New Braided T-Categories over Weak Crossed Hopf Group Coalgebras
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 .
Braided crossed categories over a group (i.e., braided ), introduced by Turaev  in the study of homotopy quantum field theories, are braided monoidal categories in Freyd-Yetter categories of crossed -sets . 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 ).
The methods of constructing braided can be found in [5–8]. Especially, in , 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 , 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 , were introduced by Van Daele and Wang . 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 .
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  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 .
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  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 ) for a comultiplication; that is, we write
Recall from Van Daele and Wang (see ) 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 ), 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  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 : for any object , and any morphism in , we set
Recall form  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
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 for all .
Proof. Firstly, we prove . For all , , we have
Secondly, we check as follows: