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-dimensional homotopy 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∈π 𝒲𝒴𝒟_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 for a weak crossed Hopf group coalgebra H. Then, we obtain a new category and show that as H is a quasitriangular and coquasitriangular weak crossed Hopf group coalgebra, then 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

()

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 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
()

for all α, β, γ ∈ π.

(ii)
The counit ε : H₁ → k is a k-linear map satisfying the identity
()

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

(called an antipode) satisfying the following equations:

()

for all h ∈ H₁, g ∈ H_α, and α ∈ π.

Let H be a weak Hopf group coalgebra. Define a family of linear maps

and

by the formulae

()

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)
, , for all g, h ∈ H₁,
(2)
, for all x ∈ H_α, α, β ∈ π,
(3)
, for all x ∈ H_α, α, β ∈ π,
(4)
, , for all x, y ∈ H₁.

Similarly, for any α ∈ π and h ∈ H₁, define

. Then, we have

(1)
, for all h ∈ H_β, α, β ∈ π,
(2)
, 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 (called a crossing) such that , 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

()

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

satisfying the following conditions:

(1)
, for all h ∈ H_αβ, α, β ∈ π,
(2)
, for all α, β, γ ∈ π,
(3)
, for all α, β, γ ∈ π,

where

for all α, β ∈ π, and such that there exists a family of

with

()

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

()

and there exists σ⁻¹ : H₁ ⊗ H₁ → k such that

()

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 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

()

Recall form [1] that a braided T-category is a crossed category 𝒞 endowed with braiding, that is a family of isomorphisms,

()

satisfying the following conditions:

(1)
for any morphism f ∈ 𝒞_α(U, U^′) with α ∈ π, g ∈ 𝒞(V, V^′), we have
()
(2)
for all U, V, W ∈ 𝒞, we have
()
(3)
for any U, V ∈ 𝒞, α ∈ π,
()

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 , where V is a left H_α-module and, for any λ ∈ π, is a k-linear morphism, such that

(1)
V is coassociative in the sense that, for any λ₁, λ₂ ∈ π, we have
()
(2)
V is counitary in the sense that
()
(3)
V is crossed in the sense that, for any λ ∈ π, h ∈ H_α,
()

where

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-modules is an H_α-linear map f : V⇀W, such that, for any λ ∈ π,

()

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:

()

for any v ∈ V, h ∈ H_αλ.

Proof. Assume that (20) and (21) hold for all h ∈ H_αλ, v ∈ V. We compute

()

as required.

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

()

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

()

This completes the proof.

Proposition 3. If (V, ρ^V) ∈ 𝒲𝒴𝒟_α(H), (W, ρ^W) ∈ 𝒲𝒴𝒟_β(H), then = Δ_α,β(1_αβ)·(V ⊗ W) ∈ 𝒲𝒴𝒟_αβ(H) with the action and coaction structures as follows:

()

for all h ∈ H_αβ, λ ∈ π,

Proof. It is easy to prove that is a left H_αβ-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 h ∈ H_αβλ, , we have

()

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

()

Then,

Proof. Obviously, ^βV is a left -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

()

Next, for all , ^βv ∈ ^βV, we get

()

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 as an object in and as an object in .

Proposition 6. Let (V, ρ^V) ∈ 𝒲𝒴𝒟_α(H); (W, ρ^W) ∈ 𝒲𝒴𝒟_β(H). Set ^VW = ^αW as an object in . Define the map

()

Then, c_V,W is H-linear, H-colinear and satisfies the following conditions:

()

Furthermore,

, for all γ ∈ π.

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

()

Secondly, we prove that c_V,W is H-linear. For all h ∈ H_αβ, we compute

()

as required.

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

()

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

()

for all v ∈ V, ^αw ∈ ^VW.

Proof. Firstly, we prove . For all v ∈ V, w ∈ W, we have

()

Secondly, we check as follows:

()

This completes the proof.

Define 𝒲𝒴𝒟(H) = ∐_α∈π 𝒲𝒴𝒟_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 .

The group homomorphism ψ : G → aut(𝒲𝒴𝒟(H)); β → ψ_β is given on components as

()

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

()

for all p_α, p_β ∈ k(G), h, g ∈ H, and the comultiplication, counit, and antipode are given by

()

Moreover, k(G) ⊗ H is a weak crossed Hopf group coalgebra with the following crossing:

()

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 , where V is a left H_α-module and, for any λ ∈ π, is a k-linear morphism, such that

(1)
V is coassociative in the sense that, for any λ₁, λ₂ ∈ π, we have
()
(2)
V is counitary in the sense that
()
(3)
V satisfies the following compatible condition:
()

where x ∈ H_α and v ∈ V.

Now, we can form the category 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 , the disjoint union of the categories for all α ∈ π.

Proposition 11. The category _H𝒲ℒ^H is a monoidal category. Moreover, for any α, β ∈ G, let and let . Set

()

Then,

with the following structures:

()

for all x ∈ H_αβ,

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-subcategory of Yetter-Drinfel’d category 𝒲𝒴𝒟(H ⊗ H) under the following action and coaction given by

()

where h ⊗ x ∈ (H ⊗ H) _α, h ∈ H₁, x ∈ H_α, v ∈ V, and

The braiding on _H𝒲ℒ^H, τ_V,W : V ⊗ W → ^VW ⊗ V is given by

()

for all

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 and λ₁, λ₂ ∈ π

()

Next, one directly shows that counitary condition (17) holds as follows:

()

Then, we have to prove that crossed condition (18) is satisfied. For all h ∈ H₁, x ∈ H_α, and , we have

()

Finally, it follows from Proposition 6, the braiding on 𝒲𝒴𝒟(H ⊗ H), that the braiding on _H𝒲ℒ^H is as the following:

()

for all

, 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).

References

1 Turaev V. G., Homotopy field theory in dimension 3 and crossed group categories, http://arxiv.org/abs/math/0005291.
Google Scholar
2 Freyd P. J. and Yetter D. N., Braided compact closed categories with applications to low-dimensional topology, Advances in Mathematics. (1989) 77, no. 2, 156–182, https://doi.org/10.1016/0001-8708(89)90018-2, MR1020583, ZBL0679.57003.
10.1016/0001-8708(89)90018-2
Web of Science® Google Scholar
3 Virelizier A., Algèbres de Hopf graduées et fibrés plats sur les 3-variétés [Ph.D. thesis], 2001, Université Louis Pasteur, Strasbourg, France.
Google Scholar
4 Virelizier A., Involutory Hopf group-coalgebras and flat bundles over 3-manifolds, Fundamenta Mathematicae. (2005) 188, 241–270, https://doi.org/10.4064/fm188-0-11, MR2191947, ZBL1118.57017.
10.4064/fm188-0-11
Web of Science® Google Scholar
5 Kirillov A. J., On G-equivariant modular categories, http://arxiv.org/abs/math/0401119.
Google Scholar
6 Wang S., New Turaev braided group categories and group Schur-Weyl duality, Applied Categorical Structures. (2013) 21, no. 2, 141–166, https://doi.org/10.1007/s10485-011-9263-2, MR3027617, ZBL06148659.
10.1007/s10485-011-9263-2
Web of Science® Google Scholar
7 Yang T. and Wang S., Constructing new braided T-categories over regular multiplier Hopf algebras, Communications in Algebra. (2011) 39, no. 9, 3073–3089, https://doi.org/10.1080/00927872.2010.514876, MR2845558, ZBL1237.16031.
10.1080/00927872.2010.514876
Web of Science® Google Scholar
8 Zunino M., Yetter-Drinfel′d modules for crossed structures, Journal of Pure and Applied Algebra. (2004) 193, no. 1-3, 313–343, https://doi.org/10.1016/j.jpaa.2004.02.014, MR2076391.
10.1016/j.jpaa.2004.02.014
Web of Science® Google Scholar
9 Van Daele A., Multiplier Hopf algebras, Transactions of the American Mathematical Society. (1994) 342, no. 2, 917–932, https://doi.org/10.2307/2154659, MR1220906, ZBL0809.16047.
10.1090/S0002-9947-1994-1220906-5
Web of Science® Google Scholar
10 Van Daele A. and Wang S. H., Weak multiplier Hopf algebras I. The main theory, Journal für die reine und angewandte Mathematik. (2013) https://doi.org/10.1515/crelle-2013-0053.
10.1515/crelle-2013-0053
Web of Science® Google Scholar
11 Van Daele A. and Wang S., New braided crossed categories and Drinfel′d quantum double for weak Hopf group coalgebras, Communications in Algebra. (2008) 36, no. 6, 2341–2386, https://doi.org/10.1080/00927870701509420, MR2418391, ZBL1154.16031.
10.1080/00927870701509420
Web of Science® Google Scholar
12 Böhm G., Nill F., and Szlachányi K., Weak Hopf algebras. I. Integral theory and C^∗-structure, Journal of Algebra. (1999) 221, no. 2, 385–438, https://doi.org/10.1006/jabr.1999.7984, MR1726707.
10.1006/jabr.1999.7984
Web of Science® Google Scholar
13 Zhou X. and Wang S., The duality theorem for weak Hopf algebra (co) actions, Communications in Algebra. (2010) 38, no. 12, 4613–4632, https://doi.org/10.1080/00927870903443303, MR2764842, ZBL1220.16024.
10.1080/00927870903443303
Web of Science® Google Scholar
14 Virelizier A., Hopf group-coalgebras, Journal of Pure and Applied Algebra. (2002) 171, no. 1, 75–122, https://doi.org/10.1016/S0022-4049(01)00125-6, MR1903398, ZBL1011.16023.
10.1016/S0022-4049(01)00125-6
Web of Science® Google Scholar
15 Sweedler M. E., Hopf Algebras, 1969, W. A. Benjamin, New York, NY, USA, Mathematics Lecture Note Series, MR0252485.
Google Scholar
16 Zhou X. and Yang T., Kegel′s theorem over weak Hopf group coalgebras, Journal of Mathematics. (2013) 33, no. 2, 228–236, MR3076229.
CAS Google Scholar

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New Braided T-Categories over Weak Crossed Hopf Group Coalgebras

Abstract

1. Introduction

2. Preliminary

2.1. Weak Crossed Hopf Group Coalgebras

2.2. Braided T-Categories

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

4. Braided T-Categories over Weak Long Dimodule Categories

Acknowledgments

References

References

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New Braided T-Categories over Weak Crossed Hopf Group Coalgebras

Abstract

1. Introduction

2. Preliminary

2.1. Weak Crossed Hopf Group Coalgebras

2.2. Braided T-Categories

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

4. Braided T-Categories over Weak Long Dimodule Categories

Acknowledgments

References

References

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