Let H 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).

1. Introduction

Braided crossed categories over a group π (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 braided T-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 Section 1, we recall definitions and properties related to braided T-categories and weak crossed Hopf group coalgebras.

In Section 2, 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 Section 3, 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, let π 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.

The comultiplication Δα,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.

The counit ε: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 over π 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.

Let H 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:

ε(gh)=ε(gε1t(h)), ε(gh)=ε(ε1s(g)h), for all g,h∈H1,

x(1,α)⊗εβt(x(2,1))=1(1,α)x⊗1(2,β), for all x∈Hα,α,β∈π,

εβs(x(1,1))⊗x(2,α)=1(1,β)⊗x1(2,α), for all x∈Hα,α,β∈π,

εαt(ε1t(x)y)=εαt(x)εαt(y), εαs(xε1s(y))=εαs(x)εαs(y), for all x,y∈H1.

Similarly, for any α∈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

ε~αs(h(1,1))⊗h(2,β)=1(1,a)⊗1(2,β)h, for all h∈Hβ,α,β∈π,

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 coalgebra H={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.

If H 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 over π 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:

Rα,βΔα,β(h)=Δ¯β-1,α-1cop(h)Rα,β, for all h∈Hαβ,α,β∈π,

(idHα⊗Δβ,γ)(Rα,βγ)=(Rα,γ)1β3(Rα,β)12γ, for all α,β,γ∈π,

(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.

We recall that a monoidal category 𝒞 is called a crossed category over group π if it consists of the following data.

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.

A group homomorphism ψ: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:

for any morphism f∈𝒞α(U,U′) with α∈π, g∈𝒞(V,V′), we have
(13)((αg)⊗f)∘cU,V=cU′,V′∘(f⊗g);

for all U,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;

for any U,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 weak α-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.

Let H 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

V is coassociative in the sense that, for any λ1,λ2∈π, we have
(16)(idV⊗Δλ1,λ2)∘ρλ1λ2V=(ρλ1V⊗idHλ2)∘ρλ2V;

V is counitary in the sense that
(17)(idV⊗ε)∘ρ1V=idV;

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 two 𝒲𝒴𝒟α-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 category 𝒲𝒴𝒟α(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 that V 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 that V⊗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 all h∈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 that cV,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 that cV,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 that cV,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 map cV,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 prove cV,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 check cV,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.

Define 𝒲𝒴𝒟(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 homomorphism ψ: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 in 𝒲𝒴𝒟(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 coalgebra H, 𝒲𝒴𝒟(H) is a braided T-category over group π.

Example 9.

Let H 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 coalgebra k(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 Theorem 8, 𝒲𝒴𝒟(k(G)⊗H) is a braided T-category.

4. Braided <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M322"><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:math></inline-formula>-Categories over Weak Long Dimodule Categories

In this section, we introduce the notion of a (left-right) weak α-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.

Let H 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

Vis coassociative in the sense that, for any λ1,λ2∈π, we have
(43)(idV⊗Δλ1,λ2)∘ρλ1λ2V=(ρλ1V⊗idHλ2)∘ρλ2V;

V is counitary in the sense that
(44)(idV⊗ε)∘ρ1V=idV;

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 category H𝒲ℒαH 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.

Let H𝒲ℒH=∐α∈π𝒲HℒαH, the disjoint union of the categories H𝒲ℒαH for all α∈π.

Proposition 11.

The category H𝒲ℒH 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 on H𝒲ℒH, 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 Proposition 6, 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.

Acknowledgments

The work was partially supported by the NNSF of China (no. 11326063), NSF for Colleges and Universities in Jiangsu Province (no. 12KJD110003), NNSF of China (no. 11226070), and NJAUF (no. LXY2012 01019, LXYQ201201103).

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