)⊗1(2,β)·w=v⊗w.
This shows that V⊗tαβW is satisfing counitary condition (17).
Then, we check the equivalent form of crossed conditions (20) and (21). In fact, for all h∈Hαβλ, v⊗w∈V⊗tαβW, we have
(27)(h(2,αβ)·(v⊗w))(0)⊗(h(2,αβ)·(v⊗w))(1,λ)φ(αβ)-1(h(1,αβλβ-1α-1))=(h(2,α)·v)(0)⊗(h(3,β)·w)(0)⊗(h(3,β)·w)(1,λ)φβ-1((h(2,α)·v)(1,βλβ-1))φβ-1α-1(h(1,αβλβ-1α-1))=h(3,α)·v(0)⊗h(6,β)·w(0)⊗h(7,λ)w(1,λ)S-1φβ-1(h(5,βλ-1β-1))φβ-1(h(4,βλβ-1)v(1,βλβ-1))φβ-1φα-1S-1×(h(2,αβλ-1β-1α-1))φβ-1α-1(h(1,αβλβ-1α-1))=h(3,α)·v(0)⊗h(6,β)·w(0)⊗h(7,λ)w(1,λ)S-1φβ-1(h(5,βλ-1β-1))φβ-1(h(4,βλβ-1)v(1,βλβ-1))φβ-1α-1S-1×(S(h(1,αβλβ-1α-1))h(2,αβλ-1β-1α-1))=h(1,α)1(2,α)·v(0)⊗h(4,β)·w(0)⊗h(5,λ)w(1,λ)S-1φβ-1(h(3,βλ-1β-1))φβ-1(h(2,βλβ-1)v(1,βλβ-1))φβ-1α-1S-1(1(1,αβλ-1β-1α-1))=h(1,α)1(2,α)·v(0)⊗h(4,β)·w(0)⊗h(5,λ)w(1,λ)S-1φβ-1(h(3,βλ-1β-1))φβ-1(h(2,βλβ-1)1(3,βλβ-1)v(1,βλβ-1)S-1×φα-1(1(1,αβλ-1β-1α-1)))=h(1,α)·v(0)⊗h(3,β)·w(0)⊗h(4,λ)w(1,λ)×φβ-1S-1(εβλ-1β-1s(h(2,1)))φβ-1(v(1,βλβ-1))=h(1,α)·v(0)⊗h(2,β)1(1,β)′1(2,β)·w(0)⊗h(3,λ)1(2,λ)′w(1,λ)S-1φβ-1(1(1,βλ-1β-1)),φβ-1(v(1,βλβ-1))=h(1,α)·v(0)⊗h(2,β)·w(0)⊗h(3,λ)w(1,λ)φβ-1(v(1,βλβ-1))=h(1,αβ)·(v⊗w)(0)⊗h(2,λ)(v⊗w)(1,λ),1(1,αβ)·(v⊗w)(0)⊗1(2,λ)(v⊗w)(1,λ)=1(1,αβ)·(v(0)⊗w(0))⊗1(2,λ)w(1,λ)φβ-1(v(1,βλβ-1))=1(1,α)·v(0)⊗1(2,β)·w(0)⊗1(3,λ)w(1,λ)φβ-1(v(1,βλβ-1))=1(1,α)·v(0)⊗1(2,β)1(1,β)′·w(0)⊗1(2,λ)′w(1,λ)×φβ-1(v(1,βλβ-1))=1(1,α)·v(0)⊗1(2,β)·w(0)⊗w(1,λ)φβ-1(v(1,βλβ-1))=v(0)⊗w(0)⊗w(1,λ)φβ-1(v(1,βλβ-1))=(v⊗w)(0)⊗(v⊗w)(1,λ).
This finishes the proof.Proposition 4.
Let(V,ρV)∈𝒲𝒴𝒟α(H), and let β∈π. Set βV=V as vector space, with action and coaction structures defined by
(28)h⊳vβ=(φβ-1(h)·v)β,∀h∈Hβαβ-1,vβ∈Vβ,ρλβV(vβ)=(v(0))β⊗φβ(v(1,β-1λβ))≔v〈0〉⊗v〈1,λ〉,∀vβ∈Vβ.
Then, Vβ∈𝒲𝒴𝒟βαβ-1(H).Proof.
Obviously,Vβ is a left Hβαβ-1-module, and conditions (16) and (17) are straightforward. Then, it remains to show that conditions (20) and (21) hold. For all vβ∈Vβ, we have
(29)1(1,βαβ-1)⊳v〈0〉⊗1(2,λ)v〈1,λ〉=(φβ-1(1(1,βαβ-1))·v(0))β⊗1(2,λ)φβ(v(1,β-1λβ))=(1(1,α)·v(0))β⊗φβ(1(2,λ))φβ(v(1,β-1λβ))=v〈0〉⊗v〈1,λ〉.
Next, for allh∈Hβαβ-1λ, vβ∈Vβ, we get
(30)(h(2,βαβ-1)⊳vβ)〈0〉⊗(h(2,βαβ-1)⊳vβ)〈1,λ〉×φβα-1β-1(h(1,βαβ-1λβα-1β-1))=((φβ-1(h(2,βαβ-1))·v)(0))β⊗φβ((φβ-1(h(2,βαβ-1))·v)(1,β-1λβ)),φβα-1β-1(h(1,βαβ-1λβα-1β-1))=((φβ-1(h)(2,α)·v)(0))β⊗φβ((φβ-1(h)(2,α)·v)(1,β-1λβ)hhhh×φα-1(φβ-1(h)(1,αβ-1λβα-1)))=(φβ-1(h)(1,α)·v(0))β⊗φβ(φβ-1(h)(2,β-1λβ)v(1,β-1λβ))=(φβ-1(h(1,βαβ-1))·v(0))β⊗φβ(φβ-1(h(2,λ))v(1,β-1λβ))=h(1,βαβ-1)⊳(v(0))β⊗φβ(φβ-1(h(2,λ))v(1,β-1λβ))=h(1,βαβ-1)⊳v〈0〉β⊗h(2,λ)v〈1,λ〉.
This completes the proof of the proposition.Remark 5.
Let(V,ρV)∈𝒲𝒴𝒟α(H) and let (W,ρW)∈𝒲𝒴𝒟β(H); then we have Vst=(tV)s as an object in 𝒲𝒴𝒟stαt-1s-1(H) and s(V⊗tαβW)=Vs⊗tsαβs-1Ws as an object in 𝒲𝒴𝒟sαβs-1(H).Proposition 6.
Let(V,ρV)∈𝒲𝒴𝒟α(H); (W,ρW)∈𝒲𝒴𝒟β(H). Set VW=Wα as an object in 𝒲𝒴𝒟αβα-1(H). Define the map
(31)cV,W:V⊗W⟶WV⊗V,cV,W(v⊗w)=(Sβ-1(v(1,β-1))·w)α⊗v(0).
Then, cV,W is H-linear, H-colinear and satisfies the following conditions:
(32)cV⊗W,U=(cV,UW⊗idW)∘(idV⊗cW,U),cV,W⊗U=(idVW⊗cV,U)∘(cV,W⊗idU).
Furthermore, cγV,γW=cV,W,  for all γ∈π.Proof.
Firstly, we need to show thatcV,W is well defined. Indeed, we have
(33)cV·W(1(1,α)·v⊗1(2,β)·w)=(Sβ-1((1(1,α)·v)(1,β-1))1(2,β)·w)α⊗(1(1,α)·v)(0)=(Sβ-1(v(1,β-1)S-1φα-1(1(1,αβα-1)))αhh×Sβ-1(1(3,β-1))1(4,β)·w)⊗1(2,α)·v(0)=(Sβ-1S-1φα-1(1(1,αβα-1))αhh×Sβ-1(v(1,β-1))εβs(1(3,1))·w)⊗1(2,α)·v(0)=(Sβ-1S-1φα-1(1(1,αβα-1))Sβ-1(1(3,β-1)v(1,β-1))·w)α⊗1(2,α)·v(0)=(Sβ-1(v(1,β-1))·w)α⊗v(0)=cV,W(v⊗w).
Secondly, we prove thatcV,W is H-linear. For all h∈Hαβ, we compute
(34)cV,W(h·(v⊗w))=(Sβ-1((h(1,α)·v)(1,β-1))h(2,β)·w)α⊗(h(1,α)·v)(0)=(Sβ-1(h(3,β-1)v(1,β-1)S-1φα-1(h(1,αβα-1)))h(4,β)·w)α⊗h(2,α)·v(0)=(Sβ-1(v(1,β-1)S-1φα-1(h(1,αβα-1)))εβs(h(3,1))·w)α⊗h(2,α)·v(0)=(Sβ-1(v(1,β-1)S-1φα-1(h(1,αβα-1)))αhhh×Sβ-1(1(2,β-1))·w)⊗h(2,α)1(1,α)·v(0)=(Sβ-1(1(3,β-1)v(1,β-1)S-1φα-1(1(1,αβα-1))αhhh×S-1φα-1(h(1,αβα-1)))·w)⊗h(2,α)1(2,α)·v(0)=(Sβ-1(v(1,β-1)S-1φα-1(h(1,αβα-1)))·w)α⊗h(2,α)·v(0)=(φα-1(h(1,αβα-1))Sβ-1(v(1,β-1))·w)α⊗h(2,α)·v(0)=h(1,αβα-1)⊳(Sβ-1(v(1,β-1))·w)α⊗h(2,α)·v=h·cV,W(v⊗w)
as required.
Finally, we check thatcV,W is satisfing the H-colinear condition. In fact,
(35)ρλW⊗VV∘cV,W(v⊗w)=((Sβ-1(v(1,β-1))·w)(0))⊗v(0)(0)α⊗v(0)(1,λ)(Sβ-1(v(1,β-1))·w)(1,λ)=(Sβ-1(v(1,β-1))(2,β)·w(0))α⊗v(0)(0)⊗v(0)(1,λ)Sβ-1(v(1,β-1))(3,λ)w(1,λ)S-1φβ-1(Sβ-1(v(1,β-1))(1,βλ-1β-1))=(Sβ-1(v(3,β-1))·w(0))α⊗v(0)⊗v(1,λ)Sλ-1(v(2,λ-1))×w(1,λ)S-1φβ-1Sβλβ-1(v(4,βλβ-1))=(Sβ-1(v(2,β-1))·w(0))α⊗v(0)⊗ελt(v(1,1))×w(1,λ)φβ-1(v(3,βλβ-1))=(Sβ-1(1(2,β-1)v(1,β-1))·w(0))α⊗v(0)⊗Sλ-1(1(1,λ-1))w(1,λ)φβ-1(v(2,βλβ-1))=(Sβ-1(1(2,β-1)v(1,β-1))·w(0))α⊗v(0)⊗Sλ-1(1(1,λ-1))w(1,λ)φβ-1S-1Sβλβ-1(1(3,βλβ-1))φβ-1(v(2,βλβ-1))=(Sβ-1(v(1,β-1))Sβ-1(1)(2,β)·w(0))α⊗v(0)⊗Sβ-1(1)(3,λ)w(1,λ),φβ-1S-1(Sβ-1(1)(1,βλ-1β-1))φβ-1(v(2,βλβ-1))=(Sβ-1(v(1,β-1))·w(0))⊗v(0)α⊗w(1,λ)φβ-1(v(2,βλβ-1))=(cV,W⊗idHλ)((v(1,βλβ-1))v(0)⊗w(0)⊗w(1,λ)φβ-1×(v(1,βλβ-1)))=(cV,W⊗idHλ)∘ρλV⊗W(v⊗w).
The rest of proof is easy to get and we omit it.Lemma 7.
The mapcV,W defined by (31) is bijective with inverse
(36)cV,W-1:WV⊗V⟶V⊗W,cV,W-1(wα⊗v)=v(0)⊗v(1,β)·w,
for all v∈V,wα∈WV.Proof.
Firstly, we provecV,W-1∘cV,W=idV⊗W. For all v∈V, w∈W, we have
(37)cV,W-1∘cV,W(v⊗w)=v(0)(0)⊗v(0)(1,β)Sβ-1(v(1,β-1))·w=v(0)⊗εβt(v(1,1))·w=1(1,α)·v(0)⊗1(2,β)εβt(v(1,1))·w=1(1,α)S-1εα-1t(v(1,1))·v(0)⊗1(2,β)·w=ε(1(2,1)′v(1,1))1(1,α)1(1,α)′·v(0)⊗1(2,β)·w=1(1,α)·v⊗1(2,β)·w=v⊗w.
Secondly, we checkcV,W∘cV,W-1=idVW⊗V as follows:
(38)cV,W∘cV,W-1(wα⊗v)=(Sβ-1(v(0)(1,β-1))v(1,β)·w)α⊗v(0)(0)=(φα-1(1(1,αβα-1))Sβ-1(v(0)(1,β-1))v(1,β)·w)α⊗1(2,α)·v(0)(0)=(φα-1(1(1,αβα-1))εβs(v(1,1))·w)α⊗1(2,α)·v(0)=(φα-1(1(1,αβα-1)εαβα-1sφα(v(1,1)))·w)α⊗1(2,α)·v(0)=(φα-1(1(1,αβα-1))·w)α⊗1(2,α)×Sα-1εαs(φα(v(1,1)))·v(0)=(φα-1(1(1,αβα-1))·w)α⊗1(2,α)εαtφαS(v(1,1))·v(0)=(φα-1(1(1,αβα-1))·w)α⊗1(2,α)×ε(φα-1(1(1,1)′)S(v(1,1)))1(2,α)′·v(0)=(φα-1(1(1,αβα-1))·w)α⊗1(2,α)εS(v(1,1))·v(0)=1(1,αβα-1)⊳wα⊗1(2,α)·v=wα⊗v.
This completes the proof.Define𝒲𝒴𝒟(H)=∐α∈π‍𝒲𝒴𝒟α(H), the disjoint union of the categories 𝒲𝒴𝒟α(H) for all α∈π. If we endow 𝒲𝒴𝒟(H) with tensor product as in Proposition 3, then 𝒲𝒴𝒟(H) becomes a monoidal category. The unit is HT={Hαt:=εαt(H1)}α∈π.The group homomorphismψ:G→aut(𝒲𝒴𝒟(H)); β→ψβ is given on components as
(39)ψβ:𝒲𝒴𝒟α(H)⟶𝒲𝒴𝒟βαβ-1(H),
where the functor ψβ acts as follows: given a morphism f:(V,ρV)→(W,ρW), for any v∈V, we set (fβ)(vβ)=(f(v))β.The braiding in𝒲𝒴𝒟(H) is given by the family {cV,W} as shown in Proposition 6. Then, we have the following theorem.Theorem 8.
For a weak crossed Hopf group coalgebraH, 𝒲𝒴𝒟(H) is a braided T-category over group π.Example 9.
LetH be a weak Hopf algebra, G a finite group, and k(G) the dual Hopf algebra of the group algebra kG.
Then, have the weak Hopf group coalgebrak(G)⊗H; the multiplication in k(G)⊗H is given by
(40)(pα⊗h)(pβ⊗g)=pαpβ⊗hg,
for all pα,pβ∈k(G), h,g∈H, and the comultiplication, counit, and antipode are given by
(41)Δu,v(pα⊗h)=∑uv=α(pu⊗h1)⊗(pv⊗h2),ε(pα⊗h)=δα,1ε(h),S(pα⊗h)=pα-1⊗S(h).Moreover,k(G)⊗H is a weak crossed Hopf group coalgebra with the following crossing:
(42)Φβ(pα⊗h)=pβ-1αβ⊗h.By Theorem8, 𝒲𝒴𝒟(k(G)⊗H) is a braided T-category.


## 4. BraidedT-Categories over Weak Long Dimodule Categories

In this section, we introduce the notion of a (left-right) weakα-Long dimodule over a weak crossed Hopf group coalgebra H and prove that the category H𝒲ℒH is a braided T-subcategory of Yetter-Drinfel’d category 𝒲𝒴𝒟(H⊗H) when H is a quasitriangular and coquasitriangular weak crossed Hopf group coalgebra.Definition 10.
LetH be a weak crossed Hopf group coalgebra over π. For a fixed element α∈π, a (left-right) weak α-Long dimodule is a couple V=(V,ρV={ρλV}λ∈π), where V is a left Hα-module and, for any λ∈π, ρλV:V→V⊗Hλ is ak-linear morphism, such that(1)
Vis coassociative in the sense that, for any λ1,λ2∈π, we have
(43)(idV⊗Δλ1,λ2)∘ρλ1λ2V=(ρλ1V⊗idHλ2)∘ρλ2V;(2)
V is counitary in the sense that
(44)(idV⊗ε)∘ρ1V=idV;(3)
V satisfies the following compatible condition:
(45)ρλV(x·v)=x·v(0)⊗v(1,λ); where x∈Hα and v∈V.Now, we can form the categoryH𝒲ℒαH of (left-right) weak α-Long dimodules where the composition of morphisms of weak α-Long dimodules is the standard composition of the underlying linear maps.LetH𝒲ℒH=∐α∈π‍𝒲HℒαH,  the disjoint union of the categories H𝒲ℒαH for all α∈π.Proposition 11.
The categoryH𝒲ℒH is a monoidal category. Moreover, for any α,β∈G, let V∈𝒲HℒαH and let W∈𝒲HℒβH. Set
(46)V⊗~W={(w(1,1)φβ-1(v(1,1)))v⊗w∈V⊗W∣v⊗whh=1(1,α)·v⊗1(2,β)·whh=ε(w(1,1)φβ-1(v(1,1)))v(0)⊗w(0)}.
Then, V⊗~W∈𝒲HℒαH  with the following structures:
(47)x·(v⊗w)=x(1,α)·v⊗x(2,β)·w,ρλV⊗~W(v⊗w)=v(0)⊗w(0)⊗w(1,λ)φβ-1(v(1,βλβ-1)),
for all x∈Hαβ, v⊗w∈V⊗~W.Proof.
It is straightforward.Let(H,σ,R) be a coquasitriangular and quasitriangular weak crossed Hopf group coalgebra with crossing φ. Define a family of vector spaces H⊗H={(H⊗H)α=H1⊗Hα}α∈π, where, the H on the left we consider its coquasitriangular structure and for the right one we consider its quasitriangular structure. Then, H⊗H is a weak crossed Hopf group coalgebra with the natural tensor product and the crossing Φ={id⊗φα}α∈π.Theorem 12.
Let(H,σ,R) be a weak crossed Hopf group coalgebra with coquasitriangular structure σ and quasitriangular structure R. Then, the category H𝒲ℒH is a braided T-subcategoryof Yetter-Drinfel’d category 𝒲𝒴𝒟(H⊗H) under the following action and coaction given by
(48)δλV(v)=aα·v(0)⊗v(1,1)⊗S-1(bλ-1)=:v[0]⊗v[1,λ],(h⊗x)⊳vα=σ(h,v(1,1))x·v(0),
where h⊗x∈(H⊗H)α, h∈H1, x∈Hα, v∈V,  and V∈H𝒲ℒαH.
The braiding onH𝒲ℒH, τV,W:V⊗W→WV⊗V is given by
(49)τV,W(v⊗w)=σ(S(v(1,1)),w(1,1))(bβ·w(0))α⊗aα·v(0),
for all V∈H𝒲ℒαH, W∈H𝒲ℒβH.Proof.
Obviously,V is a left (H⊗H)α-module. Then, we show that V satisfies the conditions in Definition 1. First, we need to check that V is coassociative. In fact, for all v∈V∈𝒲HℒαH and λ1,λ2∈π(50)(idV⊗Δλ1,λ2)∘δλ1λ2V(v)=aα·v(0)⊗v(1,1)⊗S-1(bλ2-1λ1-1(2,λ1-1))⊗v(2,1)⊗S-1(bλ2-1λ1-1(1,λ2-1))=aαaα′·v(0)⊗v(1,1)⊗S-1(bλ1-1)⊗v(2,1)⊗S-1(bλ2-1′)=aα·(aα′·v(0))(0)⊗(aα′·v(0))(1,1)⊗S-1(bλ1-1)⊗v(1,1)⊗S-1(bλ2-1′)=(δλ1V⊗id(H⊗H)λ2)(aα′·v(0)⊗v(1,1)⊗S-1(bλ2-1′))=(δλ1V⊗id(H⊗H)λ2)∘δλ2V(v).
Next, one directly shows that counitary condition (17) holds as follows:
(51)(idV⊗ε)∘δ1V(v)=aα·v(0)ε(m(1,1))εS-1(b1)=aα·vε(b1)=1α·v=v.
Then, we have to prove that crossed condition (18) is satisfied. For all h∈H1, x∈Hα, and v∈V∈𝒲HℒαH, we have
(52)(h⊗x)(2,α)·v[0]⊗(h⊗x)(3,λ)×v[1,λ]S-1Φα-1((h⊗x)(1,αλ-1α-1))=σ(h(2,1),(aα·v(0))(1,1))x(2,α)·(aα·v(0))(0)⊗h(3,1)v(1,1)S-1(h(1,1))⊗x(3,λ),S-1(bλ-1)S-1ψα-1(x(1,αλ-1α-1))=σ(h(2,1),v(1,1))x(2,α)aα·v(0)⊗h(3,1)v(2,1)S-1(h(1,1))⊗x(3,λ)S-1(bλ-1),S-1ψα-1(x(1,αλ-1α-1))=σ(h(3,1),v(2,1))x(2,α)aα·v(0)⊗v(1,1)h(2,1)S-1(h(1,1))⊗x(3,λ),S-1(ψα-1(x(1,αλ-1α-1))bλ-1)=σ(h(2,1),v(2,1))aαx(1,α)·v(0)⊗v(1,1)S-1ε1t(h(1,1))⊗x(3,λ)S-1(bλ-1(x(2,λ-1)))=σ(h(2,1),v(2,1))aαx(1,α)·v(0)⊗v(1,1)S-1ε1t(h(1,1))⊗S-1ελ-1t(x(2,1))S-1(bλ-1)=σ(1(2,1)′h,v(2,1))aα1(1,α)x·v(0)⊗v(1,1)1(1,1)′⊗S-1(1(2,λ-1))S-1(bλ-1)=σ(1(2,1)′,v(2,1))σ(h,v(3,1))aα1(1,α)x·v(0)⊗v(1,1)1(1,1)′⊗S-1(bλ-11(2,λ-1))=ε(v(2,1)1(2,1))σ(h,v(3,1))aαx·v(0)⊗v(1,1)1(1,1)⊗S-1(bλ-1)=σ(h,v(2,1))aαx·v(0)⊗v(1,1)⊗S-1(bλ-1)=σ(h,v(1,1))aα·(x·v(0))(0)⊗(x·v(0))(1,1)⊗S-1(bλ-1)=δλV((h⊗x)⊳vα).
Finally, it follows from Proposition6, the braiding on 𝒲𝒴𝒟(H⊗H), that the braiding on H𝒲ℒH is as the following:
(53)τV,W(v⊗w)=(Sβ-1(v[1,β-1]))α⊳wβ⊗v[0]=σ(S(v(1,1)),w(1,1))(bβ·w(0))α⊗aα·v(0),
for all V∈𝒲HℒαH, W∈𝒲HℒβH, v∈V, and w∈W.
This completes the proof.



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*Source: 626394-2013-12-31.xml*