New Braided T-Categories over Weak Crossed Hopf Group Coalgebras

Abstract

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 𝒲𝒴𝒟_𝛼(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 _𝐻𝒲ℒ^𝐻 for a weak crossed Hopf group coalgebra 𝐻 with quasitriangular and coquasitriangular structures and obtain that _𝐻𝒲ℒ^𝐻 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∈𝜋𝒲𝒴𝒟_𝛼(𝐻), where 𝒲𝒴𝒟_𝛼(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 _𝐻𝒲ℒ^𝐻_𝛼 for a weak crossed Hopf group coalgebra 𝐻. Then, we obtain a new category _𝐻𝒲ℒ^𝐻=∐_{𝐻𝛼∈𝜋}𝒲ℒ^𝐻_𝛼 and show that as 𝐻 is a quasitriangular and coquasitriangular weak crossed Hopf group coalgebra, then _𝐻𝒲ℒ^𝐻 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={𝐶_𝛼}_𝛼∈𝜋 together with a family of 𝑘-linear maps Δ={Δ_𝛼,𝛽∶𝐶_𝛼𝛽→𝐶_𝛼⊗𝐶_𝛽}_{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 Δ_𝛼,𝛽⊗id_𝐶𝛾Δ_{𝛼𝛽,𝛾}=id_𝐶𝛼⊗Δ_𝛽,𝛾Δ_{𝛼,𝛽𝛾},∀𝛼,𝛽,𝛾∈𝜋.id_𝐶𝛼⊗𝜀Δ_𝛼,1=id_𝐶𝛼=𝜀⊗id_𝐶𝛼Δ_1,𝛼,∀𝛼∈𝜋. (1)

We use the Sweedler-type notation (see [14]) for a comultiplication; that is, we write Δ_𝛼,𝛽(𝑐)=𝑐_(1,𝛼)⊗𝑐_(2,𝛽),forany𝛼,𝛽∈𝜋,𝑐∈𝐶_𝛼𝛽. (2)

Recall from Van Daele and Wang (see [11]) that a weak semi-Hopf group coalgebra 𝐻={𝐻_𝛼,𝑚_𝛼,1_𝛼,Δ,𝜀}_𝛼∈𝜋 is a family of algebras {𝐻_𝛼,𝑚_𝛼,1_𝛼}_𝛼∈𝜋 and at the same time a group coalgebra {𝐻_𝛼,Δ={Δ_𝛼,𝛽},𝜀}_{𝛼,𝛽∈𝜋}, such that the following conditions hold.(i)The comultiplication Δ_𝛼,𝛽∶𝐶_𝛼𝛽→𝐶_𝛼⊗𝐶_𝛽 is a homomorphism of algebras (not necessary unit preserving) such that Δ_𝛼,𝛽⊗id_𝐻𝛾Δ_{𝛼𝛽,𝛾}1_𝛼𝛽𝛾=Δ_𝛼,𝛽1_𝛼𝛽⊗1_𝛾1_𝛼⊗Δ_𝛽,𝛾1_𝛽𝛾,Δ_𝛼,𝛽⊗id_𝐻𝛾Δ_{𝛼𝛽,𝛾}1_𝛼𝛽𝛾=1_𝛼⊗Δ_𝛽,𝛾1_𝛽𝛾Δ_𝛼,𝛽1_𝛼𝛽⊗1_𝛾, (3)    for all 𝛼,𝛽,𝛾∈𝜋.(ii)The counit 𝜀∶𝐻₁→𝑘 is a 𝑘-linear map satisfying the identity 𝜀𝑔𝑥ℎ=𝜀𝑔𝑥_(2,1)𝜀𝑥_(1,1)ℎ=𝜀𝑔𝑥_(1,1)𝜀𝑥_(2,1)ℎ, (4)   for all 𝑔,ℎ,𝑥∈𝐻₁.

A weak Hopf group coalgebra over 𝜋 is a weak semi-Hopf group coalgebra 𝐻={𝐻_𝛼,𝑚_𝛼,1_𝛼,Δ,𝜀}_𝛼∈𝜋 endowed with a family of 𝑘-linear maps 𝑆={𝑆_𝛼∶𝐻_𝛼→𝐻_𝛼−1}_𝛼∈𝜋 (called an antipode) satisfying the following equations: 𝑚_𝛼𝑆_𝛼−1⊗id_𝐻𝛼Δ_{𝛼−1,𝛼}(ℎ)=1_(1,𝛼)𝜀ℎ1_(2,1),𝑚_𝛼id_𝐻𝛼⊗𝑆_𝛼−1Δ_{𝛼,𝛼−1}(ℎ)=𝜀1_(1,1)ℎ1_(2,𝛼),𝑆_𝛼𝑔_(1,𝛼)𝑔_(2,𝛼−1)𝑆_𝛼𝑔_(3,𝛼)=𝑆_𝛼𝑔, (5) for all ℎ∈𝐻₁,𝑔∈𝐻_𝛼, and 𝛼∈𝜋.

Let 𝐻 be a weak Hopf group coalgebra. Define a family of linear maps 𝜀_𝑡={𝜀^𝑡_𝛼∶𝐻₁→𝐻_𝛼}_𝛼∈𝜋 and 𝜀_𝑠={𝜀^𝑠_𝛼∶𝐻₁→𝐻_𝛼}_𝛼∈𝜋 by the formulae 𝜀^𝑡_𝛼(ℎ)=𝜀1_(1,1)ℎ1_(2,𝛼)=𝑚_𝛼id_𝐻𝛼⊗𝑆_𝛼−1Δ_{𝛼,𝛼−1}(ℎ),𝜀^𝑠_𝛼(ℎ)=1_(1,𝛼)𝜀ℎ1_(2,1)=𝑚_𝛼𝑆_𝛼−1⊗id_𝐻𝛼Δ_{𝛼−1,𝛼}(ℎ), (6)  for any ℎ∈𝐻₁, where 𝜀^𝑡 and 𝜀^𝑠 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(𝑔𝜀^𝑡₁(ℎ)), 𝜀(𝑔ℎ)=𝜀(𝜀^𝑠₁(𝑔)ℎ), for all 𝑔,ℎ∈𝐻₁,(2)𝑥_(1,𝛼)⊗𝜀^𝑡_𝛽(𝑥_(2,1))=1_(1,𝛼)𝑥⊗1_(2,𝛽), for all 𝑥∈𝐻_𝛼,𝛼,𝛽∈𝜋,(3)𝜀^𝑠_𝛽(𝑥_(1,1))⊗𝑥_(2,𝛼)=1_(1,𝛽)⊗𝑥1_(2,𝛼), for all 𝑥∈𝐻_𝛼,𝛼,𝛽∈𝜋,(4)𝜀^𝑡_𝛼(𝜀^𝑡₁(𝑥)𝑦)=𝜀^𝑡_𝛼(𝑥)𝜀^𝑡_𝛼(𝑦), 𝜀^𝑠_𝛼(𝑥𝜀^𝑠₁(𝑦))=𝜀^𝑠_𝛼(𝑥)𝜀^𝑠_𝛼(𝑦), for all 𝑥,𝑦∈𝐻₁.

Similarly, for any 𝛼∈𝜋 and ℎ∈𝐻₁, define 𝜀^𝑡_𝛼̃(ℎ)=𝜀(ℎ1_(1,1))1_(2,𝛼), 𝜀^𝑠_𝛼̃(ℎ)=1_(1,𝛼)𝜀(1_(2,1)ℎ). Then, we have(1)𝜀^𝑠_𝛼̃(ℎ_(1,1))⊗ℎ_(2,𝛽)=1_(1,𝑎)⊗1_(2,𝛽)ℎ, for all ℎ∈𝐻_𝛽,𝛼,𝛽∈𝜋,(2)𝑥_(1,𝛼)⊗𝜀^𝑡_𝛼̃(𝑥_(2,1))=𝑥1_(1,𝛼)⊗1_(2,𝛽), for all 𝑥∈𝐻_𝛼,𝛼,𝛽∈𝜋.

A weak Hopf group coalgebra 𝐻={𝐻_𝛼,𝑚_𝛼,1_𝛼,Δ,𝜀,𝑆}_𝛼∈𝜋 is called a weak crossed Hopf group coalgebra if it is endowed with a family of algebra isomorphisms 𝜑={𝜑_𝛼∶𝐻_𝛽→𝐻_{𝛼𝛽𝛼−1}}_{𝛼,𝛽∈𝜋} (called a crossing) such that (𝜑_𝛼⊗𝜑_𝛼)∘Δ_𝛽,𝛾=Δ_{𝛼𝛽𝛼−1,𝛼𝛾𝛼−1}∘𝜑_𝛼, 𝜀∘𝜑_𝛼=𝜀, and 𝜑_𝛼𝛽=𝜑_𝛼∘𝜑_𝛽 for all 𝛼,𝛽,𝛾∈𝜋.

If 𝐻 is crossed with the crossing 𝜑={𝜑_𝛼}_𝛼∈𝜋, then we have 𝜑_𝛽∘𝜀^𝑠_𝛼=𝜀^𝑠_{𝛽𝛼𝛽−1}∘𝜑_𝛽,𝜑_𝛽∘𝜀^𝑡_𝛼=𝜀^𝑡_{𝛽𝛼𝛽−1}∘𝜑_𝛽,∀𝛼,𝛽∈𝜋. (7)

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={𝑅_𝛼,𝛽∈Δ^cop_{𝛽−1,𝛼−1}(1_𝛼𝛽)(𝐻_𝛼⊗𝐻_𝛽)Δ_𝛼,𝛽(1_𝛼𝛽)} satisfying the following conditions:(1)𝑅_𝛼,𝛽Δ_𝛼,𝛽(ℎ)=Δ^cop_{𝛽−1,𝛼−1}(ℎ)𝑅_𝛼,𝛽, for all ℎ∈𝐻_𝛼𝛽,𝛼,𝛽∈𝜋,(2)(id_𝐻𝛼⊗Δ_𝛽,𝛾)(𝑅_{𝛼,𝛽𝛾})=(𝑅_𝛼,𝛾)_1𝛽3(𝑅_𝛼,𝛽)_12𝛾, for all 𝛼,𝛽,𝛾∈𝜋,(3)(Δ_𝛼,𝛽⊗id_𝐻𝛾)(𝑅_𝛽⁻¹_{𝛼−1,𝛾})=(𝑅_{𝛼−1,𝛾})_1𝛽⁻¹₃(𝑅_𝛽⁻¹_,𝛾)_𝛼−123, for all 𝛼,𝛽,𝛾∈𝜋, where Δ_𝛼,𝛽=(𝜑_𝛽⊗id_𝐻𝛽−1)∘Δ_𝛽⁻¹_{𝛼−1𝛽,𝛽}⁻¹, Δ^cop_𝛼,𝛽=𝑡_{𝐻𝛼−1,𝐻𝛽−1}∘(𝜑_𝛽⊗id_𝐻𝛽−1)∘Δ_𝛽⁻¹_{𝛼−1𝛽,𝛽}⁻¹ for all 𝛼,𝛽∈𝜋, and such that there exists a family of 𝑅={𝑅_𝛼,𝛽∈Δ_𝛼,𝛽(1_𝛼𝛽)(𝐻_𝛼⊗𝐻_𝛽)Δ^cop_{𝛽−1,𝛼−1}(1_𝛼𝛽)} with 𝑅_𝛼,𝛽𝑅_𝛼,𝛽=Δ^cop_{𝛽−1,𝛼−1}1_𝛼𝛽,𝑅_𝛼,𝛽𝑅_𝛼,𝛽=Δ_𝛼,𝛽1_𝛼𝛽,𝜑_𝛽⊗𝜑_𝛽𝑅_𝛼,𝛾=𝑅_𝛽𝛼𝛽⁻¹_{,𝛽𝛾𝛽}⁻¹, (8) for all 𝛼,𝛽,𝛾∈𝜋. In this paper, we denote 𝑅_𝛼,𝛽=𝑎_𝛼⊗𝑏_𝛽.

Recall from [16] that a coquasitriangular weak Hopf group coalgebra (𝐻,𝜎) consists of a weak Hopf group coalgebra 𝐻={𝐻_𝛼,𝑚_𝛼,1_𝛼,Δ,𝜀,𝑆}_𝛼∈𝜋 and a map 𝜎∶𝐻₁⊗𝐻₁→𝑘 satisfying 𝜎ℎ_(1,1),𝑔_(1,1)ℎ_(2,𝛼)𝑔_(2,𝛼)=𝑔_(1,𝛼)ℎ_(1,𝛼)𝜎ℎ_(2,1),𝑔_(2,1),𝜎(𝑎,𝑏𝑐)=𝜎𝑎_(1,1),𝑐𝜎𝑎_(2,1),𝜎(),𝑏𝑎𝑏,𝑐=𝜎𝑎,𝑐_(1,1)𝜎𝑏,𝑐_(2,1),𝜀𝑎_(1,1)𝑏_(1,1)𝜎𝑏_(2,1),𝑎_(2,1)𝜀𝑏_(3,1)𝑎_(3,1)=𝜎(𝑏,𝑎), (9) and there exists 𝜎⁻¹∶𝐻₁⊗𝐻₁→𝑘 such that 𝜎𝑎_(1,1),𝑏_(1,1)𝜎⁻¹𝑎_(2,1),𝑏_(2,1)(),𝜎=𝜀𝑏𝑎₋₁𝑎_(1,1),𝑏_(1,1)𝜎𝑎_(2,1),𝑏_(2,1)(),𝜀𝑎=𝜀𝑎𝑏_(1,1)𝑏_(1,1)𝜎⁻¹𝑎_(2,1),𝑏_(2,1)𝜀𝑏_(3,1)𝑎_(3,1)=𝜎⁻¹(𝑎,𝑏), (10) for all ℎ,𝑔∈𝐻_𝛼, 𝑎,𝑏,𝑐∈𝐻₁, where 𝜎⁻¹ is called a weak inverse of 𝜎.

2.2. Braided 𝑇-Categories

We recall that a monoidal category 𝒞 is called a crossed category over group 𝜋 if it consists of the following data.(1)A family of subcategories {𝒞_𝛼}_𝛼∈𝜋 such that 𝒞 is a disjoint union of this family and such that for any 𝑈∈𝒞_𝛼 and 𝑉∈𝒞_𝛽, 𝑈⊗𝑉∈𝒞_𝛼𝛽. Here, the subcategory 𝒞_𝛼 is called the 𝛼th component of 𝒞.(2)A group homomorphism 𝜓∶𝜋→aut(𝒞)∶𝛽↦𝜓_𝛽, the conjugation, (where aut(𝒞) is the group of invertible strict tensor functors from 𝒞 to itself) such that 𝜓_𝛽(𝒞_𝛼)=𝒞_𝛽𝛼𝛽⁻¹ for any 𝛼,𝛽∈𝜋. Here, the functors 𝜓_𝛽 are called conjugation isomorphisms.

We will use the Turaev’s left index notation in [1]: for any object 𝑈∈𝒞_𝛼, 𝑉,𝑊∈𝒞_𝛽 and any morphism 𝑓∶𝑉→𝑊 in 𝒞_𝛽, we set ^𝑈𝑉=𝜓_𝛼(𝑉)∈𝒞_{𝛼𝛽𝛼−1},^𝑈𝑓=𝜓_𝛼𝑓∶^𝑈𝑉⟶^𝑈𝑊. (11)

Recall form [1] that a braided 𝑇-category is a crossed category 𝒞 endowed with braiding, that is a family of isomorphisms, 𝑐=𝑐_𝑈,𝑉∈𝒞𝑈⊗𝑉,^𝑈𝑉⊗𝑈_{𝑈,𝑉∈𝒞} (12) satisfying the following conditions:(1)for any morphism 𝑓∈𝒞_𝛼(𝑈,𝑈^′) with 𝛼∈𝜋, 𝑔∈𝒞(𝑉,𝑉^′), we have (^𝛼𝑔⊗𝑓∘𝑐_𝑈,𝑉=𝑐_{𝑈′,𝑉′}∘𝑓⊗𝑔; (13)(2)for all 𝑈,𝑉,𝑊∈𝒞, we have 𝑐_{𝑈⊗𝑉,𝑊}=𝑎_{𝑈⊗𝑉𝑊,𝑈,𝑉}∘𝑐_{𝑈,𝑉𝑊}⊗id_𝑉∘𝑎⁻¹_{𝑈,𝑉𝑊,𝑉}∘id_𝑈⊗𝑐_𝑉,𝑊∘𝑎_{𝑈,𝑉,𝑊},𝑐_{𝑈,𝑉⊗𝑊}=𝑎⁻¹_{𝑈𝑉,𝑈𝑊,𝑈}∘id_𝑈𝑉⊗𝑐_𝑈,𝑊∘𝑎_{𝑈𝑉,𝑈,𝑊}∘𝑐_𝑈,𝑉⊗id_𝑊∘𝑎⁻¹_{𝑈,𝑉,𝑊}; (14)(3)for any 𝑈,𝑉∈𝒞, 𝛼∈𝜋, 𝜓_𝛼𝑐_𝑈,𝑉=𝑐_{𝜓𝛼(𝑈),𝜓𝛼(𝑉)}. (15)

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 𝒲𝒴𝒟_𝛼-module, is a couple 𝑉=(𝑉,𝜌^𝑉={𝜌^𝑉_𝜆}_𝜆∈𝜋), where 𝑉 is a left 𝐻_𝛼-module and, for any 𝜆∈𝜋, 𝜌^𝑉_𝜆∶𝑉→𝑉⊗𝐻_𝜆 is a 𝑘-linear morphism, such that(1)𝑉 is coassociative in the sense that, for any 𝜆₁,𝜆₂∈𝜋, we have id_𝑉⊗Δ_𝜆1,𝜆2∘𝜌^𝑉_𝜆1𝜆2=𝜌^𝑉_𝜆1⊗id_𝐻𝜆2∘𝜌^𝑉_𝜆2; (16)(2)𝑉 is counitary in the sense that id_𝑉⊗𝜀∘𝜌^𝑉₁=id_𝑉; (17)(3)𝑉 is crossed in the sense that, for any 𝜆∈𝜋, ℎ∈𝐻_𝛼, 𝜌^𝑉_𝜆(ℎ⋅𝑣)=ℎ_(2,𝛼)⋅𝑣₍₀₎⊗ℎ_(3,𝜆)𝑣_(1,𝜆)𝑆⁻¹𝜑_𝛼−1ℎ_(1,𝛼𝜆⁻¹_𝛼−1), (18) where 𝜌^𝑉_𝜆(𝑣)=𝑣₍₀₎⊗𝑣_(1,𝜆).

Given two 𝒲𝒴𝒟_𝛼-modules (𝑉,𝜌^𝑉) and (𝑊,𝜌^𝑊), a morphism 𝑓∶(𝑉,𝜌^𝑉)→(𝑊,𝜌^𝑊) of this two 𝒲𝒴𝒟_𝛼-modulesis an 𝐻_𝛼-linear map 𝑓∶𝑉⇀𝑊, such that, for any 𝜆∈𝜋, 𝜌^𝑊_𝜆∘𝑓=𝑓⊗id_𝐻𝜆∘𝜌^𝑉_𝜆. (19)

Then, we can form the category 𝒲𝒴𝒟_𝛼(𝐻) of 𝒲𝒴𝒟_𝛼-modules where the composition of morphisms of 𝒲𝒴𝒟_𝛼-modules is the standard composition of the underlying linear maps.

Proposition 2. Equation (18) is equivalent to the following equations: ℎ_(1,𝛼)⋅𝑣₍₀₎⊗ℎ_(2,𝜆)𝑣_(1,𝜆)=ℎ_(2,𝛼)⋅𝑣₍₀₎⊗ℎ_(2,𝛼)⋅𝑣_(1,𝜆)𝜑_𝛼−1ℎ_{(1,𝛼𝜆𝛼−1)},𝜌^𝑉_𝜆(𝑣)=𝑣₍₀₎⊗𝑣_(1,𝜆)∈𝑉⊗_𝑡𝛼𝜆𝐻_𝜆∶=Δ_𝛼,𝜆1_𝛼𝜆⋅𝑉⊗𝐻_𝜆, (20)(21) for any 𝑣∈𝑉, ℎ∈𝐻_𝛼𝜆.

Proof. Assume that (20) and (21) hold for all ℎ∈𝐻_𝛼𝜆, 𝑣∈𝑉. We compute ℎ_(2,𝛼)⋅𝑣₍₀₎⊗ℎ_(3,𝜆)𝑣_(1,𝜆)𝑆⁻¹𝜑_𝛼−1ℎ_(1,𝛼𝜆⁻¹_𝛼−1)=ℎ_(3,𝛼)⋅𝑣₍₀₎⊗ℎ_(3,𝛼)⋅𝑣_(1,𝜆)×𝜑_𝛼−1ℎ_{(2,𝛼𝜆𝛼−1)}𝑆⁻¹𝜑_𝛼−1ℎ_(1,𝛼𝜆⁻¹_𝛼−1)=ℎ_(3,𝛼)⋅𝑣₍₀₎⊗ℎ_(3,𝛼)⋅𝑣_(1,𝜆)×𝜑_𝛼−1ℎ_{(2,𝛼𝜆𝛼−1)}𝑆⁻¹ℎ_(1,𝛼𝜆⁻¹_𝛼−1)=ℎ_(2,𝛼)⋅𝑣₍₀₎⊗ℎ_(2,𝛼)⋅𝑣_(1,𝜆)×𝜑_𝛼−1𝑆⁻¹𝜀^𝑡_{𝛼𝜆−1𝛼−1}ℎ_(1,1)=1^′_()2,𝛼ℎ_(2,𝛼)⋅𝑣₍₀₎⊗1^′_()2,𝛼ℎ_(2,𝛼)⋅𝑣_(1,𝜆)×𝜑_𝛼−11_{(1,𝛼𝜆𝛼−1)}×𝜀1_(2,1)1^′_()1,1ℎ_(1,1)=1_(2,𝛼)ℎ⋅𝑣₍₀₎⊗1_(2,𝛼)ℎ⋅𝑣_(1,𝜆)𝜑_𝛼−11_{(1,𝛼𝜆𝛼−1)}=1_(1,𝛼)⋅(ℎ⋅𝑣)₍₀₎⊗1_(2,𝜆)(ℎ⋅𝑣)_(1,𝜆)=(ℎ⋅𝑣)₍₀₎⊗(ℎ⋅𝑣)_(1,𝜆) (22) as required.
Conversely, suppose that 𝑉 is crossed in the sense of (18). We first note that 𝑣₍₀₎⊗𝑣_(1,𝜆)=1_(2,𝛼)⋅𝑣₍₀₎⊗1_(3,𝜆)𝑣_(1,𝜆)𝑆⁻¹𝜑_𝛼−11_(1,𝛼𝜆⁻¹_𝛼−1)=1^′_()1,𝛼1_(2,𝛼)⋅𝑣₍₀₎⊗1^′_()2,𝜆𝑣_(1,𝜆)𝑆⁻¹×𝜑_𝛼−11_(1,𝛼𝜆⁻¹_𝛼−1)=1^′_()1,𝛼⋅1_(2,𝛼)⋅𝑣₍₀₎⊗1^′_()2,𝜆𝑣_(1,𝜆)𝑆⁻¹𝜑_𝛼−11_(1,𝛼𝜆⁻¹_𝛼−1)∈𝑉⊗_𝑡𝛼𝜆𝐻_𝜆. (23)
To show that (21) is satisfied, for all ℎ∈𝐻_𝛼𝜆, we do the following calculations: ℎ_(2,𝛼)⋅𝑣₍₀₎⊗ℎ_(2,𝛼)⋅𝑣_(1,𝜆)𝜑_𝛼−1ℎ_{(1,𝛼𝜆𝛼−1)}=ℎ_(3,𝛼)⋅𝑣₍₀₎⊗ℎ_(4,𝜆)𝑣_(1,𝜆)𝑆⁻¹𝜑_𝛼−1ℎ_(2,𝛼𝜆⁻¹_𝛼−1)×𝜑_𝛼−1ℎ_{(1,𝛼𝜆𝛼−1)}=ℎ_(2,𝛼)⋅𝑣₍₀₎⊗ℎ_(3,𝜆)𝑣_(1,𝜆)𝜑_𝛼−1𝑆⁻¹𝜀^𝑠_{𝛼𝜆−1𝛼−1}ℎ_(1,1)=ℎ_(2,𝛼)1^′_()2,𝛼⋅𝑣₍₀₎⊗ℎ_(3,𝜆)𝑣_(1,𝜆)𝜑_𝛼−1𝑆⁻¹1_(1,𝛼𝜆⁻¹_𝛼−1)×𝜀ℎ_(1,1)1^′_()1,11_(2,1)=ℎ_(1,𝛼)1_(2,𝛼)⋅𝑣₍₀₎⊗ℎ_(2,𝜆)𝑣_(1,𝜆)𝜑_𝛼−1𝑆⁻¹1_(1,𝛼𝜆⁻¹_𝛼−1)=ℎ_(1,𝛼)1^′_()1,𝛼1_(2,𝛼)⋅𝑣₍₀₎⊗ℎ_(2,𝜆)1^′_()2,𝜆𝑣_(1,𝜆)𝑆⁻¹×𝜑_𝛼−11_(1,𝛼𝜆⁻¹_𝛼−1)=ℎ_(1,𝛼)1_(2,𝛼)⋅𝑣₍₀₎⊗ℎ_(2,𝜆)1_(3,𝜆)𝑣_(1,𝜆)𝑆⁻¹×𝜑_𝛼−11_(1,𝛼𝜆⁻¹_𝛼−1)=ℎ_(1,𝛼)⋅𝑣₍₀₎⊗ℎ_(2,𝜆)𝑣_(1,𝜆). (24)
This completes the proof.

Proposition 3. If (𝑉,𝜌^𝑉)∈𝒲𝒴𝒟_𝛼(𝐻), (𝑊,𝜌^𝑊)∈𝒲𝒴𝒟_𝛽(𝐻), then 𝑉⊗_𝑡𝛼𝛽𝑊 = Δ_𝛼,𝛽(1_𝛼𝛽)⋅(𝑉⊗𝑊)∈𝒲𝒴𝒟_𝛼𝛽(𝐻) with the action and coaction structures as follows: ℎ⋅(𝑣⊗𝑤)=ℎ_(1,𝛼)⋅𝑣⊗ℎ_(2,𝛽)𝜌⋅𝑤,_{𝑉⊗𝑡𝛼𝛽𝑊𝜆}(𝑣⊗𝑤)=𝑣₍₀₎⊗𝑤₍₀₎⊗𝑤_(1,𝜆)𝜑_𝛽⁻¹𝑣_{(1,𝛽𝜆𝛽}⁻¹₎, (25) for all ℎ∈𝐻_𝛼𝛽, 𝜆∈𝜋, 𝑣⊗𝑤∈𝑉⊗_𝑡𝛼𝛽𝑊.

Proof. It is easy to prove that 𝑉⊗_𝑡𝛼𝛽𝑊 is a left 𝐻_𝛼𝛽-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 id_{𝑉⊗𝑡𝛼𝛽𝑊}⊗𝜀∘𝜌^{𝑉⊗𝑡𝛼𝛽𝑊}₁()𝑤𝑣⊗𝑤=𝜀_(1,1)𝜑_𝛽⁻¹1^′_(2,1)×𝜀𝜑_𝛽⁻¹1^′_(1,1)𝜑_𝛽⁻¹𝑣_(1,1)𝑣₍₀₎⊗𝑤₍₀₎=𝜀1_(2,𝛽)⋅𝑤_(1,1)𝜑_𝛽⁻¹1_(1,1)𝜑_𝛽⁻¹1^′_()2,1×𝜀𝜑_𝛽⁻¹1^′_()1,1𝜑_𝛽⁻¹𝑣_(1,1)𝑣₍₀₎⊗1_(2,𝛽)⋅𝑤₍₀₎=𝜀1_(5,1)𝑤_(1,1)𝑆⁻¹𝜑_𝛽⁻¹1_(3,1)𝜑_𝛽⁻¹1_(2,1)×𝜀1_(1,1)𝑣_(1,1)𝑣₍₀₎⊗1_(4,𝛽)⋅𝑤₍₀₎=𝜀1_(4,1)𝑤_(1,1)𝜑_𝛽⁻¹𝑆⁻¹𝜀^𝑠₁1_(2,1)𝜀1_(1,1)𝑣_(1,1)𝑣₍₀₎⊗1_(3,𝛽)⋅𝑤₍₀₎=𝜀1_(4,1)𝑤_(1,1)𝜀^𝑡₁𝑆⁻¹𝜑_𝛽⁻¹1_(2,1)𝜀1_(1,1)𝑣_(1,1)𝑣₍₀₎⊗1_(3,𝛽)⋅𝑤₍₀₎=𝜀1_(2,𝛽)⋅𝑤_(1,1)𝜀1_(1,1)𝑣_(1,1)𝑣₍₀₎⊗1_(2,𝛽)⋅𝑤₍₀₎=𝜀1_(2,𝛼)⋅𝑣_(1,1)𝜑_𝛼−11_(1,1)1_(2,𝛼)⋅𝑣₍₀₎⊗1_(3,𝛽)1⋅𝑤=𝜀_(3,1)𝑣_(1,1)𝜑_𝛼−1𝑆⁻¹𝜀^𝑠₁1_(1,1)1_(2,𝛼)⋅𝑣₍₀₎⊗1_(4,𝛽)1⋅𝑤=𝜀_(1,𝛼)⋅𝑣_(1,1)1_(1,𝛼)⋅𝑣₍₀₎⊗1_(2,𝛽)⋅𝑤=𝑣⊗𝑤. (26) 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∈𝐻_𝛼𝛽𝜆, 𝑣⊗𝑤∈𝑉⊗_𝑡𝛼𝛽𝑊, we have ℎ_(2,𝛼𝛽)⋅(𝑣⊗𝑤)₍₀₎⊗ℎ_(2,𝛼𝛽)⋅(𝑣⊗𝑤)_(1,𝜆)𝜑_(𝛼𝛽)⁻¹ℎ_{(1,𝛼𝛽𝜆𝛽}⁻¹_𝛼−1)=ℎ_(2,𝛼)⋅𝑣₍₀₎⊗ℎ_(3,𝛽)⋅𝑤₍₀₎⊗ℎ_(3,𝛽)⋅𝑤_(1,𝜆)𝜑_𝛽⁻¹ℎ_(2,𝛼)⋅𝑣_{(1,𝛽𝜆𝛽−1)}𝜑_𝛽⁻¹_𝛼−1ℎ_{(1,𝛼𝛽𝜆𝛽}⁻¹_𝛼−1)=ℎ_(3,𝛼)⋅𝑣₍₀₎⊗ℎ_(6,𝛽)⋅𝑤₍₀₎⊗ℎ_(7,𝜆)𝑤_(1,𝜆)𝑆⁻¹𝜑_𝛽⁻¹ℎ_(5,𝛽𝜆⁻¹_𝛽⁻¹₎𝜑_𝛽⁻¹ℎ_{(4,𝛽𝜆𝛽}⁻¹₎𝑣_{(1,𝛽𝜆𝛽}⁻¹₎𝜑_𝛽⁻¹𝜑_𝛼−1𝑆⁻¹×ℎ_{(2,𝛼𝛽𝜆}⁻¹_𝛽⁻¹_𝛼−1)𝜑_𝛽⁻¹_𝛼−1ℎ_{(1,𝛼𝛽𝜆𝛽}⁻¹_𝛼−1)=ℎ_(3,𝛼)⋅𝑣₍₀₎⊗ℎ_(6,𝛽)⋅𝑤₍₀₎⊗ℎ_(7,𝜆)𝑤_(1,𝜆)𝑆⁻¹𝜑_𝛽⁻¹ℎ_(5,𝛽𝜆⁻¹_𝛽⁻¹₎𝜑_𝛽⁻¹ℎ_{(4,𝛽𝜆𝛽}⁻¹₎𝑣_{(1,𝛽𝜆𝛽}⁻¹₎𝜑_𝛽⁻¹_𝛼−1𝑆⁻¹×𝑆ℎ_{(1,𝛼𝛽𝜆𝛽}⁻¹_𝛼−1)ℎ_{(2,𝛼𝛽𝜆}⁻¹_𝛽⁻¹_𝛼−1)=ℎ_(1,𝛼)1_(2,𝛼)⋅𝑣₍₀₎⊗ℎ_(4,𝛽)⋅𝑤₍₀₎⊗ℎ_(5,𝜆)𝑤_(1,𝜆)𝑆⁻¹𝜑_𝛽⁻¹ℎ_(3,𝛽𝜆⁻¹_𝛽⁻¹₎𝜑_𝛽⁻¹ℎ_{(2,𝛽𝜆𝛽}⁻¹₎𝑣_{(1,𝛽𝜆𝛽}⁻¹₎𝜑_𝛽⁻¹_𝛼−1𝑆⁻¹1_{(1,𝛼𝛽𝜆}⁻¹_𝛽⁻¹_𝛼−1)=ℎ_(1,𝛼)1_(2,𝛼)⋅𝑣₍₀₎⊗ℎ_(4,𝛽)⋅𝑤₍₀₎⊗ℎ_(5,𝜆)𝑤_(1,𝜆)𝑆⁻¹𝜑_𝛽⁻¹ℎ_(3,𝛽𝜆⁻¹_𝛽⁻¹₎𝜑_𝛽⁻¹ℎ_{(2,𝛽𝜆𝛽}⁻¹₎1_{(3,𝛽𝜆𝛽}⁻¹₎𝑣_{(1,𝛽𝜆𝛽}⁻¹₎𝑆⁻¹×𝜑_𝛼−11_{(1,𝛼𝛽𝜆}⁻¹_𝛽⁻¹_𝛼−1)=ℎ_(1,𝛼)⋅𝑣₍₀₎⊗ℎ_(3,𝛽)⋅𝑤₍₀₎⊗ℎ_(4,𝜆)𝑤_(1,𝜆)×𝜑_𝛽⁻¹𝑆⁻¹𝜀^𝑠_{𝛽𝜆−1𝛽−1}ℎ_(2,1)𝜑_𝛽⁻¹𝑣_{(1,𝛽𝜆𝛽}⁻¹₎=ℎ_(1,𝛼)⋅𝑣₍₀₎⊗ℎ_(2,𝛽)1^′_1,𝛽1_(2,𝛽)⋅𝑤₍₀₎⊗ℎ_(3,𝜆)1^′_()2,𝜆𝑤_(1,𝜆)𝑆⁻¹𝜑_𝛽⁻¹1_(1,𝛽𝜆⁻¹_𝛽⁻¹₎,𝜑_𝛽⁻¹𝑣_{(1,𝛽𝜆𝛽}⁻¹₎=ℎ_(1,𝛼)⋅𝑣₍₀₎⊗ℎ_(2,𝛽)⋅𝑤₍₀₎⊗ℎ_(3,𝜆)𝑤_(1,𝜆)𝜑_𝛽⁻¹𝑣_{(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)𝜑_𝛽⁻¹𝑣_{(1,𝛽𝜆𝛽}⁻¹₎=1_(1,𝛼)⋅𝑣₍₀₎⊗1_(2,𝛽)⋅𝑤₍₀₎⊗1_(3,𝜆)𝑤_(1,𝜆)𝜑_𝛽⁻¹𝑣_{(1,𝛽𝜆𝛽}⁻¹₎=1_(1,𝛼)⋅𝑣₍₀₎⊗1_(2,𝛽)1^′_1,𝛽⋅𝑤₍₀₎⊗1^′_()2,𝜆𝑤_(1,𝜆)×𝜑_𝛽⁻¹𝑣_{(1,𝛽𝜆𝛽}⁻¹₎=1_(1,𝛼)⋅𝑣₍₀₎⊗1_(2,𝛽)⋅𝑤₍₀₎⊗𝑤_(1,𝜆)𝜑_𝛽⁻¹𝑣_{(1,𝛽𝜆𝛽}⁻¹₎=𝑣₍₀₎⊗𝑤₍₀₎⊗𝑤_(1,𝜆)𝜑_𝛽⁻¹𝑣_{(1,𝛽𝜆𝛽}⁻¹₎=(𝑣⊗𝑤)₍₀₎⊗(𝑣⊗𝑤)_(1,𝜆). (27)
This finishes the proof.

Proposition 4. Let (𝑉,𝜌^𝑉)∈𝒲𝒴𝒟_𝛼(𝐻), and let 𝛽∈𝜋. Set ^𝛽𝑉=𝑉 as vector space, with action and coaction structures defined by ℎ⊳^𝛽𝑣=^𝛽𝜑_𝛽⁻¹(ℎ)⋅𝑣,∀ℎ∈𝐻_𝛽𝛼𝛽⁻¹,^𝛽𝑣∈^𝛽𝜌𝑉,_{𝛽𝑉𝜆𝛽}𝑣=^𝛽𝑣₍₀₎⊗𝜑_𝛽𝑣_(1,𝛽⁻¹_𝜆𝛽)∶=𝑣_⟨0⟩⊗𝑣_⟨1,𝜆⟩,∀^𝛽𝑣∈^𝛽𝑉. (28) Then, ^𝛽𝑉∈𝒲𝒴𝒟_𝛽𝛼𝛽⁻¹(𝐻).

Proof. Obviously, ^𝛽𝑉 is a left 𝐻_𝛽𝛼𝛽⁻¹-module, and conditions (16) and (17) are straightforward. Then, it remains to show that conditions (20) and (21) hold. For all ^𝛽𝑣∈^𝛽𝑉, we have 1_{(1,𝛽𝛼𝛽}⁻¹₎⊳𝑣_⟨0⟩⊗1_(2,𝜆)𝑣_⟨1,𝜆⟩=^𝛽𝜑_𝛽⁻¹1_{(1,𝛽𝛼𝛽}⁻¹₎⋅𝑣₍₀₎⊗1_(2,𝜆)𝜑_𝛽𝑣_(1,𝛽⁻¹_𝜆𝛽)=^𝛽1_(1,𝛼)⋅𝑣₍₀₎⊗𝜑_𝛽1_(2,𝜆)𝜑_𝛽𝑣_(1,𝛽⁻¹_𝜆𝛽)=𝑣_⟨0⟩⊗𝑣_⟨1,𝜆⟩. (29)
Next, for all ℎ∈𝐻_𝛽𝛼𝛽⁻¹_𝜆, ^𝛽𝑣∈^𝛽𝑉, we get ℎ_{(2,𝛽𝛼𝛽}⁻¹₎⊳^𝛽𝑣_⟨0⟩⊗ℎ_{(2,𝛽𝛼𝛽}⁻¹₎⊳^𝛽𝑣_⟨1,𝜆⟩×𝜑_{𝛽𝛼−1𝛽}⁻¹ℎ_{(1,𝛽𝛼𝛽}⁻¹_{𝜆𝛽𝛼−1𝛽}⁻¹₎=^𝛽𝜑_𝛽⁻¹ℎ_{(2,𝛽𝛼𝛽}⁻¹₎⋅𝑣₍₀₎⊗𝜑_𝛽𝜑_𝛽⁻¹ℎ_{(2,𝛽𝛼𝛽}⁻¹₎⋅𝑣_{(1,𝛽−1𝜆𝛽)},𝜑_{𝛽𝛼−1𝛽}⁻¹ℎ_{(1,𝛽𝛼𝛽}⁻¹_{𝜆𝛽𝛼−1𝛽}⁻¹₎=^𝛽𝜑_𝛽⁻¹(ℎ)_(2,𝛼)⋅𝑣₍₀₎⊗𝜑_𝛽𝜑_𝛽⁻¹(ℎ)_(2,𝛼)⋅𝑣_{(1,𝛽−1𝜆𝛽)}×𝜑_𝛼−1𝜑_𝛽⁻¹(ℎ)_(1,𝛼𝛽⁻¹_{𝜆𝛽𝛼−1)}=^𝛽𝜑_𝛽⁻¹(ℎ)_(1,𝛼)⋅𝑣₍₀₎⊗𝜑_𝛽𝜑_𝛽⁻¹(ℎ)_(2,𝛽⁻¹_𝜆𝛽)𝑣_(1,𝛽⁻¹_𝜆𝛽)=^𝛽𝜑_𝛽⁻¹ℎ_{(1,𝛽𝛼𝛽}⁻¹₎⋅𝑣₍₀₎⊗𝜑_𝛽𝜑_𝛽⁻¹ℎ_(2,𝜆)𝑣_(1,𝛽⁻¹_𝜆𝛽)=ℎ_{(1,𝛽𝛼𝛽}⁻¹₎⊳^𝛽𝑣₍₀₎⊗𝜑_𝛽𝜑_𝛽⁻¹ℎ_(2,𝜆)𝑣_(1,𝛽⁻¹_𝜆𝛽)=ℎ_{(1,𝛽𝛼𝛽}⁻¹₎⊳^𝛽𝑣_⟨0⟩⊗ℎ_(2,𝜆)𝑣_⟨1,𝜆⟩. (30)
This completes the proof of the proposition.

Remark 5. Let (𝑉,𝜌^𝑉)∈𝒲𝒴𝒟_𝛼(𝐻) and let (𝑊,𝜌^𝑊)∈𝒲𝒴𝒟_𝛽(𝐻); then we have ^𝑠𝑡𝑉=^𝑠(^𝑡𝑉) as an object in 𝒲𝒴𝒟_{𝑠𝑡𝛼𝑡−1𝑠−1}(𝐻) and ^𝑠(𝑉⊗_𝑡𝛼𝛽𝑊)=^𝑠𝑉⊗_{𝑡𝑠𝛼𝛽𝑠−1}^𝑠𝑊 as an object in 𝒲𝒴𝒟_{𝑠𝛼𝛽𝑠−1}(𝐻).

Proposition 6. Let (𝑉,𝜌^𝑉)∈𝒲𝒴𝒟_𝛼(𝐻); (𝑊,𝜌^𝑊)∈𝒲𝒴𝒟_𝛽(𝐻). Set ^𝑉𝑊=^𝛼𝑊 as an object in 𝒲𝒴𝒟_{𝛼𝛽𝛼−1}(𝐻). Define the map 𝑐_𝑉,𝑊∶𝑉⊗𝑊⟶^𝑉𝑐𝑊⊗𝑉,_𝑉,𝑊(𝑣⊗𝑤)=^𝛼𝑆_𝛽⁻¹𝑣_(1,𝛽⁻¹₎⋅𝑤⊗𝑣₍₀₎. (31) Then, 𝑐_𝑉,𝑊 is 𝐻-linear, 𝐻-colinear and satisfies the following conditions: 𝑐_{𝑉⊗𝑊,𝑈}=𝑐_{𝑉,𝑊𝑈}⊗id_𝑊∘id_𝑉⊗𝑐_𝑊,𝑈,𝑐_{𝑉,𝑊⊗𝑈}=id_𝑉𝑊⊗𝑐_𝑉,𝑈∘𝑐_𝑉,𝑊⊗id_𝑈. (32) Furthermore, 𝑐_{𝛾𝑉,𝛾𝑊}=𝑐_𝑉,𝑊, for all 𝛾∈𝜋.

Proof. Firstly, we need to show that 𝑐_𝑉,𝑊 is well defined. Indeed, we have 𝑐_𝑉⋅𝑊1_(1,𝛼)⋅𝑣⊗1_(2,𝛽)=⋅𝑤_𝛼𝑆_𝛽⁻¹1_(1,𝛼)⋅𝑣_(1,𝛽−1)1_(2,𝛽)⋅𝑤⊗1_(1,𝛼)⋅𝑣₍₀₎=^𝛼𝑆_𝛽⁻¹𝑣_(1,𝛽⁻¹₎𝑆⁻¹𝜑_𝛼−11_{(1,𝛼𝛽𝛼−1)}×𝑆_𝛽⁻¹1_(3,𝛽⁻¹₎1_(4,𝛽)⋅𝑤⊗1_(2,𝛼)⋅𝑣₍₀₎=^𝛼𝑆_𝛽⁻¹𝑆⁻¹𝜑_𝛼−11_{(1,𝛼𝛽𝛼−1)}×𝑆_𝛽⁻¹𝑣_(1,𝛽⁻¹₎𝜀^𝑠_𝛽1_(3,1)⋅𝑤⊗1_(2,𝛼)⋅𝑣₍₀₎=^𝛼𝑆_𝛽⁻¹𝑆⁻¹𝜑_𝛼−11_{(1,𝛼𝛽𝛼−1)}𝑆_𝛽⁻¹1_(3,𝛽⁻¹₎𝑣_(1,𝛽⁻¹₎⋅𝑤⊗1_(2,𝛼)⋅𝑣₍₀₎=^𝛼𝑆_𝛽⁻¹𝑣_(1,𝛽⁻¹₎⋅𝑤⊗𝑣₍₀₎=𝑐_𝑉,𝑊(𝑣⊗𝑤). (33)
Secondly, we prove that 𝑐_𝑉,𝑊 is 𝐻-linear. For all ℎ∈𝐻_𝛼𝛽, we compute 𝑐_𝑉,𝑊((=ℎ⋅𝑣⊗𝑤))_𝛼𝑆_𝛽⁻¹ℎ_(1,𝛼)⋅𝑣_(1,𝛽−1)ℎ_(2,𝛽)⋅𝑤⊗ℎ_(1,𝛼)⋅𝑣₍₀₎=^𝛼𝑆_𝛽⁻¹ℎ_(3,𝛽⁻¹₎𝑣_(1,𝛽⁻¹₎𝑆⁻¹𝜑_𝛼−1ℎ_{(1,𝛼𝛽𝛼−1)}ℎ_(4,𝛽)⋅𝑤⊗ℎ_(2,𝛼)⋅𝑣₍₀₎=^𝛼𝑆_𝛽⁻¹𝑣_(1,𝛽⁻¹₎𝑆⁻¹𝜑_𝛼−1ℎ_{(1,𝛼𝛽𝛼−1)}𝜀^𝑠_𝛽ℎ_(3,1)⋅𝑤⊗ℎ_(2,𝛼)⋅𝑣₍₀₎=^𝛼𝑆_𝛽⁻¹𝑣_(1,𝛽⁻¹₎𝑆⁻¹𝜑_𝛼−1ℎ_{(1,𝛼𝛽𝛼−1)}×𝑆_𝛽⁻¹1_(2,𝛽⁻¹₎⋅𝑤⊗ℎ_(2,𝛼)1_(1,𝛼)⋅𝑣₍₀₎=^𝛼𝑆_𝛽⁻¹1_(3,𝛽⁻¹₎𝑣_(1,𝛽⁻¹₎𝑆⁻¹𝜑_𝛼−11_{(1,𝛼𝛽𝛼−1)}×𝑆⁻¹𝜑_𝛼−1ℎ_{(1,𝛼𝛽𝛼−1)}⋅𝑤⊗ℎ_(2,𝛼)1_(2,𝛼)⋅𝑣₍₀₎=^𝛼𝑆_𝛽⁻¹𝑣_(1,𝛽⁻¹₎𝑆⁻¹𝜑_𝛼−1ℎ_{(1,𝛼𝛽𝛼−1)}⋅𝑤⊗ℎ_(2,𝛼)⋅𝑣₍₀₎=^𝛼𝜑_𝛼−1ℎ_{(1,𝛼𝛽𝛼−1)}𝑆_𝛽⁻¹𝑣_(1,𝛽⁻¹₎⋅𝑤⊗ℎ_(2,𝛼)⋅𝑣₍₀₎=ℎ_{(1,𝛼𝛽𝛼−1)}⊳^𝛼𝑆_𝛽⁻¹𝑣_(1,𝛽⁻¹₎⋅𝑤⊗ℎ_(2,𝛼)⋅𝑣=ℎ⋅𝑐_𝑉,𝑊(𝑣⊗𝑤) (34) as required.
Finally, we check that 𝑐_𝑉,𝑊 is satisfing the 𝐻-colinear condition. In fact, 𝜌^𝑉_𝜆^𝑊⊗𝑉∘𝑐_𝑉,𝑊()=𝑣⊗𝑤_𝛼𝑆_𝛽⁻¹𝑣_(1,𝛽⁻¹₎⋅𝑤₍₀₎⊗𝑣₍₀₎₍₀₎⊗𝑣_(0)(1,𝜆)𝑆_𝛽⁻¹𝑣_(1,𝛽⁻¹₎⋅𝑤_(1,𝜆)=^𝛼𝑆_𝛽⁻¹𝑣_(1,𝛽⁻¹_)(2,𝛽)⋅𝑤₍₀₎⊗𝑣₍₀₎₍₀₎⊗𝑣_(0)(1,𝜆)𝑆_𝛽⁻¹𝑣_(1,𝛽⁻¹_)(3,𝜆)𝑤_(1,𝜆)𝑆⁻¹𝜑_𝛽⁻¹𝑆_𝛽⁻¹𝑣_(1,𝛽⁻¹_{)(1,𝛽𝜆−1𝛽−1)}=^𝛼𝑆_𝛽⁻¹𝑣_(3,𝛽⁻¹₎⋅𝑤₍₀₎⊗𝑣₍₀₎⊗𝑣_(1,𝜆)𝑆_𝜆⁻¹𝑣_(2,𝜆⁻¹₎×𝑤_(1,𝜆)𝑆⁻¹𝜑_𝛽⁻¹𝑆_𝛽𝜆𝛽⁻¹𝑣_{(4,𝛽𝜆𝛽}⁻¹₎=^𝛼𝑆_𝛽⁻¹𝑣_(2,𝛽⁻¹₎⋅𝑤₍₀₎⊗𝑣₍₀₎⊗𝜀^𝑡_𝜆𝑣_(1,1)×𝑤_(1,𝜆)𝜑_𝛽⁻¹𝑣_{(3,𝛽𝜆𝛽}⁻¹₎=^𝛼𝑆_𝛽⁻¹1_(2,𝛽⁻¹₎𝑣_(1,𝛽⁻¹₎⋅𝑤₍₀₎⊗𝑣₍₀₎⊗𝑆_𝜆⁻¹1_(1,𝜆⁻¹₎𝑤_(1,𝜆)𝜑_𝛽⁻¹𝑣_{(2,𝛽𝜆𝛽}⁻¹₎=^𝛼𝑆_𝛽⁻¹1_(2,𝛽⁻¹₎𝑣_(1,𝛽⁻¹₎⋅𝑤₍₀₎⊗𝑣₍₀₎⊗𝑆_𝜆⁻¹1_(1,𝜆⁻¹₎𝑤_(1,𝜆)𝜑_𝛽⁻¹𝑆⁻¹𝑆_𝛽𝜆𝛽⁻¹1_{(3,𝛽𝜆𝛽}⁻¹₎𝜑_𝛽⁻¹𝑣_{(2,𝛽𝜆𝛽}⁻¹₎=^𝛼𝑆_𝛽⁻¹𝑣_(1,𝛽⁻¹₎𝑆_𝛽⁻¹(1)_(2,𝛽)⋅𝑤₍₀₎⊗𝑣₍₀₎⊗𝑆_𝛽⁻¹(1)_(3,𝜆)𝑤_(1,𝜆),𝜑_𝛽⁻¹𝑆⁻¹𝑆_𝛽⁻¹(1)_(1,𝛽𝜆⁻¹_𝛽⁻¹₎𝜑_𝛽⁻¹𝑣_{(2,𝛽𝜆𝛽}⁻¹₎=^𝛼𝑆_𝛽⁻¹𝑣_(1,𝛽⁻¹₎⋅𝑤₍₀₎⊗𝑣₍₀₎⊗𝑤_(1,𝜆)𝜑_𝛽⁻¹𝑣_{(2,𝛽𝜆𝛽}⁻¹₎=𝑐_𝑉,𝑊⊗id_𝐻𝜆𝑣₍₀₎⊗𝑤₍₀₎⊗𝑤_(1,𝜆)𝜑_𝛽⁻¹×𝑣_{(1,𝛽𝜆𝛽}⁻¹₎=𝑐_𝑉,𝑊⊗id_𝐻𝜆∘𝜌_𝜆^𝑉⊗𝑊(𝑣⊗𝑤). (35)
The rest of proof is easy to get and we omit it.

Lemma 7. The map 𝑐_𝑉,𝑊 defined by (31) is bijective with inverse 𝑐⁻¹_𝑉,𝑊∶^𝑉𝑐𝑊⊗𝑉⟶𝑉⊗𝑊,_{−1𝑉,𝑊𝛼}𝑤⊗𝑣=𝑣₍₀₎⊗𝑣_(1,𝛽)⋅𝑤, (36) for all 𝑣∈𝑉,^𝛼𝑤∈^𝑉𝑊.

Proof. Firstly, we prove 𝑐⁻¹_𝑉,𝑊∘𝑐_𝑉,𝑊=id_𝑉⊗𝑊. For all 𝑣∈𝑉, 𝑤∈𝑊, we have 𝑐⁻¹_𝑉,𝑊∘𝑐_𝑉,𝑊(𝑣⊗𝑤)=𝑣₍₀₎₍₀₎⊗𝑣_(0)(1,𝛽)𝑆_𝛽⁻¹𝑣_(1,𝛽⁻¹₎⋅𝑤=𝑣₍₀₎⊗𝜀^𝑡_𝛽𝑣_(1,1)⋅𝑤=1_(1,𝛼)⋅𝑣₍₀₎⊗1_(2,𝛽)𝜀^𝑡_𝛽𝑣_(1,1)⋅𝑤=1_(1,𝛼)𝑆⁻¹𝜀^𝑡_𝛼−1𝑣_(1,1)⋅𝑣₍₀₎⊗1_(2,𝛽)1⋅𝑤=𝜀^′_()2,1𝑣_(1,1)1_(1,𝛼)1^′_()1,𝛼⋅𝑣₍₀₎⊗1_(2,𝛽)⋅𝑤=1_(1,𝛼)⋅𝑣⊗1_(2,𝛽)⋅𝑤=𝑣⊗𝑤. (37)
Secondly, we check 𝑐_𝑉,𝑊∘𝑐⁻¹_𝑉,𝑊=id_{𝑉𝑊⊗𝑉} as follows: 𝑐_𝑉,𝑊∘𝑐⁻¹_𝑉,𝑊^𝛼=𝑤⊗𝑣_𝛼𝑆_𝛽⁻¹𝑣_(0)(1,𝛽⁻¹₎𝑣_(1,𝛽)⋅𝑤⊗𝑣₍₀₎₍₀₎=^𝛼𝜑_𝛼−11_{(1,𝛼𝛽𝛼−1)}𝑆_𝛽⁻¹𝑣_(0)(1,𝛽⁻¹₎𝑣_(1,𝛽)⋅𝑤⊗1_(2,𝛼)⋅𝑣₍₀₎₍₀₎=^𝛼𝜑_𝛼−11_{(1,𝛼𝛽𝛼−1)}𝜀^𝑠_𝛽𝑣_(1,1)⋅𝑤⊗1_(2,𝛼)⋅𝑣₍₀₎=^𝛼𝜑_𝛼−11_{(1,𝛼𝛽𝛼−1)}𝜀^𝑠_{𝛼𝛽𝛼−1}𝜑_𝛼𝑣_(1,1)⋅𝑤⊗1_(2,𝛼)⋅𝑣₍₀₎=^𝛼𝜑_𝛼−11_{(1,𝛼𝛽𝛼−1)}⋅𝑤⊗1_(2,𝛼)×𝑆_𝛼−1𝜀^𝑠_𝛼𝜑_𝛼𝑣_(1,1)⋅𝑣₍₀₎=^𝛼𝜑_𝛼−11_{(1,𝛼𝛽𝛼−1)}⋅𝑤⊗1_(2,𝛼)𝜀^𝑡_𝛼𝜑_𝛼𝑆𝑣_(1,1)⋅𝑣₍₀₎=^𝛼𝜑_𝛼−11_{(1,𝛼𝛽𝛼−1)}⋅𝑤⊗1_(2,𝛼)×𝜀𝜑_𝛼−11^′_()1,1𝑆𝑣_(1,1)1^′_()2,𝛼⋅𝑣₍₀₎=^𝛼𝜑_𝛼−11_{(1,𝛼𝛽𝛼−1)}⋅𝑤⊗1_(2,𝛼)𝜀𝑆𝑣_(1,1)⋅𝑣₍₀₎=1_{(1,𝛼𝛽𝛼−1)}⊳^𝛼𝑤⊗1_(2,𝛼)=⋅𝑣_𝛼𝑤⊗𝑣. (38)
This completes the proof.

Define 𝒲𝒴𝒟(𝐻)=∐_𝛼∈𝜋𝒲𝒴𝒟_𝛼(𝐻), the disjoint union of the categories 𝒲𝒴𝒟_𝛼(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 𝐻^𝑇={𝐻^𝑡_𝛼∶=𝜀^𝑡_𝛼(𝐻₁)}_𝛼∈𝜋.

The group homomorphism 𝜓∶𝐺→aut(𝒲𝒴𝒟(𝐻)); 𝛽→𝜓_𝛽 is given on components as 𝜓_𝛽∶𝒲𝒴𝒟_𝛼(𝐻)⟶𝒲𝒴𝒟_𝛽𝛼𝛽⁻¹(𝐻), (39) where the functor 𝜓_𝛽 acts as follows: given a morphism 𝑓∶(𝑉,𝜌^𝑉)→(𝑊,𝜌^𝑊), for any 𝑣∈𝑉, we set (^𝛽𝑓)(^𝛽𝑣)=^𝛽(𝑓(𝑣)).

The braiding in 𝒲𝒴𝒟(𝐻) is given by the family {𝑐_𝑉,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 𝑝_𝛼⊗ℎ𝑝_𝛽⊗𝑔=𝑝_𝛼𝑝_𝛽⊗ℎ𝑔, (40) for all 𝑝_𝛼,𝑝_𝛽∈𝑘(𝐺), ℎ,𝑔∈𝐻, and the comultiplication, counit, and antipode are given by Δ_𝑢,𝑣𝑝_𝛼⊗ℎ=_{𝑢𝑣=𝛼}𝑝_𝑢⊗ℎ₁⊗𝑝_𝑣⊗ℎ₂,𝜀𝑝_𝛼⊗ℎ=𝛿_𝛼,1𝜀(ℎ),𝑆𝑝_𝛼⊗ℎ=𝑝_𝛼−1⊗𝑆(ℎ). (41)

Moreover, 𝑘(𝐺)⊗𝐻 is a weak crossed Hopf group coalgebra with the following crossing: Φ_𝛽𝑝_𝛼⊗ℎ=𝑝_𝛽⁻¹_𝛼𝛽⊗ℎ. (42)

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

4. Braided 𝑇-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 _𝐻𝒲ℒ^𝐻 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,𝜌^𝑉={𝜌^𝑉_𝜆}_𝜆∈𝜋), where 𝑉 is a left 𝐻_𝛼-module and, for any 𝜆∈𝜋, 𝜌^𝑉_𝜆∶𝑉→𝑉⊗𝐻_𝜆 is a𝑘-linear morphism, such that(1)𝑉is coassociative in the sense that, for any 𝜆₁,𝜆₂∈𝜋, we have id_𝑉⊗Δ_𝜆1,𝜆2∘𝜌^𝑉_𝜆1𝜆2=𝜌^𝑉_𝜆1⊗id_𝐻𝜆2∘𝜌^𝑉_𝜆2; (43)(2)𝑉 is counitary in the sense that id_𝑉⊗𝜀∘𝜌^𝑉₁=id_𝑉; (44)(3)𝑉 satisfies the following compatible condition: 𝜌^𝑉_𝜆(𝑥⋅𝑣)=𝑥⋅𝑣₍₀₎⊗𝑣_(1,𝜆); (45) where 𝑥∈𝐻_𝛼 and 𝑣∈𝑉.

Now, we can form the category _𝐻𝒲ℒ^𝐻_𝛼 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 _𝐻𝒲ℒ^𝐻=∐_{𝐻𝛼∈𝜋}𝒲ℒ^𝐻_𝛼, the disjoint union of the categories _𝐻𝒲ℒ^𝐻_𝛼 for all 𝛼∈𝜋.

Proposition 11. The category _𝐻𝒲ℒ^𝐻 is a monoidal category. Moreover, for any 𝛼,𝛽∈𝐺, let 𝑉∈_𝐻𝒲ℒ^𝐻_𝛼 and let 𝑊∈_𝐻𝒲ℒ^𝐻_𝛽. Set 𝑉⊗𝑊=𝑣⊗𝑤∈𝑉⊗𝑊∣𝑣⊗𝑤=1_(1,𝛼)⋅𝑣⊗1_(2,𝛽)𝑤⋅𝑤=𝜀_(1,1)𝜑_𝛽⁻¹𝑣_(1,1)𝑣₍₀₎⊗𝑤₍₀₎. (46) Then, 𝑉⊗𝑊∈_𝐻𝒲ℒ^𝐻_𝛼 with the following structures: 𝑥⋅(𝑣⊗𝑤)=𝑥_(1,𝛼)⋅𝑣⊗𝑥_(2,𝛽)𝜌⋅𝑤,_𝑉𝑊⊗_𝜆(𝑣⊗𝑤)=𝑣₍₀₎⊗𝑤₍₀₎⊗𝑤_(1,𝜆)𝜑_𝛽⁻¹𝑣_{(1,𝛽𝜆𝛽}⁻¹₎, (47) for all 𝑥∈𝐻_𝛼𝛽, 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⊗𝐻_𝛼}_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⊗𝜑_𝛼}_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 _𝐻𝒲ℒ^𝐻 is a braided 𝑇-subcategoryof Yetter-Drinfel’d category 𝒲𝒴𝒟(𝐻⊗𝐻) under the following action and coaction given by 𝛿^𝑉_𝜆(𝑣)=𝑎_𝛼⋅𝑣₍₀₎⊗𝑣_(1,1)⊗𝑆⁻¹𝑏_𝜆⁻¹=∶𝑣_[0]⊗𝑣_[1,𝜆],(ℎ⊗𝑥)⊳_𝛼𝑣=𝜎ℎ,𝑣_(1,1)𝑥⋅𝑣₍₀₎, (48) where ℎ⊗𝑥∈(𝐻⊗𝐻)_𝛼, ℎ∈𝐻₁, 𝑥∈𝐻_𝛼, 𝑣∈𝑉, and 𝑉∈_𝐻𝒲ℒ^𝐻_𝛼.
The braiding on _𝐻𝒲ℒ^𝐻, 𝜏_𝑉,𝑊∶𝑉⊗𝑊→^𝑉𝑊⊗𝑉 is given by 𝜏_𝑉,𝑊(𝑣⊗𝑤)=𝜎𝑆𝑣_(1,1),𝑤_(1,1)^𝛼𝑏_𝛽⋅𝑤₍₀₎⊗𝑎_𝛼⋅𝑣₍₀₎, (49) for all 𝑉∈_𝐻𝒲ℒ^𝐻_𝛼, 𝑊∈_𝐻𝒲ℒ^𝐻_𝛽.

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∈_𝐻𝒲ℒ^𝐻_𝛼 and 𝜆₁,𝜆₂∈𝜋id_𝑉⊗Δ_𝜆1,𝜆2∘𝛿^𝑉_𝜆1𝜆2(𝑣)=𝑎_𝛼⋅𝑣₍₀₎⊗𝑣_(1,1)⊗𝑆⁻¹𝑏_𝜆2⁻¹_𝜆1⁻¹_(2,𝜆1⁻¹₎⊗𝑣_(2,1)⊗𝑆⁻¹𝑏_𝜆2⁻¹_𝜆1⁻¹_(1,𝜆2⁻¹₎=𝑎_𝛼𝑎^′_𝛼⋅𝑣₍₀₎⊗𝑣_(1,1)⊗𝑆⁻¹𝑏_𝜆1⁻¹⊗𝑣_(2,1)⊗𝑆⁻¹𝑏^′_𝜆2−1=𝑎_𝛼⋅𝑎^′_𝛼⋅𝑣₍₀₎₍₀₎⊗𝑎^′_𝛼⋅𝑣_(0)(1,1)⊗𝑆⁻¹𝑏_𝜆1⁻¹⊗𝑣_(1,1)⊗𝑆⁻¹𝑏^′_𝜆2−1=𝛿^𝑉_𝜆1⊗id_{(𝐻⊗𝐻)𝜆2}𝑎^′_𝛼⋅𝑣₍₀₎⊗𝑣_(1,1)⊗𝑆⁻¹𝑏^′_𝜆2−1=𝛿^𝑉_𝜆1⊗id_{(𝐻⊗𝐻)𝜆2}∘𝛿^𝑉_𝜆2(𝑣). (50) Next, one directly shows that counitary condition (17) holds as follows: id_𝑉⊗𝜀∘𝛿^𝑉₁(𝑣)=𝑎_𝛼⋅𝑣₍₀₎𝜀𝑚_(1,1)𝜀𝑆⁻¹𝑏₁=𝑎_𝛼⋅𝑣𝜀𝑏₁=1_𝛼⋅𝑣=𝑣. (51)
Then, we have to prove that crossed condition (18) is satisfied. For all ℎ∈𝐻₁, 𝑥∈𝐻_𝛼, and 𝑣∈𝑉∈_𝐻𝒲ℒ^𝐻_𝛼, we have (ℎ⊗𝑥)_(2,𝛼)⋅𝑣_[0]⊗(ℎ⊗𝑥)_(3,𝜆)×𝑣_[1,𝜆]𝑆⁻¹Φ_𝛼−1(ℎ⊗𝑥)_(1,𝛼𝜆⁻¹_𝛼−1)=𝜎ℎ_(2,1),𝑎_𝛼⋅𝑣_(0)(1,1)𝑥_(2,𝛼)⋅𝑎_𝛼⋅𝑣₍₀₎₍₀₎⊗ℎ_(3,1)𝑣_(1,1)𝑆⁻¹ℎ_(1,1)⊗𝑥_(3,𝜆),𝑆⁻¹𝑏_𝜆⁻¹𝑆⁻¹𝜓_𝛼−1𝑥_(1,𝛼𝜆⁻¹_𝛼−1)=𝜎ℎ_(2,1),𝑣_(1,1)𝑥_(2,𝛼)𝑎_𝛼⋅𝑣₍₀₎⊗ℎ_(3,1)𝑣_(2,1)𝑆⁻¹ℎ_(1,1)⊗𝑥_(3,𝜆)𝑆⁻¹𝑏_𝜆⁻¹,𝑆⁻¹𝜓_𝛼−1𝑥_(1,𝛼𝜆⁻¹_𝛼−1)=𝜎ℎ_(3,1),𝑣_(2,1)𝑥_(2,𝛼)𝑎_𝛼⋅𝑣₍₀₎⊗𝑣_(1,1)ℎ_(2,1)𝑆⁻¹ℎ_(1,1)⊗𝑥_(3,𝜆),𝑆⁻¹𝜓_𝛼−1𝑥_(1,𝛼𝜆⁻¹_𝛼−1)𝑏_𝜆⁻¹=𝜎ℎ_(2,1),𝑣_(2,1)𝑎_𝛼𝑥_(1,𝛼)⋅𝑣₍₀₎⊗𝑣_(1,1)𝑆⁻¹𝜀^𝑡₁ℎ_(1,1)⊗𝑥_(3,𝜆)𝑆⁻¹𝑏_𝜆⁻¹𝑥_(2,𝜆⁻¹₎=𝜎ℎ_(2,1),𝑣_(2,1)𝑎_𝛼𝑥_(1,𝛼)⋅𝑣₍₀₎⊗𝑣_(1,1)𝑆⁻¹𝜀^𝑡₁ℎ_(1,1)⊗𝑆⁻¹𝜀^𝑡_𝜆−1𝑥_(2,1)𝑆⁻¹𝑏_𝜆⁻¹=𝜎1^′_()2,1ℎ,𝑣_(2,1)𝑎_𝛼1_(1,𝛼)𝑥⋅𝑣₍₀₎⊗𝑣_(1,1)1^′_()1,1⊗𝑆⁻¹1_(2,𝜆⁻¹₎𝑆⁻¹𝑏_𝜆⁻¹=𝜎1^′_()2,1,𝑣_(2,1)𝜎ℎ,𝑣_(3,1)𝑎_𝛼1_(1,𝛼)𝑥⋅𝑣₍₀₎⊗𝑣_(1,1)1^′_()1,1⊗𝑆⁻¹𝑏_𝜆⁻¹1_(2,𝜆⁻¹₎=𝜀𝑣_(2,1)1_(2,1)𝜎ℎ,𝑣_(3,1)𝑎_𝛼𝑥⋅𝑣₍₀₎⊗𝑣_(1,1)1_(1,1)⊗𝑆⁻¹𝑏_𝜆⁻¹=𝜎ℎ,𝑣_(2,1)𝑎_𝛼𝑥⋅𝑣₍₀₎⊗𝑣_(1,1)⊗𝑆⁻¹𝑏_𝜆⁻¹=𝜎ℎ,𝑣_(1,1)𝑎_𝛼⋅𝑥⋅𝑣₍₀₎₍₀₎⊗𝑥⋅𝑣_(0)(1,1)⊗𝑆⁻¹𝑏_𝜆⁻¹=𝛿^𝑉_𝜆(ℎ⊗𝑥)⊳_𝛼𝑣. (52)
Finally, it follows from Proposition 6, the braiding on 𝒲𝒴𝒟(𝐻⊗𝐻), that the braiding on _𝐻𝒲ℒ^𝐻 is as the following: 𝜏_𝑉,𝑊(𝑣⊗𝑤)=^𝛼𝑆_𝛽⁻¹𝑣_[1,𝛽⁻¹_]⊳_𝛽𝑤⊗𝑣_[0]=𝜎𝑆𝑣_(1,1),𝑤_(1,1)^𝛼𝑏_𝛽⋅𝑤₍₀₎⊗𝑎_𝛼⋅𝑣₍₀₎, (53) for all 𝑉∈_𝐻𝒲ℒ^𝐻_𝛼, 𝑊∈_𝐻𝒲ℒ^𝐻_𝛽, 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).