Research Article | Open Access

# New Braided *T*-Categories over Weak Crossed Hopf Group Coalgebras

**Academic Editor:**Jaan Janno

#### Abstract

Let be a weak crossed Hopf group coalgebra over group ; we first introduce a kind of new *α*-Yetter-Drinfel’d module categories for and use it to construct a braided -category . As an application, we give the concept of a Long dimodule category for a weak crossed Hopf group coalgebra with quasitriangular and coquasitriangular structures and obtain that is a braided -category by translating it into a weak Yetter-Drinfel'd module subcategory .

#### 1. Introduction

Braided crossed categories over a group (i.e., braided ), introduced by Turaev [1] in the study of homotopy quantum field theories, are braided monoidal categories in Freyd-Yetter categories of crossed -sets [2]. Such categories play an important role in the construction of homotopy invariants. By using braided Virelizier [3, 4] constructed Hennings-type invariants of flat group bundles over complements of links in the 3-sphere. Braided also provide suitable mathematical formalism to describe the orbifold models of rational conformal field theory (see [5]).

The methods of constructing braided can be found in [5–8]. Especially, in [8], Zunino gave the definition of -Yetter-Drinfel’d modules over Hopf group coalgebras and constructed a braided , then proved that both the category of Yetter-Drinfel’d modules and the center of the category of representations of as well as the category of representations of the quantum double of are isomorphic as braided 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 of Yetter-Drinfel’d category .

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 .

This paper is organized as follows. In Section 1, we recall definitions and properties related to braided -categories and weak crossed Hopf group coalgebras.

In Section 2, let be a weak crossed Hopf group coalgebra over group ; is a fixed element in . We first introduce the concept of a (left-right) weak -Yetter-Drinfel’d module and define the category , where is the category of (left-right) weak -Yetter-Drinfel’d modules. Then, we show that the category is a braided -category.

In Section 3, we introduce a (left-right) weak -Long dimodule category for a weak crossed Hopf group coalgebra . Then, we obtain a new category and show that as is a quasitriangular and coquasitriangular weak crossed Hopf group coalgebra, then is a braided of Yetter-Drinfel’d category .

#### 2. Preliminary

Throughout the paper, let be a group with the unit and let be a field. All algebras, vector spaces, and so forth are supposed to be over . We use the Sweedler-type notation [15] for the comultiplication and coaction, for the flip map, and 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 is a family of -spaces together with a family of -linear maps (called a comultiplication) and a -linear map (called a counit), such that is coassociative in the sense that

We use the Sweedler-type notation (see [14]) for a comultiplication; that is, we write

Recall from Van Daele and Wang (see [11]) that a weak semi-Hopf group coalgebra is a family of algebras and at the same time a group coalgebra , such that the following conditions hold.(i)The comultiplication is a homomorphism of algebras (not necessary unit preserving) such that for all .(ii)The counit is a -linear map satisfying the identity for all .

A weak Hopf group coalgebra over is a weak semi-Hopf group coalgebra endowed with a family of -linear maps (called an antipode) satisfying the following equations: for all , and .

Let be a weak Hopf group coalgebra. Define a family of linear maps and by the formulae for any , where and are called the and counital maps.

By Van Daele and Wang (see [11]), let be a weak semi-Hopf group coalgebra. Then, we have the following equations:(1), , for all ,(2), for all ,(3), for all ,(4), , for all .

Similarly, for any and , define , . Then, we have(1), for all ,(2), for all .

A weak Hopf group coalgebra is called a weak crossed Hopf group coalgebra if it is endowed with a family of algebra isomorphisms (called a crossing) such that , , and for all .

If is crossed with the crossing , then we have

A quasitriangular weak crossed Hopf group coalgebra over is a pair where is a weak crossed Hopf group coalgebra together with a family of maps satisfying the following conditions:(1), for all ,(2), for all ,(3), for all , where , for all , and such that there exists a family of with for all . In this paper, we denote .

Recall from [16] that a coquasitriangular weak Hopf group coalgebra consists of a weak Hopf group coalgebra and a map satisfying and there exists such that for all , , where is called a weak inverse of .

##### 2.2. Braided -Categories

We recall that a monoidal category is called a crossed category over group if it consists of the following data.(1)A family of subcategories such that is a disjoint union of this family and such that for any and , . Here, the subcategory is called the th component of .(2)A group homomorphism , the conjugation, (where is the group of invertible strict tensor functors from to itself) such that for any . Here, the functors are called conjugation isomorphisms.

We will use the Turaev’s left index notation in [1]: for any object , and any morphism in , we set

Recall form [1] that a braided -category is a crossed category endowed with braiding, that is a family of isomorphisms, satisfying the following conditions:(1)for any morphism with , , we have (2)for all , we have (3)for any , ,

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

In this section, we first introduce the definition of weak -Drinfel’d modules over a weak crossed Hopf group coalgebra and then use it to construct a class of braided -categories.

*Definition 1. *Let be a weak crossed Hopf group coalgebra over group and let be a fixed element in . A (left-right) weak -Yetter-Drinfel’d module, or simply a -module, is a couple , where is a left -module and, for any , is a -linear morphism, such that(1) is coassociative in the sense that, for any , we have
(2) is counitary in the sense that
(3) is crossed in the sense that, for any , ,
where .

Given two -modules and , a morphism of this two is an -linear map , such that, for any ,

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

Proposition 2. *Equation (18) is equivalent to the following equations:
**
for any , .*

*Proof. *Assume that (20) and (21) hold for all , . We compute
as required.

Conversely, suppose that is crossed in the sense of (18). We first note that

To show that (21) is satisfied, for all , we do the following calculations:

This completes the proof.

Proposition 3. *If , , then = with the action and coaction structures as follows:
**
for all , , .*

*Proof. *It is easy to prove that is a left -module, and the proof of coassociativity of is similar to the Hopf group coalgebra case. For all , we have
This shows that is satisfing counitary condition (17).

Then, we check the equivalent form of crossed conditions (20) and (21). In fact, for all , , we have

This finishes the proof.

Proposition 4. *Let , and let . Set as vector space, with action and coaction structures defined by
**
Then, .*

*Proof. *Obviously, is a left -module, and conditions (16) and (17) are straightforward. Then, it remains to show that conditions (20) and (21) hold. For all , we have

Next, for all , , we get

This completes the proof of the proposition.

*Remark 5. *Let and let ; then we have as an object in and as an object in .

Proposition 6. *Let ; . Set as an object in . Define the map
**
Then, is -linear, -colinear and satisfies the following conditions:
**
Furthermore, , for all .*

*Proof. *Firstly, we need to show that is well defined. Indeed, we have

Secondly, we prove that is -linear. For all , we compute
as required.

Finally, we check that is satisfing the -colinear condition. In fact,

The rest of proof is easy to get and we omit it.

Lemma 7. *The map defined by (31) is bijective with inverse
*