Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 626394 10 pageshttpdxdoiorg1011552013626394

Research ArticleNew Braided T-Categories over Weak Crossed HopfGroup Coalgebras

Xuan Zhou1 and Tao Yang2

1 Mathematics and Information Technology School Jiangsu Second Normal University Nanjing Jiangsu 210013 China2Department of Mathematics Nanjing Agricultural University Nanjing Jiangsu 210095 China

Correspondence should be addressed to Xuan Zhou zhouxuanseu126com

Received 29 August 2013 Accepted 4 October 2013

Academic Editor Jaan Janno

Copyright copy 2013 X Zhou and T YangThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Let119867 be a weak crossed Hopf group coalgebra over group 120587 we first introduce a kind of new 120572-Yetter-Drinfelrsquod module categoriesWYD

120572(119867) for 120572 isin 120587 and use it to construct a braided 119879-category WYD(119867) As an application we give the concept of a Long

dimodule category119867WL119867 for a weak crossed Hopf group coalgebra119867 with quasitriangular and coquasitriangular structures and

obtain that119867WL119867 is a braided 119879-category by translating it into a weak Yetter-Drinfelrsquod module subcategoryWYD(119867 otimes 119867)

1 Introduction

Braided crossed categories over a group 120587 (ie braided119879-categories) introduced by Turaev [1] in the study of3-dimensional homotopy quantum field theories are braidedmonoidal categories in Freyd-Yetter categories of crossed120587-sets [2] Such categories play an important role in theconstruction of homotopy invariants By using braided119879-categories Virelizier [3 4] constructed Hennings-typeinvariants of flat group bundles over complements of linksin the 3-sphere Braided 119879-categories also provide suitablemathematical formalism to describe the orbifold models ofrational conformal field theory (see [5])

Themethods of constructing braided 119879-categories can befound in [5ndash8] Especially in [8] Zunino gave the definitionof 120572-Yetter-Drinfelrsquod modules over Hopf group coalgebrasand constructed a braided 119879-category then proved that boththe category of Yetter-Drinfelrsquod modules YD(119867) and thecenter of the category of representations of 119867 as well asthe category of representations of the quantum double of119867 are isomorphic as braided 119879-categories Furthermore in[6] Wang considered the dual setting of Zuninorsquos partialresults formed the category of Long dimodules over Hopfgroup algebras and proved that the category is a braided119879-subcategory of Yetter-Drinfelrsquod categoryYD(119867 otimes 119861)

Weak multiplier Hopf algebras as a further developmentof the notion of the well-known multiplier Hopf algebras[9] were introduced by Van Daele and Wang [10] Examplesof such weak multiplier Hopf algebras can be constructedfrom weak Hopf group coalgebras [10 11] Furthermore theconcepts of weak Hopf group coalgebras are also regard as anatural generalization ofweakHopf algebras [12 13] andHopfgroup coalgebras [14]

In this paper we mainly generalize the above construc-tions shown in [6 8] replacing their Hopf group coalgebras(or Hopf group algebras) by weak crossedHopf group coalge-bras [11] and provide new examples of braided 119879-categories

This paper is organized as follows In Section 1 we recalldefinitions and properties related to braided119879-categories andweak crossed Hopf group coalgebras

In Section 2 let119867 be a weak crossed Hopf group coalge-bra over group 120587 120572 is a fixed element in 120587 We first introducethe concept of a (left-right) weak 120572-Yetter-Drinfelrsquod moduleand define the category WYD(119867) = ∐

120572isin120587WYD

120572(119867)

where WYD120572(119867) is the category of (left-right) weak 120572-

Yetter-Drinfelrsquod modules Then we show that the categoryWYD(119867) is a braided 119879-category

In Section 3 we introduce a (left-right) weak 120572-Longdimodule category

119867WL119867

120572for a weak crossed Hopf group

2 Abstract and Applied Analysis

coalgebra 119867 Then we obtain a new category119867WL119867

=

∐120572isin120587 119867

WL119867

120572and show that as 119867 is a quasitriangular

and coquasitriangular weak crossed Hopf group coalgebrathen

119867WL119867 is a braided 119879-subcategory of Yetter-Drinfelrsquod

categoryWYD(119867 otimes 119867)

2 Preliminary

Throughout the paper let 120587 be a group with the unit 1 andlet 119896 be a field All algebras vector spaces and so forth aresupposed to be over 119896 We use the Sweedler-type notation[15] for the comultiplication and coaction 119905 for the flip mapand id for the identity map In the section we will recall somebasic definitions and results related to our paper

21 Weak Crossed Hopf Group Coalgebras Recall fromTuraev and Virelizier (see [1 14]) that a group coalgebra over120587 is a family of 119896-spaces 119862 = 119862

120572120572isin120587

together with a familyof 119896-linear maps Δ = Δ

120572120573 119862120572120573

rarr 119862120572otimes 119862120573120572120573isin120587

(calleda comultiplication) and a 119896-linear map 120576 119862

1rarr 119896 (called a

counit) such that Δ is coassociative in the sense that

(Δ120572120573

otimes id119862120574)Δ120572120573120574

= (id119862120572

otimes Δ120573120574

) Δ120572120573120574

forall120572 120573 120574 isin 120587

(id119862120572

otimes 120576) Δ1205721

= id119862120572

= (120576 otimes id119862120572) Δ1120572

forall120572 isin 120587

(1)

We use the Sweedler-type notation (see [14]) for a comul-tiplication that is we write

Δ120572120573

(119888) = 119888(1120572)

otimes 119888(2120573)

for any 120572 120573 isin 120587 119888 isin 119862120572120573 (2)

Recall from Van Daele and Wang (see [11]) that a weaksemi-Hopf group coalgebra 119867 = 119867

120572 119898120572 1120572 Δ 120576120572isin120587

isa family of algebras 119867

120572 119898120572 1120572120572isin120587

and at the same timea group coalgebra 119867

120572 Δ = Δ

120572120573 120576120572120573isin120587

such that thefollowing conditions hold

(i) The comultiplication Δ120572120573

119862120572120573

rarr 119862120572otimes 119862120573is a

homomorphism of algebras (not necessary unit pre-serving) such that

(Δ120572120573

otimes id119867120574)Δ120572120573120574

(1120572120573120574

)

= (Δ120572120573

(1120572120573) otimes 1120574) (1120572otimes Δ120573120574

(1120573120574))

(Δ120572120573

otimes id119867120574)Δ120572120573120574

(1120572120573120574

)

= (1120572otimes Δ120573120574

(1120573120574)) (Δ

120572120573(1120572120573) otimes 1120574)

(3)

for all 120572 120573 120574 isin 120587

(ii) The counit 120576 1198671

rarr 119896 is a 119896-linear map satisfyingthe identity

120576 (119892119909ℎ) = 120576 (119892119909(21)

) 120576 (119909(11)

ℎ) = 120576 (119892119909(11)

) 120576 (119909(21)

ℎ) (4)

for all 119892 ℎ 119909 isin 1198671

A weakHopf group coalgebra over 120587 is a weak semi-Hopfgroup coalgebra 119867 = 119867

120572 119898120572 1120572 Δ 120576120572isin120587

endowed with afamily of 119896-linear maps 119878 = 119878

120572 119867120572rarr 119867120572minus1120572isin120587

(called anantipode) satisfying the following equations

119898120572(119878120572minus1 otimes id

119867120572) Δ120572minus1120572(ℎ) = 1

(1120572)120576 (ℎ1(21)

)

119898120572(id119867120572

otimes 119878120572minus1) Δ120572120572minus1 (ℎ) = 120576 (1

(11)ℎ) 1(2120572)

119878120572(119892(1120572)

) 119892(2120572minus1)119878120572(119892(3120572)

) = 119878120572(119892)

(5)

for all ℎ isin 1198671 119892 isin 119867

120572 and 120572 isin 120587

Let119867 be a weak Hopf group coalgebra Define a family oflinear maps 120576

119905= 120576119905

120572 1198671rarr 119867

120572120572isin120587

and 120576119904= 120576119904

120572 1198671rarr

119867120572120572isin120587

by the formulae

120576119905

120572(ℎ) = 120576 (1

(11)ℎ) 1(2120572)

= 119898120572(id119867120572

otimes 119878120572minus1) Δ120572120572minus1 (ℎ)

120576119904

120572(ℎ) = 1

(1120572)120576 (ℎ1(21)

) = 119898120572(119878120572minus1 otimes id

119867120572) Δ120572minus1120572(ℎ)

(6)

for any ℎ isin 1198671 where 120576

119905 and 120576119904 are called the 120587-target and

120587-source counital mapsBy Van Daele and Wang (see [11]) let119867 be a weak semi-

Hopf group coalgebraThen we have the following equations

(1) 120576(119892ℎ) = 120576(119892120576119905

1(ℎ)) 120576(119892ℎ) = 120576(120576

119904

1(119892)ℎ) for all 119892 ℎ isin

1198671

(2) 119909(1120572)

otimes120576119905

120573(119909(21)

) = 1(1120572)

119909otimes1(2120573)

for all 119909 isin 119867120572 120572 120573 isin

120587(3) 120576119904120573(119909(11)

)otimes119909(2120572)

= 1(1120573)

otimes1199091(2120572)

for all 119909 isin 119867120572 120572 120573 isin

120587(4) 120576119905120572(120576119905

1(119909)119910) = 120576

119905

120572(119909)120576119905

120572(119910) 120576119904120572(119909120576119904

1(119910)) = 120576

119904

120572(119909)120576119904

120572(119910) for

all 119909 119910 isin 1198671

Similarly for any 120572 isin 120587 and ℎ isin 1198671 define 120576

119905

120572(ℎ) =

120576(ℎ1(11)

)1(2120572)

120576119904120572(ℎ) = 1

(1120572)120576(1(21)

ℎ) Then we have

(1) 120576119904120572(ℎ(11)

)otimesℎ(2120573)

= 1(1119886)

otimes1(2120573)

ℎ for all ℎ isin 119867120573 120572 120573 isin

120587(2) 119909(1120572)

otimes120576119905

120572(119909(21)

) = 1199091(1120572)

otimes1(2120573)

for all 119909 isin 119867120572 120572 120573 isin

120587

AweakHopf group coalgebra119867 = 119867120572119898120572 1120572 Δ 120576 119878

120572isin120587

is called a weak crossedHopf group coalgebra if it is endowedwith a family of algebra isomorphisms 120593 = 120593

120572 119867120573

rarr

119867120572120573120572minus1120572120573isin120587

(called a crossing) such that (120593120572otimes 120593120572) ∘ Δ120573120574

=

Δ120572120573120572minus1120572120574120572minus1 ∘120593120572 120576∘120593120572= 120576 and120593

120572120573= 120593120572∘120593120573for all 120572 120573 120574 isin 120587

If 119867 is crossed with the crossing 120593 = 120593120572120572isin120587

then wehave

120593120573∘ 120576119904

120572= 120576119904

120573120572120573minus1 ∘ 120593120573 120593120573∘ 120576119905

120572

= 120576119905

120573120572120573minus1 ∘ 120593120573 forall120572 120573 isin 120587

(7)

A quasitriangular weak crossed Hopf group coalgebraover 120587 is a pair (119867 119877) where 119867 is a weak crossed Hopfgroup coalgebra together with a family of maps 119877 = 119877

120572120573isin

Δcop120573minus1120572minus1(1120572120573)(119867120572

otimes 119867120573)Δ120572120573

(1120572120573) satisfying the following

conditions

Abstract and Applied Analysis 3

(1) 119877120572120573

Δ120572120573

(ℎ) = Δcop120573minus1120572minus1(ℎ)119877

120572120573 for all ℎ isin 119867

120572120573 120572 120573 isin

120587(2) (id

119867120572otimes Δ120573120574

)(119877120572120573120574

) = (119877120572120574

)11205733

(119877120572120573

)12120574

for all 120572 120573120574 isin 120587

(3) (Δ120572120573

otimesid119867120574) (119877120573minus1120572minus1120574) = (119877

120572minus1120574)1120573minus13(119877120573minus1120574)120572minus123 for

all 120572 120573 120574 isin 120587

whereΔ120572120573

= (120593120573otimesid119867120573minus1)∘Δ120573minus1120572minus1120573120573minus1 Δcop120572120573

= 119905119867120572minus1 119867120573minus1∘(120593120573otimes

id119867120573minus1) ∘Δ120573minus1120572minus1120573120573minus1 for all 120572 120573 isin 120587 and such that there exists

a family of119877 = 119877120572120573

isin Δ120572120573

(1120572120573)(119867120572otimes119867120573)Δ

cop120573minus1120572minus1(1120572120573)with

119877120572120573

119877120572120573

= Δcop120573minus1120572minus1 (1120572120573) 119877

120572120573119877120572120573

= Δ120572120573

(1120572120573)

(120593120573otimes 120593120573) (119877120572120574

) = 119877120573120572120573minus1120573120574120573minus1

(8)

for all 120572 120573 120574 isin 120587 In this paper we denote 119877120572120573

= 119886120572otimes 119887120573

Recall from [16] that a coquasitriangular weak Hopfgroup coalgebra (119867 120590) consists of a weak Hopf groupcoalgebra 119867 = 119867

120572 119898120572 1120572 Δ 120576 119878

120572isin120587and a map 120590 119867

1otimes

1198671rarr 119896 satisfying

120590 (ℎ(11)

119892(11)

) ℎ(2120572)

119892(2120572)

= 119892(1120572)

ℎ(1120572)

120590 (ℎ(21)

119892(21)

)

120590 (119886 119887119888) = 120590 (119886(11)

119888) 120590 (119886(21)

119887)

120590 (119886119887 119888) = 120590 (119886 119888(11)

) 120590 (119887 119888(21)

)

120576 (119886(11)

119887(11)

) 120590 (119887(21)

119886(21)

) 120576 (119887(31)

119886(31)

) = 120590 (119887 119886)

(9)

and there exists 120590minus1 1198671otimes 1198671rarr 119896 such that

120590 (119886(11)

119887(11)

) 120590minus1

(119886(21)

119887(21)

) = 120576 (119887119886)

120590minus1

(119886(11)

119887(11)

) 120590 (119886(21)

119887(21)

) = 120576 (119886119887)

120576 (119886(11)

119887(11)

) 120590minus1

(119886(21)

119887(21)

) 120576 (119887(31)

119886(31)

) = 120590minus1

(119886 119887)

(10)

for all ℎ 119892 isin 119867120572 119886 119887 119888 isin 119867

1 where 120590

minus1 is called a weakinverse of 120590

22 Braided 119879-Categories We recall that a monoidal cate-goryC is called a crossed category over group 120587 if it consistsof the following data

(1) A family of subcategories C120572120572isin120587

such that C is adisjoint union of this family and such that for any119880 isin C

120572and 119881 isin C

120573 119880 otimes 119881 isin C

120572120573 Here the

subcategoryC120572is called the 120572th component ofC

(2) A group homomorphism 120595 120587 rarr aut(C) 120573 997891rarr

120595120573 the conjugation (where aut(C) is the group of

invertible strict tensor functors fromC to itself) suchthat 120595

120573(C120572) = C

120573120572120573minus1 for any 120572 120573 isin 120587 Here the

functors 120595120573are called conjugation isomorphisms

We will use the Turaevrsquos left index notation in [1] for anyobject 119880 isin C

120572 119881119882 isin C

120573and any morphism 119891 119881 rarr 119882

inC120573 we set119880

119881 = 120595120572(119881) isin C

120572120573120572minus1 119880

119891 = 120595120572(119891)

119880

119881 997888rarr119880

119882

(11)

Recall form [1] that a braided 119879-category is a crossedcategory C endowed with braiding that is a family ofisomorphisms

119888 = 119888119880119881

isin C (119880 otimes 119881 (119880

119881) otimes 119880) 119880119881isinC

(12)

satisfying the following conditions

(1) for any morphism 119891 isin C120572(119880 1198801015840

) with 120572 isin 120587 119892 isin

C(119881 1198811015840

) we have

( (120572

119892) otimes 119891) ∘ 119888119880119881

= 11988811988010158401198811015840 ∘ (119891 otimes 119892) (13)

(2) for all 119880119881119882 isin C we have

119888119880otimes119881119882

= 119886 119880otimes119881119882119880119881

∘ (119888119880119881119882

otimes id119881) ∘ 119886minus1

119880119881119882119881

∘ (id119880otimes 119888119881119882

) ∘ 119886119880119881119882

119888119880119881otimes119882

= 119886minus1

119880119881119880119882119880

∘ (id119880119881

otimes 119888119880119882

) ∘ 119886 119880119881119880119882

∘ (119888119880119881

otimes id119882) ∘ 119886minus1

119880119881119882

(14)

(3) for any 119880119881 isin C 120572 isin 120587

120595120572(119888119880119881

) = 119888120595120572(119880)120595120572(119881)

(15)

3 Yetter-Drinfelrsquod Categories for WeakCrossed Hopf Group Coalgebras

In this section we first introduce the definition of weak120572-Yetter-Drinfelrsquod modules over a weak crossed Hopf groupcoalgebra 119867 and then use it to construct a class of braided119879-categories

Definition 1 Let 119867 be a weak crossed Hopf group coalgebraover group 120587 and let 120572 be a fixed element in 120587 A (left-right) weak 120572-Yetter-Drinfelrsquod module or simply a WYD

120572-

module is a couple 119881 = (119881 120588119881

= 120588119881

120582120582isin120587

) where 119881 is a left119867120572-module and for any 120582 isin 120587 120588119881

120582 119881 rarr 119881otimes119867

120582is a 119896-linear

morphism such that

(1) 119881 is coassociative in the sense that for any 1205821 1205822isin 120587

we have

(id119881otimes Δ1205821 1205822

) ∘ 120588119881

12058211205822

= (120588119881

1205821

otimes id1198671205822

) ∘ 120588119881

1205822

(16)

(2) 119881 is counitary in the sense that

(id119881otimes 120576) ∘ 120588

119881

1= id119881 (17)

(3) 119881 is crossed in the sense that for any 120582 isin 120587 ℎ isin 119867120572

120588119881

120582(ℎ sdot V) = ℎ

(2120572)sdot V(0)

otimes ℎ(3120582)

V(1120582)

119878minus1

120593120572minus1 (ℎ(1120572120582minus1120572minus1))

(18)

where 120588119881120582(V) = V

(0)otimes V(1120582)

4 Abstract and Applied Analysis

Given two WYD120572-modules (119881 120588

119881

) and (119882 120588119882

) amorphism 119891 (119881 120588

119881

) rarr (119882 120588119882

) of this two WYD120572-

modules is an 119867120572-linear map 119891 119881 119882 such that for any

120582 isin 120587

120588119882

120582∘ 119891 = (119891 otimes id

119867120582) ∘ 120588119881

120582 (19)

Then we can form the categoryWYD120572(119867) ofWYD

120572-

modules where the composition of morphisms of WYD120572-

modules is the standard composition of the underlying linearmaps

Proposition 2 Equation (18) is equivalent to the followingequations

ℎ(1120572)

sdot V(0)

otimes ℎ(2120582)

V(1120582)

= (ℎ(2120572)

sdot V)(0)

otimes (ℎ(2120572)

sdot V)(1120582)

120593120572minus1 (ℎ(1120572120582120572

minus1))

(20)

120588119881

120582(V) = V

(0)otimes V(1120582)

isin 119881otimes119905120572120582119867120582= Δ120572120582

(1120572120582) sdot (119881 otimes 119867

120582)

(21)

for any V isin 119881 ℎ isin 119867120572120582

Proof Assume that (20) and (21) hold for all ℎ isin 119867120572120582 V isin 119881

We compute

ℎ(2120572)

sdot V(0)

otimes ℎ(3120582)

V(1120582)

119878minus1

120593120572minus1 (ℎ(1120572120582minus1120572minus1))

= (ℎ(3120572)

sdot V)(0)

otimes (ℎ(3120572)

sdot V)(1120582)

times 120593120572minus1 (ℎ(2120572120582120572

minus1)) 119878minus1

120593120572minus1 (ℎ(1120572120582minus1120572minus1))

= (ℎ(3120572)

sdot V)(0)

otimes (ℎ(3120572)

sdot V)(1120582)

times 120593120572minus1 (ℎ(2120572120582120572

minus1)119878minus1

(ℎ(1120572120582minus1120572minus1)))

= (ℎ(2120572)

sdot V)(0)

otimes (ℎ(2120572)

sdot V)(1120582)

times 120593120572minus1119878minus1

(120576119905

120572120582minus1120572minus1 (ℎ(11)

))

= (11015840

(2120572)ℎ(2120572)

sdot V)(0)

otimes (11015840

(2120572)ℎ(2120572)

sdot V)(1120582)

times 120593120572minus1 (1(1120572120582120572

minus1)) times 120576 (1

(21)11015840

(11)ℎ(11)

)

= (1(2120572)

ℎ sdot V)(0)

otimes (1(2120572)

ℎ sdot V)(1120582)

120593120572minus1 (1(1120572120582120572

minus1))

= 1(1120572)

sdot (ℎ sdot V)(0)

otimes 1(2120582)

(ℎ sdot V)(1120582)

= (ℎ sdot V)(0)

otimes (ℎ sdot V)(1120582)

(22)

as requiredConversely suppose that119881 is crossed in the sense of (18)

We first note that

V(0)

otimes V(1120582)

= 1(2120572)

sdot V(0)

otimes 1(3120582)

V(1120582)

119878minus1

120593120572minus1 (1(1120572120582minus1120572minus1))

= 11015840

(1120572)1(2120572)

sdot V(0)

otimes 11015840

(2120582)V(1120582)

119878minus1

times 120593120572minus1 (1(1120572120582minus1120572minus1))

= 11015840

(1120572)sdot (1(2120572)

sdot V(0))

otimes 11015840

(2120582)[V(1120582)

119878minus1

120593120572minus1 (1(1120572120582minus1120572minus1))]

isin 119881otimes119905120572120582119867120582

(23)

To show that (21) is satisfied for all ℎ isin 119867120572120582 we do the

following calculations

(ℎ(2120572)

sdot V)(0)

otimes (ℎ(2120572)

sdot V)(1120582)

120593120572minus1 (ℎ(1120572120582120572

minus1))

= ℎ(3120572)

sdot V(0)

otimes ℎ(4120582)

V(1120582)

119878minus1

120593120572minus1 (ℎ(2120572120582minus1120572minus1))

times 120593120572minus1 (ℎ(1120572120582120572

minus1))

= ℎ(2120572)

sdot V(0)

otimes ℎ(3120582)

V(1120582)

120593120572minus1119878minus1

(120576119904

120572120582minus1120572minus1 (ℎ(11)

))

= ℎ(2120572)

11015840

(2120572)sdot V(0)

otimes ℎ(3120582)

V(1120582)

120593120572minus1119878minus1

(1(1120572120582minus1120572minus1))

times 120576 (ℎ(11)

11015840

(11)1(21)

)

= ℎ(1120572)

1(2120572)

sdot V(0)

otimes ℎ(2120582)

V(1120582)

120593120572minus1119878minus1

(1(1120572120582minus1120572minus1))

= ℎ(1120572)

11015840

(1120572)1(2120572)

sdot V(0)

otimes ℎ(2120582)

11015840

(2120582)V(1120582)

119878minus1

times 120593120572minus1 (1(1120572120582minus1120572minus1))

= ℎ(1120572)

1(2120572)

sdot V(0)

otimes ℎ(2120582)

1(3120582)

V(1120582)

119878minus1

times 120593120572minus1 (1(1120572120582minus1120572minus1))

= ℎ(1120572)

sdot V(0)

otimes ℎ(2120582)

V(1120582)

(24)

This completes the proof

Proposition 3 If (119881 120588119881

) isin WYD120572(119867) (119882 120588

119882

) isin

WYD120573(119867) then 119881otimes

119905120572120573119882 = Δ

120572120573(1120572120573) sdot (119881 otimes 119882) isin

WYD120572120573(119867) with the action and coaction structures as

follows

ℎ sdot (V otimes 119908) = ℎ(1120572)

sdot V otimes ℎ(2120573)

sdot 119908

120588119881otimes119905120572120573119882

120582(V otimes 119908) = V

(0)otimes 119908(0)

otimes 119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

(25)

for all ℎ isin 119867120572120573 120582 isin 120587 V otimes 119908 isin 119881otimes

119905120572120573119882

Proof It is easy to prove that 119881otimes119905120572120573119882 is a left 119867

120572120573-module

and the proof of coassociativity of 119881otimes119905120572120573119882 is similar to the

Hopf group coalgebra case For all V otimes 119908 isin 119881otimes119905120572120573119882 we have

(id119881otimes119905120572120573119882

otimes 120576) ∘ 120588119881otimes119905120572120573119882

1(V otimes 119908)

= 120576 (119908(11)

120593120573minus1(11015840

)(21)

)

times 120576 (120593120573minus1(11015840

)(11)

120593120573minus1 (V(11)

)) V(0)

otimes 119908(0)

Abstract and Applied Analysis 5

= 120576 ((1(2120573)

sdot 119908)(11)

120593120573minus1 (1(11)

) 120593120573minus1 (11015840

(21)))

times 120576 (120593120573minus1 (11015840

(11)) 120593120573minus1 (V(11)

)) V(0)

otimes (1(2120573)

sdot 119908)(0)

= 120576 (1(51)

119908(11)

119878minus1

120593120573minus1 (1(31)

) 120593120573minus1 (1(21)

))

times 120576 (1(11)

V(11)

) V(0)

otimes 1(4120573)

sdot 119908(0)

= 120576 (1(41)

119908(11)

120593120573minus1119878minus1

(120576119904

1(1(21)

))) 120576 (1(11)

V(11)

) V(0)

otimes 1(3120573)

sdot 119908(0)

= 120576 (1(41)

119908(11)

120576119905

1119878minus1

120593120573minus1 (1(21)

)) 120576 (1(11)

V(11)

) V(0)

otimes 1(3120573)

sdot 119908(0)

= 120576 ((1(2120573)

sdot 119908)(11)

) 120576 (1(11)

V(11)

) V(0)

otimes (1(2120573)

sdot 119908)(0)

= 120576 ((1(2120572)

sdot V)(11)

120593120572minus1 (1(11)

)) (1(2120572)

sdot V)(0)

otimes 1(3120573)

sdot 119908

= 120576 (1(31)

V(11)

120593120572minus1119878minus1

120576119904

1(1(11)

)) 1(2120572)

sdot V(0)

otimes 1(4120573)

sdot 119908

= 120576 ((1(1120572)

sdot V)(11)

) (1(1120572)

sdot V)(0)

otimes 1(2120573)

sdot 119908

= V otimes 119908

(26)

This shows that119881otimes119905120572120573119882 is satisfing counitary condition (17)

Then we check the equivalent form of crossed conditions(20) and (21) In fact for all ℎ isin 119867

120572120573120582 V otimes 119908 isin 119881otimes

119905120572120573119882 we

have

(ℎ(2120572120573)

sdot (V otimes 119908))(0)

otimes (ℎ(2120572120573)

sdot (V otimes 119908))(1120582)

120593(120572120573)minus1 (ℎ(1120572120573120582120573

minus1120572minus1))

= (ℎ(2120572)

sdot V)(0)

otimes (ℎ(3120573)

sdot 119908)(0)

otimes (ℎ(3120573)

sdot 119908)(1120582)

120593120573minus1 ((ℎ(2120572)

sdot V)(1120573120582120573

minus1))

120593120573minus1120572minus1 (ℎ(1120572120573120582120573

minus1120572minus1))

= ℎ(3120572)

sdot V(0)

otimes ℎ(6120573)

sdot 119908(0)

otimes ℎ(7120582)

119908(1120582)

119878minus1

120593120573minus1 (ℎ(5120573120582minus1120573minus1))

120593120573minus1 (ℎ(4120573120582120573

minus1)V(1120573120582120573

minus1)) 120593120573minus1120593120572minus1119878minus1

times (ℎ(2120572120573120582

minus1120573minus1120572minus1)) 120593120573minus1120572minus1 (ℎ(1120572120573120582120573

minus1120572minus1))

= ℎ(3120572)

sdot V(0)

otimes ℎ(6120573)

sdot 119908(0)

otimes ℎ(7120582)

119908(1120582)

119878minus1

120593120573minus1 (ℎ(5120573120582minus1120573minus1))

120593120573minus1 (ℎ(4120573120582120573

minus1)V(1120573120582120573

minus1)) 120593120573minus1120572minus1119878minus1

times (119878 (ℎ(1120572120573120582120573

minus1120572minus1)) ℎ(2120572120573120582

minus1120573minus1120572minus1))

= ℎ(1120572)

1(2120572)

sdot V(0)

otimes ℎ(4120573)

sdot 119908(0)

otimes ℎ(5120582)

119908(1120582)

119878minus1

120593120573minus1 (ℎ(3120573120582minus1120573minus1))

120593120573minus1 (ℎ(2120573120582120573

minus1)V(1120573120582120573

minus1)) 120593120573minus1120572minus1119878minus1

(1(1120572120573120582

minus1120573minus1120572minus1))

= ℎ(1120572)

1(2120572)

sdot V(0)

otimes ℎ(4120573)

sdot 119908(0)

otimes ℎ(5120582)

119908(1120582)

119878minus1

120593120573minus1 (ℎ(3120573120582minus1120573minus1))

120593120573minus1 (ℎ(2120573120582120573

minus1)1(3120573120582120573

minus1)V(1120573120582120573

minus1)119878minus1

times120593120572minus1 (1(1120572120573120582

minus1120573minus1120572minus1)))

= ℎ(1120572)

sdot V(0)

otimes ℎ(3120573)

sdot 119908(0)

otimes ℎ(4120582)

119908(1120582)

times 120593120573minus1119878minus1

(120576119904

120573120582minus1120573minus1 (ℎ(21)

)) 120593120573minus1 (V(1120573120582120573

minus1))

= ℎ(1120572)

sdot V(0)

otimes ℎ(2120573)

11015840

(1120573)1(2120573)

sdot 119908(0)

otimes ℎ(3120582)

11015840

(2120582)119908(1120582)

119878minus1

120593120573minus1 (1(1120573120582minus1120573minus1))

120593120573minus1 (V(1120573120582120573

minus1))

= ℎ(1120572)

sdot V(0)

otimes ℎ(2120573)

sdot 119908(0)

otimes ℎ(3120582)

119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

= ℎ(1120572120573)

sdot (V otimes 119908)(0)

otimes ℎ(2120582)

(V otimes 119908)(1120582)

1(1120572120573)

sdot (V otimes 119908)(0)

otimes 1(2120582)

(V otimes 119908)(1120582)

= 1(1120572120573)

sdot (V(0)

otimes 119908(0)) otimes 1(2120582)

119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

= 1(1120572)

sdot V(0)

otimes 1(2120573)

sdot 119908(0)

otimes 1(3120582)

119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

= 1(1120572)

sdot V(0)

otimes 1(2120573)

11015840

(1120573)sdot 119908(0)

otimes 11015840

(2120582)119908(1120582)

times 120593120573minus1 (V(1120573120582120573

minus1))

= 1(1120572)

sdot V(0)

otimes 1(2120573)

sdot 119908(0)

otimes 119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

= V(0)

otimes 119908(0)

otimes 119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

= (V otimes 119908)(0)

otimes (V otimes 119908)(1120582)

(27)

This finishes the proof

Proposition 4 Let (119881 120588119881) isin WYD120572(119867) and let 120573 isin 120587 Set

120573

119881 = 119881 as vector space with action and coaction structuresdefined by

ℎ ⊳120573V =120573

(120593120573minus1 (ℎ) sdot V) forallℎ isin 119867

120573120572120573minus1 120573V isin120573

119881

120588120573119881

120582(120573V) =

120573

(V(0)) otimes 120593120573(V(1120573minus1120582120573)

)

= V⟨0⟩

otimes V⟨1120582⟩

forall120573V isin120573

119881

(28)

Then120573

119881 isin WYD120573120572120573minus1(119867)

6 Abstract and Applied Analysis

Proof Obviously 120573119881 is a left119867120573120572120573minus1-module and conditions

(16) and (17) are straightforward Then it remains to showthat conditions (20) and (21) hold For all 120573V isin

120573

119881 we have

1(1120573120572120573

minus1)⊳ V⟨0⟩

otimes 1(2120582)

V⟨1120582⟩

=120573

(120593120573minus1 (1(1120573120572120573

minus1)) sdot V(0)) otimes1(2120582)

120593120573(V(1120573minus1120582120573)

)

=120573

(1(1120572)

sdot V(0)) otimes 120593120573(1(2120582)

) 120593120573(V(1120573minus1120582120573)

)

= V⟨0⟩

otimes V⟨1120582⟩

(29)

Next for all ℎ isin 119867120573120572120573minus1120582 120573V isin

120573

119881 we get

(ℎ(2120573120572120573

minus1)⊳120573V)⟨0⟩

otimes (ℎ(2120573120572120573

minus1)⊳120573V)⟨1120582⟩

times 120593120573120572minus1120573minus1 (ℎ(1120573120572120573

minus1120582120573120572minus1120573minus1))

=

120573

((120593120573minus1 (ℎ(2120573120572120573

minus1)) sdot V)(0)

)

otimes 120593120573((120593120573minus1 (ℎ(2120573120572120573

minus1)) sdot V)(1120573minus1120582120573)

)

120593120573120572minus1120573minus1 (ℎ(1120573120572120573

minus1120582120573120572minus1120573minus1))

=

120573

((120593120573minus1(ℎ)(2120572)

sdot V)(0)

)

otimes 120593120573((120593120573minus1(ℎ)(2120572)

sdot V)(1120573minus1120582120573)

times 120593120572minus1 (120593120573minus1(ℎ)(1120572120573minus1120582120573120572minus1)))

=120573

(120593120573minus1(ℎ)(1120572)

sdot V(0))

otimes 120593120573(120593120573minus1(ℎ)(2120573minus1120582120573)

V(1120573minus1120582120573)

)

=120573

(120593120573minus1 (ℎ(1120573120572120573

minus1)) sdot V(0))

otimes 120593120573(120593120573minus1 (ℎ(2120582)

) V(1120573minus1120582120573)

)

= ℎ(1120573120572120573

minus1)⊳120573

(V(0))

otimes 120593120573(120593120573minus1 (ℎ(2120582)

) V(1120573minus1120582120573)

)

= ℎ(1120573120572120573

minus1)⊳120573V⟨0⟩

otimes ℎ(2120582)

V⟨1120582⟩

(30)

This completes the proof of the proposition

Remark 5 Let (119881 120588119881

) isin WYD120572(119867) and let (119882 120588

119882

) isin

WYD120573(119867) then we have 119904119905119881 =

119904

(119905

119881) as an object inWYD

119904119905120572119905minus1119904minus1(119867) and 119904(119881otimes

119905120572120573119882) =

119904

119881otimes119905119904120572120573119904minus1

119904

119882 as anobject inWYD

119904120572120573119904minus1(119867)

Proposition 6 Let (119881 120588119881

) isin WYD120572(119867) (119882 120588

119882

) isin

WYD120573(119867) Set 119881119882 =

120572

119882 as an object inWYD120572120573120572minus1(119867)

Define the map

119888119881119882

119881 otimes119882 997888rarr119881

119882otimes119881

119888119881119882

(V otimes 119908) =120572

(119878120573minus1 (V(1120573minus1)) sdot 119908) otimesV

(0)

(31)

Then 119888119881119882

is 119867-linear 119867-colinear and satisfies the followingconditions

119888119881otimes119882119880

= (119888119881

119882

119880otimes id119882) ∘ (id

119881otimes 119888119882119880

)

119888119881119882otimes119880

= (id119881119882

otimes 119888119881119880

) ∘ (119888119881119882

otimes id119880)

(32)

Furthermore 119888120574119881120574119882

= 119888119881119882

for all 120574 isin 120587

Proof Firstly we need to show that 119888119881119882

is well definedIndeed we have

119888119881sdot119882

(1(1120572)

sdot V otimes 1(2120573)

sdot 119908)

=

120572

(119878120573minus1 ((1(1120572)

sdot V)(1120573minus1)) 1(2120573)

sdot 119908) otimes(1(1120572)

sdot V)(0)

=120572

(119878120573minus1 (V(1120573minus1)119878minus1

120593120572minus1 (1(1120572120573120572

minus1)))

times 119878120573minus1 (1(3120573minus1)) 1(4120573)

sdot 119908) otimes 1(2120572)

sdot V(0)

=120572

(119878120573minus1119878minus1

120593120572minus1 (1(1120572120573120572

minus1))

times 119878120573minus1 (V(1120573minus1)) 120576119904

120573(1(31)

) sdot 119908) otimes 1(2120572)

sdot V(0)

=120572

(119878120573minus1119878minus1

120593120572minus1 (1(1120572120573120572

minus1)) 119878120573minus1 (1(3120573minus1)V(1120573minus1)) sdot 119908)

otimes 1(2120572)

sdot V(0)

=120572

(119878120573minus1 (V(1120573minus1)) sdot 119908) otimes V

(0)

= 119888119881119882

(V otimes 119908)

(33)

Secondly we prove that 119888119881119882

is119867-linear For all ℎ isin 119867120572120573

we compute

119888119881119882

(ℎ sdot (V otimes 119908))

=

120572

(119878120573minus1 ((ℎ(1120572)

sdot V)(1120573minus1)) ℎ(2120573)

sdot 119908) otimes (ℎ(1120572)

sdot V)(0)

=120572

(119878120573minus1 (ℎ(3120573minus1)V(1120573minus1)119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) ℎ(4120573)

sdot 119908)

otimes ℎ(2120572)

sdot V(0)

=120572

(119878120573minus1 (V(1120573minus1)119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) 120576119904

120573(ℎ(31)

) sdot 119908)

otimes ℎ(2120572)

sdot V(0)

=120572

(119878120573minus1 (V(1120573minus1)119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1)))

times 119878120573minus1 (1(2120573minus1)) sdot 119908) otimes ℎ

(2120572)1(1120572)

sdot V(0)

Abstract and Applied Analysis 7

=120572

(119878120573minus1 (1(3120573minus1)V(1120573minus1)119878minus1

120593120572minus1 (1(1120572120573120572

minus1))

times 119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) sdot 119908) otimes ℎ

(2120572)1(2120572)

sdot V(0)

=120572

(119878120573minus1 (V(1120573minus1)119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) sdot 119908)

otimes ℎ(2120572)

sdot V(0)

=120572

(120593120572minus1 (ℎ(1120572120573120572

minus1)) 119878120573minus1 (V(1120573minus1)) sdot 119908) otimes ℎ

(2120572)sdot V(0)

= ℎ(1120572120573120572

minus1)⊳120572

(119878120573minus1 (V(1120573minus1)) sdot 119908) otimes ℎ

(2120572)sdot V

= ℎ sdot 119888119881119882

(V otimes 119908)

(34)as required

Finally we check that 119888119881119882

is satisfing the 119867-colinearcondition In fact

120588

119881

119882otimes119881

120582∘ 119888119881119882

(V otimes 119908)

=

120572

((119878120573minus1 (V(1120573minus1)) sdot 119908)

(0)

) otimes V(0)(0)

otimes V(0)(1120582)

(119878120573minus1 (V(1120573minus1)) sdot 119908)

(1120582)

=

120572

(119878120573minus1(V(1120573minus1))(2120573)

sdot 119908(0)) otimes V

(0)(0)

otimes V(0)(1120582)

119878120573minus1(V(1120573minus1))(3120582)

119908(1120582)

119878minus1

120593120573minus1

(119878120573minus1(V(1120573minus1))(1120573120582minus1120573minus1)

)

=120572

(119878120573minus1 (V(3120573minus1)) sdot 119908(0)) otimes V(0)

otimes V(1120582)

119878120582minus1 (V(2120582minus1))

times 119908(1120582)

119878minus1

120593120573minus1119878120573120582120573minus1 (V(4120573120582120573

minus1))

=120572

(119878120573minus1 (V(2120573minus1)) sdot 119908(0)) otimes V(0)

otimes 120576119905

120582(V(11)

)

times 119908(1120582)

120593120573minus1 (V(3120573120582120573

minus1))

=120572

(119878120573minus1 (1(2120573minus1)V(1120573minus1)) sdot 119908(0)) otimes V(0)

otimes 119878120582minus1 (1(1120582minus1)) 119908(1120582)

120593120573minus1 (V(2120573120582120573

minus1))

=120572

(119878120573minus1 (1(2120573minus1)V(1120573minus1)) sdot 119908(0)) otimes V(0)

otimes 119878120582minus1 (1(1120582minus1)) 119908(1120582)

120593120573minus1119878minus1

119878120573120582120573minus1 (1(3120573120582120573

minus1)) 120593120573minus1 (V(2120573120582120573

minus1))

=120572

(119878120573minus1 (V(1120573minus1)) 119878120573minus1(1)(2120573)

sdot 119908(0))

otimes V(0)

otimes 119878120573minus1(1)(3120582)

119908(1120582)

120593120573minus1119878minus1

(119878120573minus1(1)(1120573120582minus1120573minus1)) 120593120573minus1 (V(2120573120582120573

minus1))

=120572

(119878120573minus1 (V(1120573minus1)) sdot 119908(0)) otimes V(0)

otimes 119908(1120582)

120593120573minus1 (V(2120573120582120573

minus1))

= (119888119881119882

otimes id119867120582

) (V(0)

otimes 119908(0)

otimes 119908(1120582)

120593120573minus1

times (V(1120573120582120573

minus1)))

= (119888119881119882

otimes id119867120582

) ∘ 120588119881otimes119882

120582(V otimes 119908)

(35)

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

Lemma 7 The map 119888119881119882

defined by (31) is bijective withinverse

119888minus1

119881119882119881

119882otimes119881 997888rarr 119881 otimes119882

119888minus1

119881119882(120572

119908otimes V) = V(0)

otimes V(1120573)

sdot 119908

(36)

for all V isin 119881120572

119908 isin119881

119882

Proof Firstly we prove 119888minus1

119881119882∘ 119888119881119882

= id119881otimes119882

For all V isin 119881119908 isin 119882 we have

119888minus1

119881119882∘ 119888119881119882

(V otimes 119908)

= V(0)(0)

otimes V(0)(1120573)

119878120573minus1 (V(1120573minus1)) sdot 119908

= V(0)

otimes 120576119905

120573(V(11)

) sdot 119908 = 1(1120572)

sdot V(0)

otimes 1(2120573)

120576119905

120573(V(11)

) sdot 119908

= 1(1120572)

119878minus1

120576119905

120572minus1 (V(11)

) sdot V(0)

otimes 1(2120573)

sdot 119908

= 120576 (11015840

(21)V(11)

) 1(1120572)

11015840

(1120572)sdot V(0)

otimes 1(2120573)

sdot 119908

= 1(1120572)

sdot V otimes 1(2120573)

sdot 119908 = V otimes 119908

(37)

Secondly we check 119888119881119882

∘ 119888minus1

119881119882= id 119881

119882otimes119881as follows

119888119881119882

∘ 119888minus1

119881119882(120572

119908otimesV)

=120572

(119878120573minus1 (V(0)(1120573

minus1)) V(1120573)

sdot 119908) otimesV(0)(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) 119878120573minus1 (V(0)(1120573

minus1)) V(1120573)

sdot 119908)

otimes 1(2120572)

sdot V(0)(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) 120576119904

120573(V(11)

) sdot 119908) otimes1(2120572)

sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)120576119904

120572120573120572minus1120593120572(V(11)

)) sdot 119908) otimes1(2120572)

sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes1

(2120572)

times 119878120572minus1120576119904

120572(120593120572(V(11)

)) sdot V(0)

8 Abstract and Applied Analysis

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes 1

(2120572)120576119905

120572120593120572119878 (V(11)

) sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes1

(2120572)

times 120576 (120593120572minus1 (11015840

(11)) 119878 (V(11)

)) 11015840

(2120572)sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes 1

(2120572)120576119878 (V(11)

) sdot V(0)

= 1(1120572120573120572

minus1)⊳120572

119908otimes1(2120572)

sdot V

=120572

119908otimes V(38)

This completes the proof

Define WYD(119867) = ∐120572isin120587

WYD120572(119867) the disjoint

union of the categories WYD120572(119867) for all 120572 isin 120587 If we

endow WYD(119867) with tensor product as in Proposition 3then WYD(119867) becomes a monoidal category The unit is119867119879

= 119867119905

120572= 120576119905

120572(1198671)120572isin120587

The group homomorphism 120595 119866 rarr aut(WYD(119867))

120573 rarr 120595120573is given on components as

120595120573 WYD

120572(119867) 997888rarr WYD

120573120572120573minus1 (119867) (39)

where the functor 120595120573acts as follows given a morphism 119891

(119881 120588119881

) rarr (119882 120588119882

) for any V isin 119881 we set ( 120573119891)( 120573V) =

120573

(119891(V))The braiding in WYD(119867) is given by the family 119888

119881119882

as shown in Proposition 6 Then we have the followingtheorem

Theorem 8 For a weak crossed Hopf group coalgebra 119867WYD(119867) is a braided 119879-category over group 120587

Example 9 Let 119867 be a weak Hopf algebra 119866 a finite groupand 119896(119866) the dual Hopf algebra of the group algebra 119896119866

Then have the weak Hopf group coalgebra 119896(119866) otimes119867 themultiplication in 119896(119866) otimes 119867 is given by

(119901120572otimes ℎ) (119901

120573otimes 119892) = 119901

120572119901120573otimes ℎ119892 (40)

for all 119901120572 119901120573

isin 119896(119866) ℎ 119892 isin 119867 and the comultiplicationcounit and antipode are given by

Δ119906V (119901120572 otimes ℎ) = sum

119906V=120572(119901119906otimes ℎ1) otimes (119901V otimes ℎ

2)

120576 (119901120572otimes ℎ) = 120575

1205721120576 (ℎ)

119878 (119901120572otimes ℎ) = 119901

120572minus1 otimes 119878 (ℎ)

(41)

Moreover 119896(119866) otimes 119867 is a weak crossed Hopf groupcoalgebra with the following crossing

Φ120573(119901120572otimes ℎ) = 119901

120573minus1120572120573

otimes ℎ (42)

ByTheorem 8WYD(119896(119866)otimes119867) is a braided 119879-category

4 Braided 119879-Categories over Weak LongDimodule Categories

In this section we introduce the notion of a (left-right)weak 120572-Long dimodule over a weak crossed Hopf groupcoalgebra119867 and prove that the category

119867WL119867 is a braided

119879-subcategory of Yetter-Drinfelrsquod category WYD(119867 otimes 119867)

when 119867 is a quasitriangular and coquasitriangular weakcrossed Hopf group coalgebra

Definition 10 Let119867 be a weak crossed Hopf group coalgebraover 120587 For a fixed element 120572 isin 120587 a (left-right) weak 120572-Longdimodule is a couple 119881 = (119881 120588

119881

= 120588119881

120582120582isin120587

) where 119881 is aleft 119867

120572-module and for any 120582 isin 120587 120588119881

120582 119881 rarr 119881 otimes 119867

120582is

a 119896-linear morphism such that

(1) 119881 is coassociative in the sense that for any 1205821 1205822isin 120587

we have

(id119881otimes Δ1205821 1205822

) ∘ 120588119881

12058211205822

= (120588119881

1205821

otimes id1198671205822

) ∘ 120588119881

1205822

(43)

(2) 119881 is counitary in the sense that

(id119881otimes 120576) ∘ 120588

119881

1= id119881 (44)

(3) 119881 satisfies the following compatible condition

120588119881

120582(119909 sdot V) = 119909 sdot V

(0)otimes V(1120582)

(45)

where 119909 isin 119867120572and V isin 119881

Now we can form the category119867WL119867120572of (left-right)

weak 120572-Long dimodules where the composition of mor-phisms of weak 120572-Long dimodules is the standard compo-sition of the underlying linear maps

Let119867WL119867 = ∐

120572isin120587 119867WL119867120572 the disjoint union of the

categories119867WL119867120572for all 120572 isin 120587

Proposition 11 Thecategory119867WL119867 is amonoidal category

Moreover for any 120572 120573 isin 119866 let 119881 isin119867WL119867120572

and let119882isin119867WL119867120573 Set

119881otimes119882 = V otimes 119908 isin 119881 otimes119882 | V otimes 119908

= 1(1120572)

sdot V otimes 1(2120573)

sdot 119908

= 120576 (119908(11)

120593120573minus1 (V(11)

)) V(0)

otimes 119908(0)

(46)

Then 119881 otimes119882isin119867WL119867120572with the following structures

119909 sdot (V otimes 119908) = 119909(1120572)

sdot V otimes 119909(2120573)

sdot 119908

120588119881otimes119882

120582(V otimes 119908) = V

(0)otimes 119908(0)

otimes 119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

(47)

for all 119909 isin 119867120572120573 V otimes 119908 isin 119881otimes119882

Proof It is straightforward

Abstract and Applied Analysis 9

Let (119867 120590 119877) be a coquasitriangular and quasitriangularweak crossed Hopf group coalgebra with crossing 120593 Define afamily of vector spaces 119867 otimes 119867 = (119867 otimes 119867)

120572= 1198671otimes 119867120572120572isin120587

where the 119867 on the left we consider its coquasitriangularstructure and for the right one we consider its quasitriangularstructureThen119867otimes119867 is aweak crossedHopf group coalgebrawith the natural tensor product and the crossing Φ = id otimes

120593120572120572isin120587

Theorem 12 Let (119867 120590 119877) be a weak crossed Hopf groupcoalgebra with coquasitriangular structure 120590 and quasitrian-gular structure 119877 Then the category

119867WL119867 is a braided

119879-subcategory of Yetter-Drinfelrsquod category WYD(119867 otimes 119867)

under the following action and coaction given by

120575119881

120582(V) = 119886

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1) = V

[0]otimes V[1120582]

(ℎ otimes 119909) ⊳120572V = 120590 (ℎ V

(11)) 119909 sdot V(0)

(48)

where ℎ otimes 119909 isin (119867 otimes 119867)120572 ℎ isin 119867

1 119909 isin 119867

120572 V isin 119881 and

119881isin119867WL119867120572

The braiding on119867WL119867 120591

119881119882 119881 otimes 119882 rarr

119881

119882otimes119881 isgiven by

120591119881119882

(V otimes 119908) = 120590 (119878 (V(11)

) 119908(11)

)120572

(119887120573sdot 119908(0)) otimes 119886120572sdot V(0)

(49)

for all 119881isin119867WL119867120572119882isin

119867WL119867120573

Proof Obviously119881 is a left (119867otimes119867)120572-moduleThen we show

that119881 satisfies the conditions inDefinition 1 First we need tocheck that119881 is coassociative In fact for all V isin 119881 isin

119867WL119867120572

and 1205821 1205822isin 120587

(id119881otimes Δ1205821 1205822

) ∘ 120575119881

12058211205822(V)

= 119886120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1

2120582minus1

1(2120582minus1

1))

otimes V(21)

otimes 119878minus1

(119887120582minus1

2120582minus1

1(1120582minus1

2))

= 1198861205721198861015840

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1

1

)

otimes V(21)

otimes 119878minus1

(1198871015840

120582minus1

2

)

= 119886120572sdot (1198861015840

120572sdot V(0))(0)

otimes (1198861015840

120572sdot V(0))(11)

otimes 119878minus1

(119887120582minus1

1

) otimes V(11)

otimes 119878minus1

(1198871015840

120582minus1

2

)

= (120575119881

1205821

otimes id(119867otimes119867)1205822

) (1198861015840

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(1198871015840

120582minus1

2

))

= (120575119881

1205821

otimes id(119867otimes119867)1205822

) ∘ 120575119881

1205822(V)

(50)

Next one directly shows that counitary condition (17) holdsas follows

(id119881otimes 120576) ∘ 120575

119881

1(V) = 119886

120572sdot V(0)120576 (119898(11)

) 120576119878minus1

(1198871)

= 119886120572sdot V120576 (119887

1) = 1120572sdot V = V

(51)

Then we have to prove that crossed condition (18) issatisfied For all ℎ isin 119867

1 119909 isin 119867

120572 and V isin 119881isin

119867WL119867120572 we

have(ℎ otimes 119909)

(2120572)sdot V[0]

otimes (ℎ otimes 119909)(3120582)

times V[1120582]

119878minus1

Φ120572minus1 ((ℎ otimes 119909)

(1120572120582minus1120572minus1))

= 120590 (ℎ(21)

(119886120572sdot V(0))(11)

) 119909(2120572)

sdot (119886120572sdot V(0))(0)

otimes ℎ(31)

V(11)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(119887120582minus1) 119878minus1

120595120572minus1 (119909(1120572120582minus1120572minus1))

= 120590 (ℎ(21)

V(11)

) 119909(2120572)

119886120572sdot V(0)

otimes ℎ(31)

V(21)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(119887120582minus1)

119878minus1

120595120572minus1 (119909(1120572120582minus1120572minus1))

= 120590 (ℎ(31)

V(21)

) 119909(2120572)

119886120572sdot V(0)

otimes V(11)

ℎ(21)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(120595120572minus1 (119909(1120572120582minus1120572minus1)) 119887120582minus1)

= 120590 (ℎ(21)

V(21)

) 119886120572119909(1120572)

sdot V(0)

otimes V(11)

119878minus1

120576119905

1(ℎ(11)

)

otimes 119909(3120582)

119878minus1

(119887120582minus1 (119909(2120582minus1)))

= 120590 (ℎ(21)

V(21)

) 119886120572119909(1120572)

sdot V(0)

otimes V(11)

119878minus1

120576119905

1(ℎ(11)

)

otimes 119878minus1

120576119905

120582minus1 (119909(21)

) 119878minus1

(119887120582minus1)

= 120590 (11015840

(21)ℎ V(21)

) 1198861205721(1120572)

119909 sdot V(0)

otimes V(11)

11015840

(11)

otimes 119878minus1

(1(2120582minus1)) 119878minus1

(119887120582minus1)

= 120590 (11015840

(21) V(21)

) 120590 (ℎ V(31)

) 1198861205721(1120572)

119909 sdot V(0)

otimes V(11)

11015840

(11)otimes 119878minus1

(119887120582minus11(2120582minus1))

= 120576 (V(21)

1(21)

) 120590 (ℎ V(31)

) 119886120572119909 sdot V(0)

otimes V(11)

1(11)

otimes 119878minus1

(119887120582minus1)

= 120590 (ℎ V(21)

) 119886120572119909 sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1)

= 120590 (ℎ V(11)

) 119886120572sdot (119909 sdot V

(0))(0)

otimes (119909 sdot V(0))(11)

otimes 119878minus1

(119887120582minus1)

= 120575119881

120582((ℎ otimes 119909) ⊳

120572V)

(52)

Finally it follows from Proposition 6 the braiding onWYD(119867 otimes 119867) that the braiding on

119867WL119867 is as the

following

120591119881119882

(V otimes 119908) =120572

(119878120573minus1 (V[1120573minus1])) ⊳120573119908otimesV[0]

= 120590 (119878 (V(11)

) 119908(11)

)120572

(119887120573sdot 119908(0)) otimes119886120572sdot V(0)

(53)

for all 119881isin119867WL119867120572119882isin119867WL119867120573 V isin 119881 and 119908 isin 119882

This completes the proof

10 Abstract and Applied Analysis

Acknowledgments

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

References

[1] V G Turaev ldquoHomotopy field theory in dimension 3 andcrossed group categoriesrdquo httparxivorgabsmath0005291

[2] P J Freyd and D N Yetter ldquoBraided compact closed categorieswith applications to low-dimensional topologyrdquo Advances inMathematics vol 77 no 2 pp 156ndash182 1989

[3] A Virelizier Algebres de Hopf graduees et fibres plats sur les3-varietes [PhD thesis] Universite Louis Pasteur StrasbourgFrance 2001

[4] A Virelizier ldquoInvolutory Hopf group-coalgebras and flat bun-dles over 3-manifoldsrdquo Fundamenta Mathematicae vol 188 pp241ndash270 2005

[5] A J Kirillov ldquoOn G-equivariant modular categoriesrdquohttparxivorgabsmath0401119

[6] S Wang ldquoNew Turaev braided group categories and groupSchur-Weyl dualityrdquo Applied Categorical Structures vol 21 no2 pp 141ndash166 2013

[7] T Yang and S Wang ldquoConstructing new braided 119879-categoriesover regular multiplier Hopf algebrasrdquo Communications inAlgebra vol 39 no 9 pp 3073ndash3089 2011

[8] M Zunino ldquoYetter-Drinfelrsquod modules for crossed structuresrdquoJournal of Pure and Applied Algebra vol 193 no 1-3 pp 313ndash343 2004

[9] A Van Daele ldquoMultiplier Hopf algebrasrdquo Transactions of theAmerican Mathematical Society vol 342 no 2 pp 917ndash9321994

[10] A Van Daele and S H Wang ldquoWeak multiplier Hopf algebrasI The main theoryrdquo Journal fur die reine und angewandteMathematik 2013

[11] A Van Daele and S Wang ldquoNew braided crossed categoriesandDrinfelrsquod quantumdouble for weakHopf group coalgebrasrdquoCommunications in Algebra vol 36 no 6 pp 2341ndash2386 2008

[12] G Bohm F Nill and K Szlachanyi ldquoWeak Hopf algebras IIntegral theory and 119862

lowast-structurerdquo Journal of Algebra vol 221no 2 pp 385ndash438 1999

[13] X Zhou and S Wang ldquoThe duality theorem for weak Hopfalgebra (co) actionsrdquo Communications in Algebra vol 38 no12 pp 4613ndash4632 2010

[14] A Virelizier ldquoHopf group-coalgebrasrdquo Journal of Pure andApplied Algebra vol 171 no 1 pp 75ndash122 2002

[15] M E Sweedler Hopf Algebras Mathematics Lecture NoteSeries W A Benjamin New York NY USA 1969

[16] X Zhou and T Yang ldquoKegelrsquos theorem over weak Hopf groupcoalgebrasrdquo Journal of Mathematics vol 33 no 2 pp 228ndash2362013

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

coalgebra 119867 Then we obtain a new category119867WL119867

=

∐120572isin120587 119867

WL119867

120572and show that as 119867 is a quasitriangular

and coquasitriangular weak crossed Hopf group coalgebrathen

119867WL119867 is a braided 119879-subcategory of Yetter-Drinfelrsquod

categoryWYD(119867 otimes 119867)

2 Preliminary

Throughout the paper let 120587 be a group with the unit 1 andlet 119896 be a field All algebras vector spaces and so forth aresupposed to be over 119896 We use the Sweedler-type notation[15] for the comultiplication and coaction 119905 for the flip mapand id for the identity map In the section we will recall somebasic definitions and results related to our paper

21 Weak Crossed Hopf Group Coalgebras Recall fromTuraev and Virelizier (see [1 14]) that a group coalgebra over120587 is a family of 119896-spaces 119862 = 119862

120572120572isin120587

together with a familyof 119896-linear maps Δ = Δ

120572120573 119862120572120573

rarr 119862120572otimes 119862120573120572120573isin120587

(calleda comultiplication) and a 119896-linear map 120576 119862

1rarr 119896 (called a

counit) such that Δ is coassociative in the sense that

(Δ120572120573

otimes id119862120574)Δ120572120573120574

= (id119862120572

otimes Δ120573120574

) Δ120572120573120574

forall120572 120573 120574 isin 120587

(id119862120572

otimes 120576) Δ1205721

= id119862120572

= (120576 otimes id119862120572) Δ1120572

forall120572 isin 120587

(1)

We use the Sweedler-type notation (see [14]) for a comul-tiplication that is we write

Δ120572120573

(119888) = 119888(1120572)

otimes 119888(2120573)

for any 120572 120573 isin 120587 119888 isin 119862120572120573 (2)

Recall from Van Daele and Wang (see [11]) that a weaksemi-Hopf group coalgebra 119867 = 119867

120572 119898120572 1120572 Δ 120576120572isin120587

isa family of algebras 119867

120572 119898120572 1120572120572isin120587

and at the same timea group coalgebra 119867

120572 Δ = Δ

120572120573 120576120572120573isin120587

such that thefollowing conditions hold

(i) The comultiplication Δ120572120573

119862120572120573

rarr 119862120572otimes 119862120573is a

homomorphism of algebras (not necessary unit pre-serving) such that

(Δ120572120573

otimes id119867120574)Δ120572120573120574

(1120572120573120574

)

= (Δ120572120573

(1120572120573) otimes 1120574) (1120572otimes Δ120573120574

(1120573120574))

(Δ120572120573

otimes id119867120574)Δ120572120573120574

(1120572120573120574

)

= (1120572otimes Δ120573120574

(1120573120574)) (Δ

120572120573(1120572120573) otimes 1120574)

(3)

for all 120572 120573 120574 isin 120587

(ii) The counit 120576 1198671

rarr 119896 is a 119896-linear map satisfyingthe identity

120576 (119892119909ℎ) = 120576 (119892119909(21)

) 120576 (119909(11)

ℎ) = 120576 (119892119909(11)

) 120576 (119909(21)

ℎ) (4)

for all 119892 ℎ 119909 isin 1198671

A weakHopf group coalgebra over 120587 is a weak semi-Hopfgroup coalgebra 119867 = 119867

120572 119898120572 1120572 Δ 120576120572isin120587

endowed with afamily of 119896-linear maps 119878 = 119878

120572 119867120572rarr 119867120572minus1120572isin120587

(called anantipode) satisfying the following equations

119898120572(119878120572minus1 otimes id

119867120572) Δ120572minus1120572(ℎ) = 1

(1120572)120576 (ℎ1(21)

)

119898120572(id119867120572

otimes 119878120572minus1) Δ120572120572minus1 (ℎ) = 120576 (1

(11)ℎ) 1(2120572)

119878120572(119892(1120572)

) 119892(2120572minus1)119878120572(119892(3120572)

) = 119878120572(119892)

(5)

for all ℎ isin 1198671 119892 isin 119867

120572 and 120572 isin 120587

Let119867 be a weak Hopf group coalgebra Define a family oflinear maps 120576

119905= 120576119905

120572 1198671rarr 119867

120572120572isin120587

and 120576119904= 120576119904

120572 1198671rarr

119867120572120572isin120587

by the formulae

120576119905

120572(ℎ) = 120576 (1

(11)ℎ) 1(2120572)

= 119898120572(id119867120572

otimes 119878120572minus1) Δ120572120572minus1 (ℎ)

120576119904

120572(ℎ) = 1

(1120572)120576 (ℎ1(21)

) = 119898120572(119878120572minus1 otimes id

119867120572) Δ120572minus1120572(ℎ)

(6)

for any ℎ isin 1198671 where 120576

119905 and 120576119904 are called the 120587-target and

120587-source counital mapsBy Van Daele and Wang (see [11]) let119867 be a weak semi-

Hopf group coalgebraThen we have the following equations

(1) 120576(119892ℎ) = 120576(119892120576119905

1(ℎ)) 120576(119892ℎ) = 120576(120576

119904

1(119892)ℎ) for all 119892 ℎ isin

1198671

(2) 119909(1120572)

otimes120576119905

120573(119909(21)

) = 1(1120572)

119909otimes1(2120573)

for all 119909 isin 119867120572 120572 120573 isin

120587(3) 120576119904120573(119909(11)

)otimes119909(2120572)

= 1(1120573)

otimes1199091(2120572)

for all 119909 isin 119867120572 120572 120573 isin

120587(4) 120576119905120572(120576119905

1(119909)119910) = 120576

119905

120572(119909)120576119905

120572(119910) 120576119904120572(119909120576119904

1(119910)) = 120576

119904

120572(119909)120576119904

120572(119910) for

all 119909 119910 isin 1198671

Similarly for any 120572 isin 120587 and ℎ isin 1198671 define 120576

119905

120572(ℎ) =

120576(ℎ1(11)

)1(2120572)

120576119904120572(ℎ) = 1

(1120572)120576(1(21)

ℎ) Then we have

(1) 120576119904120572(ℎ(11)

)otimesℎ(2120573)

= 1(1119886)

otimes1(2120573)

ℎ for all ℎ isin 119867120573 120572 120573 isin

120587(2) 119909(1120572)

otimes120576119905

120572(119909(21)

) = 1199091(1120572)

otimes1(2120573)

for all 119909 isin 119867120572 120572 120573 isin

120587

AweakHopf group coalgebra119867 = 119867120572119898120572 1120572 Δ 120576 119878

120572isin120587

is called a weak crossedHopf group coalgebra if it is endowedwith a family of algebra isomorphisms 120593 = 120593

120572 119867120573

rarr

119867120572120573120572minus1120572120573isin120587

(called a crossing) such that (120593120572otimes 120593120572) ∘ Δ120573120574

=

Δ120572120573120572minus1120572120574120572minus1 ∘120593120572 120576∘120593120572= 120576 and120593

120572120573= 120593120572∘120593120573for all 120572 120573 120574 isin 120587

If 119867 is crossed with the crossing 120593 = 120593120572120572isin120587

then wehave

120593120573∘ 120576119904

120572= 120576119904

120573120572120573minus1 ∘ 120593120573 120593120573∘ 120576119905

120572

= 120576119905

120573120572120573minus1 ∘ 120593120573 forall120572 120573 isin 120587

(7)

A quasitriangular weak crossed Hopf group coalgebraover 120587 is a pair (119867 119877) where 119867 is a weak crossed Hopfgroup coalgebra together with a family of maps 119877 = 119877

120572120573isin

Δcop120573minus1120572minus1(1120572120573)(119867120572

otimes 119867120573)Δ120572120573

(1120572120573) satisfying the following

conditions

Abstract and Applied Analysis 3

(1) 119877120572120573

Δ120572120573

(ℎ) = Δcop120573minus1120572minus1(ℎ)119877

120572120573 for all ℎ isin 119867

120572120573 120572 120573 isin

120587(2) (id

119867120572otimes Δ120573120574

)(119877120572120573120574

) = (119877120572120574

)11205733

(119877120572120573

)12120574

for all 120572 120573120574 isin 120587

(3) (Δ120572120573

otimesid119867120574) (119877120573minus1120572minus1120574) = (119877

120572minus1120574)1120573minus13(119877120573minus1120574)120572minus123 for

all 120572 120573 120574 isin 120587

whereΔ120572120573

= (120593120573otimesid119867120573minus1)∘Δ120573minus1120572minus1120573120573minus1 Δcop120572120573

= 119905119867120572minus1 119867120573minus1∘(120593120573otimes

id119867120573minus1) ∘Δ120573minus1120572minus1120573120573minus1 for all 120572 120573 isin 120587 and such that there exists

a family of119877 = 119877120572120573

isin Δ120572120573

(1120572120573)(119867120572otimes119867120573)Δ

cop120573minus1120572minus1(1120572120573)with

119877120572120573

119877120572120573

= Δcop120573minus1120572minus1 (1120572120573) 119877

120572120573119877120572120573

= Δ120572120573

(1120572120573)

(120593120573otimes 120593120573) (119877120572120574

) = 119877120573120572120573minus1120573120574120573minus1

(8)

for all 120572 120573 120574 isin 120587 In this paper we denote 119877120572120573

= 119886120572otimes 119887120573

Recall from [16] that a coquasitriangular weak Hopfgroup coalgebra (119867 120590) consists of a weak Hopf groupcoalgebra 119867 = 119867

120572 119898120572 1120572 Δ 120576 119878

120572isin120587and a map 120590 119867

1otimes

1198671rarr 119896 satisfying

120590 (ℎ(11)

119892(11)

) ℎ(2120572)

119892(2120572)

= 119892(1120572)

ℎ(1120572)

120590 (ℎ(21)

119892(21)

)

120590 (119886 119887119888) = 120590 (119886(11)

119888) 120590 (119886(21)

119887)

120590 (119886119887 119888) = 120590 (119886 119888(11)

) 120590 (119887 119888(21)

)

120576 (119886(11)

119887(11)

) 120590 (119887(21)

119886(21)

) 120576 (119887(31)

119886(31)

) = 120590 (119887 119886)

(9)

and there exists 120590minus1 1198671otimes 1198671rarr 119896 such that

120590 (119886(11)

119887(11)

) 120590minus1

(119886(21)

119887(21)

) = 120576 (119887119886)

120590minus1

(119886(11)

119887(11)

) 120590 (119886(21)

119887(21)

) = 120576 (119886119887)

120576 (119886(11)

119887(11)

) 120590minus1

(119886(21)

119887(21)

) 120576 (119887(31)

119886(31)

) = 120590minus1

(119886 119887)

(10)

for all ℎ 119892 isin 119867120572 119886 119887 119888 isin 119867

1 where 120590

minus1 is called a weakinverse of 120590

22 Braided 119879-Categories We recall that a monoidal cate-goryC is called a crossed category over group 120587 if it consistsof the following data

(1) A family of subcategories C120572120572isin120587

such that C is adisjoint union of this family and such that for any119880 isin C

120572and 119881 isin C

120573 119880 otimes 119881 isin C

120572120573 Here the

subcategoryC120572is called the 120572th component ofC

(2) A group homomorphism 120595 120587 rarr aut(C) 120573 997891rarr

120595120573 the conjugation (where aut(C) is the group of

invertible strict tensor functors fromC to itself) suchthat 120595

120573(C120572) = C

120573120572120573minus1 for any 120572 120573 isin 120587 Here the

functors 120595120573are called conjugation isomorphisms

We will use the Turaevrsquos left index notation in [1] for anyobject 119880 isin C

120572 119881119882 isin C

120573and any morphism 119891 119881 rarr 119882

inC120573 we set119880

119881 = 120595120572(119881) isin C

120572120573120572minus1 119880

119891 = 120595120572(119891)

119880

119881 997888rarr119880

119882

(11)

Recall form [1] that a braided 119879-category is a crossedcategory C endowed with braiding that is a family ofisomorphisms

119888 = 119888119880119881

isin C (119880 otimes 119881 (119880

119881) otimes 119880) 119880119881isinC

(12)

satisfying the following conditions

(1) for any morphism 119891 isin C120572(119880 1198801015840

) with 120572 isin 120587 119892 isin

C(119881 1198811015840

) we have

( (120572

119892) otimes 119891) ∘ 119888119880119881

= 11988811988010158401198811015840 ∘ (119891 otimes 119892) (13)

(2) for all 119880119881119882 isin C we have

119888119880otimes119881119882

= 119886 119880otimes119881119882119880119881

∘ (119888119880119881119882

otimes id119881) ∘ 119886minus1

119880119881119882119881

∘ (id119880otimes 119888119881119882

) ∘ 119886119880119881119882

119888119880119881otimes119882

= 119886minus1

119880119881119880119882119880

∘ (id119880119881

otimes 119888119880119882

) ∘ 119886 119880119881119880119882

∘ (119888119880119881

otimes id119882) ∘ 119886minus1

119880119881119882

(14)

(3) for any 119880119881 isin C 120572 isin 120587

120595120572(119888119880119881

) = 119888120595120572(119880)120595120572(119881)

(15)

3 Yetter-Drinfelrsquod Categories for WeakCrossed Hopf Group Coalgebras

In this section we first introduce the definition of weak120572-Yetter-Drinfelrsquod modules over a weak crossed Hopf groupcoalgebra 119867 and then use it to construct a class of braided119879-categories

Definition 1 Let 119867 be a weak crossed Hopf group coalgebraover group 120587 and let 120572 be a fixed element in 120587 A (left-right) weak 120572-Yetter-Drinfelrsquod module or simply a WYD

120572-

module is a couple 119881 = (119881 120588119881

= 120588119881

120582120582isin120587

) where 119881 is a left119867120572-module and for any 120582 isin 120587 120588119881

120582 119881 rarr 119881otimes119867

120582is a 119896-linear

morphism such that

(1) 119881 is coassociative in the sense that for any 1205821 1205822isin 120587

we have

(id119881otimes Δ1205821 1205822

) ∘ 120588119881

12058211205822

= (120588119881

1205821

otimes id1198671205822

) ∘ 120588119881

1205822

(16)

(2) 119881 is counitary in the sense that

(id119881otimes 120576) ∘ 120588

119881

1= id119881 (17)

(3) 119881 is crossed in the sense that for any 120582 isin 120587 ℎ isin 119867120572

120588119881

120582(ℎ sdot V) = ℎ

(2120572)sdot V(0)

otimes ℎ(3120582)

V(1120582)

119878minus1

120593120572minus1 (ℎ(1120572120582minus1120572minus1))

(18)

where 120588119881120582(V) = V

(0)otimes V(1120582)

4 Abstract and Applied Analysis

Given two WYD120572-modules (119881 120588

119881

) and (119882 120588119882

) amorphism 119891 (119881 120588

119881

) rarr (119882 120588119882

) of this two WYD120572-

modules is an 119867120572-linear map 119891 119881 119882 such that for any

120582 isin 120587

120588119882

120582∘ 119891 = (119891 otimes id

119867120582) ∘ 120588119881

120582 (19)

Then we can form the categoryWYD120572(119867) ofWYD

120572-

modules where the composition of morphisms of WYD120572-

modules is the standard composition of the underlying linearmaps

Proposition 2 Equation (18) is equivalent to the followingequations

ℎ(1120572)

sdot V(0)

otimes ℎ(2120582)

V(1120582)

= (ℎ(2120572)

sdot V)(0)

otimes (ℎ(2120572)

sdot V)(1120582)

120593120572minus1 (ℎ(1120572120582120572

minus1))

(20)

120588119881

120582(V) = V

(0)otimes V(1120582)

isin 119881otimes119905120572120582119867120582= Δ120572120582

(1120572120582) sdot (119881 otimes 119867

120582)

(21)

for any V isin 119881 ℎ isin 119867120572120582

Proof Assume that (20) and (21) hold for all ℎ isin 119867120572120582 V isin 119881

We compute

ℎ(2120572)

sdot V(0)

otimes ℎ(3120582)

V(1120582)

119878minus1

120593120572minus1 (ℎ(1120572120582minus1120572minus1))

= (ℎ(3120572)

sdot V)(0)

otimes (ℎ(3120572)

sdot V)(1120582)

times 120593120572minus1 (ℎ(2120572120582120572

minus1)) 119878minus1

120593120572minus1 (ℎ(1120572120582minus1120572minus1))

= (ℎ(3120572)

sdot V)(0)

otimes (ℎ(3120572)

sdot V)(1120582)

times 120593120572minus1 (ℎ(2120572120582120572

minus1)119878minus1

(ℎ(1120572120582minus1120572minus1)))

= (ℎ(2120572)

sdot V)(0)

otimes (ℎ(2120572)

sdot V)(1120582)

times 120593120572minus1119878minus1

(120576119905

120572120582minus1120572minus1 (ℎ(11)

))

= (11015840

(2120572)ℎ(2120572)

sdot V)(0)

otimes (11015840

(2120572)ℎ(2120572)

sdot V)(1120582)

times 120593120572minus1 (1(1120572120582120572

minus1)) times 120576 (1

(21)11015840

(11)ℎ(11)

)

= (1(2120572)

ℎ sdot V)(0)

otimes (1(2120572)

ℎ sdot V)(1120582)

120593120572minus1 (1(1120572120582120572

minus1))

= 1(1120572)

sdot (ℎ sdot V)(0)

otimes 1(2120582)

(ℎ sdot V)(1120582)

= (ℎ sdot V)(0)

otimes (ℎ sdot V)(1120582)

(22)

as requiredConversely suppose that119881 is crossed in the sense of (18)

We first note that

V(0)

otimes V(1120582)

= 1(2120572)

sdot V(0)

otimes 1(3120582)

V(1120582)

119878minus1

120593120572minus1 (1(1120572120582minus1120572minus1))

= 11015840

(1120572)1(2120572)

sdot V(0)

otimes 11015840

(2120582)V(1120582)

119878minus1

times 120593120572minus1 (1(1120572120582minus1120572minus1))

= 11015840

(1120572)sdot (1(2120572)

sdot V(0))

otimes 11015840

(2120582)[V(1120582)

119878minus1

120593120572minus1 (1(1120572120582minus1120572minus1))]

isin 119881otimes119905120572120582119867120582

(23)

To show that (21) is satisfied for all ℎ isin 119867120572120582 we do the

following calculations

(ℎ(2120572)

sdot V)(0)

otimes (ℎ(2120572)

sdot V)(1120582)

120593120572minus1 (ℎ(1120572120582120572

minus1))

= ℎ(3120572)

sdot V(0)

otimes ℎ(4120582)

V(1120582)

119878minus1

120593120572minus1 (ℎ(2120572120582minus1120572minus1))

times 120593120572minus1 (ℎ(1120572120582120572

minus1))

= ℎ(2120572)

sdot V(0)

otimes ℎ(3120582)

V(1120582)

120593120572minus1119878minus1

(120576119904

120572120582minus1120572minus1 (ℎ(11)

))

= ℎ(2120572)

11015840

(2120572)sdot V(0)

otimes ℎ(3120582)

V(1120582)

120593120572minus1119878minus1

(1(1120572120582minus1120572minus1))

times 120576 (ℎ(11)

11015840

(11)1(21)

)

= ℎ(1120572)

1(2120572)

sdot V(0)

otimes ℎ(2120582)

V(1120582)

120593120572minus1119878minus1

(1(1120572120582minus1120572minus1))

= ℎ(1120572)

11015840

(1120572)1(2120572)

sdot V(0)

otimes ℎ(2120582)

11015840

(2120582)V(1120582)

119878minus1

times 120593120572minus1 (1(1120572120582minus1120572minus1))

= ℎ(1120572)

1(2120572)

sdot V(0)

otimes ℎ(2120582)

1(3120582)

V(1120582)

119878minus1

times 120593120572minus1 (1(1120572120582minus1120572minus1))

= ℎ(1120572)

sdot V(0)

otimes ℎ(2120582)

V(1120582)

(24)

This completes the proof

Proposition 3 If (119881 120588119881

) isin WYD120572(119867) (119882 120588

119882

) isin

WYD120573(119867) then 119881otimes

119905120572120573119882 = Δ

120572120573(1120572120573) sdot (119881 otimes 119882) isin

WYD120572120573(119867) with the action and coaction structures as

follows

ℎ sdot (V otimes 119908) = ℎ(1120572)

sdot V otimes ℎ(2120573)

sdot 119908

120588119881otimes119905120572120573119882

120582(V otimes 119908) = V

(0)otimes 119908(0)

otimes 119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

(25)

for all ℎ isin 119867120572120573 120582 isin 120587 V otimes 119908 isin 119881otimes

119905120572120573119882

Proof It is easy to prove that 119881otimes119905120572120573119882 is a left 119867

120572120573-module

and the proof of coassociativity of 119881otimes119905120572120573119882 is similar to the

Hopf group coalgebra case For all V otimes 119908 isin 119881otimes119905120572120573119882 we have

(id119881otimes119905120572120573119882

otimes 120576) ∘ 120588119881otimes119905120572120573119882

1(V otimes 119908)

= 120576 (119908(11)

120593120573minus1(11015840

)(21)

)

times 120576 (120593120573minus1(11015840

)(11)

120593120573minus1 (V(11)

)) V(0)

otimes 119908(0)

Abstract and Applied Analysis 5

= 120576 ((1(2120573)

sdot 119908)(11)

120593120573minus1 (1(11)

) 120593120573minus1 (11015840

(21)))

times 120576 (120593120573minus1 (11015840

(11)) 120593120573minus1 (V(11)

)) V(0)

otimes (1(2120573)

sdot 119908)(0)

= 120576 (1(51)

119908(11)

119878minus1

120593120573minus1 (1(31)

) 120593120573minus1 (1(21)

))

times 120576 (1(11)

V(11)

) V(0)

otimes 1(4120573)

sdot 119908(0)

= 120576 (1(41)

119908(11)

120593120573minus1119878minus1

(120576119904

1(1(21)

))) 120576 (1(11)

V(11)

) V(0)

otimes 1(3120573)

sdot 119908(0)

= 120576 (1(41)

119908(11)

120576119905

1119878minus1

120593120573minus1 (1(21)

)) 120576 (1(11)

V(11)

) V(0)

otimes 1(3120573)

sdot 119908(0)

= 120576 ((1(2120573)

sdot 119908)(11)

) 120576 (1(11)

V(11)

) V(0)

otimes (1(2120573)

sdot 119908)(0)

= 120576 ((1(2120572)

sdot V)(11)

120593120572minus1 (1(11)

)) (1(2120572)

sdot V)(0)

otimes 1(3120573)

sdot 119908

= 120576 (1(31)

V(11)

120593120572minus1119878minus1

120576119904

1(1(11)

)) 1(2120572)

sdot V(0)

otimes 1(4120573)

sdot 119908

= 120576 ((1(1120572)

sdot V)(11)

) (1(1120572)

sdot V)(0)

otimes 1(2120573)

sdot 119908

= V otimes 119908

(26)

This shows that119881otimes119905120572120573119882 is satisfing counitary condition (17)

Then we check the equivalent form of crossed conditions(20) and (21) In fact for all ℎ isin 119867

120572120573120582 V otimes 119908 isin 119881otimes

119905120572120573119882 we

have

(ℎ(2120572120573)

sdot (V otimes 119908))(0)

otimes (ℎ(2120572120573)

sdot (V otimes 119908))(1120582)

120593(120572120573)minus1 (ℎ(1120572120573120582120573

minus1120572minus1))

= (ℎ(2120572)

sdot V)(0)

otimes (ℎ(3120573)

sdot 119908)(0)

otimes (ℎ(3120573)

sdot 119908)(1120582)

120593120573minus1 ((ℎ(2120572)

sdot V)(1120573120582120573

minus1))

120593120573minus1120572minus1 (ℎ(1120572120573120582120573

minus1120572minus1))

= ℎ(3120572)

sdot V(0)

otimes ℎ(6120573)

sdot 119908(0)

otimes ℎ(7120582)

119908(1120582)

119878minus1

120593120573minus1 (ℎ(5120573120582minus1120573minus1))

120593120573minus1 (ℎ(4120573120582120573

minus1)V(1120573120582120573

minus1)) 120593120573minus1120593120572minus1119878minus1

times (ℎ(2120572120573120582

minus1120573minus1120572minus1)) 120593120573minus1120572minus1 (ℎ(1120572120573120582120573

minus1120572minus1))

= ℎ(3120572)

sdot V(0)

otimes ℎ(6120573)

sdot 119908(0)

otimes ℎ(7120582)

119908(1120582)

119878minus1

120593120573minus1 (ℎ(5120573120582minus1120573minus1))

120593120573minus1 (ℎ(4120573120582120573

minus1)V(1120573120582120573

minus1)) 120593120573minus1120572minus1119878minus1

times (119878 (ℎ(1120572120573120582120573

minus1120572minus1)) ℎ(2120572120573120582

minus1120573minus1120572minus1))

= ℎ(1120572)

1(2120572)

sdot V(0)

otimes ℎ(4120573)

sdot 119908(0)

otimes ℎ(5120582)

119908(1120582)

119878minus1

120593120573minus1 (ℎ(3120573120582minus1120573minus1))

120593120573minus1 (ℎ(2120573120582120573

minus1)V(1120573120582120573

minus1)) 120593120573minus1120572minus1119878minus1

(1(1120572120573120582

minus1120573minus1120572minus1))

= ℎ(1120572)

1(2120572)

sdot V(0)

otimes ℎ(4120573)

sdot 119908(0)

otimes ℎ(5120582)

119908(1120582)

119878minus1

120593120573minus1 (ℎ(3120573120582minus1120573minus1))

120593120573minus1 (ℎ(2120573120582120573

minus1)1(3120573120582120573

minus1)V(1120573120582120573

minus1)119878minus1

times120593120572minus1 (1(1120572120573120582

minus1120573minus1120572minus1)))

= ℎ(1120572)

sdot V(0)

otimes ℎ(3120573)

sdot 119908(0)

otimes ℎ(4120582)

119908(1120582)

times 120593120573minus1119878minus1

(120576119904

120573120582minus1120573minus1 (ℎ(21)

)) 120593120573minus1 (V(1120573120582120573

minus1))

= ℎ(1120572)

sdot V(0)

otimes ℎ(2120573)

11015840

(1120573)1(2120573)

sdot 119908(0)

otimes ℎ(3120582)

11015840

(2120582)119908(1120582)

119878minus1

120593120573minus1 (1(1120573120582minus1120573minus1))

120593120573minus1 (V(1120573120582120573

minus1))

= ℎ(1120572)

sdot V(0)

otimes ℎ(2120573)

sdot 119908(0)

otimes ℎ(3120582)

119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

= ℎ(1120572120573)

sdot (V otimes 119908)(0)

otimes ℎ(2120582)

(V otimes 119908)(1120582)

1(1120572120573)

sdot (V otimes 119908)(0)

otimes 1(2120582)

(V otimes 119908)(1120582)

= 1(1120572120573)

sdot (V(0)

otimes 119908(0)) otimes 1(2120582)

119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

= 1(1120572)

sdot V(0)

otimes 1(2120573)

sdot 119908(0)

otimes 1(3120582)

119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

= 1(1120572)

sdot V(0)

otimes 1(2120573)

11015840

(1120573)sdot 119908(0)

otimes 11015840

(2120582)119908(1120582)

times 120593120573minus1 (V(1120573120582120573

minus1))

= 1(1120572)

sdot V(0)

otimes 1(2120573)

sdot 119908(0)

otimes 119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

= V(0)

otimes 119908(0)

otimes 119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

= (V otimes 119908)(0)

otimes (V otimes 119908)(1120582)

(27)

This finishes the proof

Proposition 4 Let (119881 120588119881) isin WYD120572(119867) and let 120573 isin 120587 Set

120573

119881 = 119881 as vector space with action and coaction structuresdefined by

ℎ ⊳120573V =120573

(120593120573minus1 (ℎ) sdot V) forallℎ isin 119867

120573120572120573minus1 120573V isin120573

119881

120588120573119881

120582(120573V) =

120573

(V(0)) otimes 120593120573(V(1120573minus1120582120573)

)

= V⟨0⟩

otimes V⟨1120582⟩

forall120573V isin120573

119881

(28)

Then120573

119881 isin WYD120573120572120573minus1(119867)

6 Abstract and Applied Analysis

Proof Obviously 120573119881 is a left119867120573120572120573minus1-module and conditions

(16) and (17) are straightforward Then it remains to showthat conditions (20) and (21) hold For all 120573V isin

120573

119881 we have

1(1120573120572120573

minus1)⊳ V⟨0⟩

otimes 1(2120582)

V⟨1120582⟩

=120573

(120593120573minus1 (1(1120573120572120573

minus1)) sdot V(0)) otimes1(2120582)

120593120573(V(1120573minus1120582120573)

)

=120573

(1(1120572)

sdot V(0)) otimes 120593120573(1(2120582)

) 120593120573(V(1120573minus1120582120573)

)

= V⟨0⟩

otimes V⟨1120582⟩

(29)

Next for all ℎ isin 119867120573120572120573minus1120582 120573V isin

120573

119881 we get

(ℎ(2120573120572120573

minus1)⊳120573V)⟨0⟩

otimes (ℎ(2120573120572120573

minus1)⊳120573V)⟨1120582⟩

times 120593120573120572minus1120573minus1 (ℎ(1120573120572120573

minus1120582120573120572minus1120573minus1))

=

120573

((120593120573minus1 (ℎ(2120573120572120573

minus1)) sdot V)(0)

)

otimes 120593120573((120593120573minus1 (ℎ(2120573120572120573

minus1)) sdot V)(1120573minus1120582120573)

)

120593120573120572minus1120573minus1 (ℎ(1120573120572120573

minus1120582120573120572minus1120573minus1))

=

120573

((120593120573minus1(ℎ)(2120572)

sdot V)(0)

)

otimes 120593120573((120593120573minus1(ℎ)(2120572)

sdot V)(1120573minus1120582120573)

times 120593120572minus1 (120593120573minus1(ℎ)(1120572120573minus1120582120573120572minus1)))

=120573

(120593120573minus1(ℎ)(1120572)

sdot V(0))

otimes 120593120573(120593120573minus1(ℎ)(2120573minus1120582120573)

V(1120573minus1120582120573)

)

=120573

(120593120573minus1 (ℎ(1120573120572120573

minus1)) sdot V(0))

otimes 120593120573(120593120573minus1 (ℎ(2120582)

) V(1120573minus1120582120573)

)

= ℎ(1120573120572120573

minus1)⊳120573

(V(0))

otimes 120593120573(120593120573minus1 (ℎ(2120582)

) V(1120573minus1120582120573)

)

= ℎ(1120573120572120573

minus1)⊳120573V⟨0⟩

otimes ℎ(2120582)

V⟨1120582⟩

(30)

This completes the proof of the proposition

Remark 5 Let (119881 120588119881

) isin WYD120572(119867) and let (119882 120588

119882

) isin

WYD120573(119867) then we have 119904119905119881 =

119904

(119905

119881) as an object inWYD

119904119905120572119905minus1119904minus1(119867) and 119904(119881otimes

119905120572120573119882) =

119904

119881otimes119905119904120572120573119904minus1

119904

119882 as anobject inWYD

119904120572120573119904minus1(119867)

Proposition 6 Let (119881 120588119881

) isin WYD120572(119867) (119882 120588

119882

) isin

WYD120573(119867) Set 119881119882 =

120572

119882 as an object inWYD120572120573120572minus1(119867)

Define the map

119888119881119882

119881 otimes119882 997888rarr119881

119882otimes119881

119888119881119882

(V otimes 119908) =120572

(119878120573minus1 (V(1120573minus1)) sdot 119908) otimesV

(0)

(31)

Then 119888119881119882

is 119867-linear 119867-colinear and satisfies the followingconditions

119888119881otimes119882119880

= (119888119881

119882

119880otimes id119882) ∘ (id

119881otimes 119888119882119880

)

119888119881119882otimes119880

= (id119881119882

otimes 119888119881119880

) ∘ (119888119881119882

otimes id119880)

(32)

Furthermore 119888120574119881120574119882

= 119888119881119882

for all 120574 isin 120587

Proof Firstly we need to show that 119888119881119882

is well definedIndeed we have

119888119881sdot119882

(1(1120572)

sdot V otimes 1(2120573)

sdot 119908)

=

120572

(119878120573minus1 ((1(1120572)

sdot V)(1120573minus1)) 1(2120573)

sdot 119908) otimes(1(1120572)

sdot V)(0)

=120572

(119878120573minus1 (V(1120573minus1)119878minus1

120593120572minus1 (1(1120572120573120572

minus1)))

times 119878120573minus1 (1(3120573minus1)) 1(4120573)

sdot 119908) otimes 1(2120572)

sdot V(0)

=120572

(119878120573minus1119878minus1

120593120572minus1 (1(1120572120573120572

minus1))

times 119878120573minus1 (V(1120573minus1)) 120576119904

120573(1(31)

) sdot 119908) otimes 1(2120572)

sdot V(0)

=120572

(119878120573minus1119878minus1

120593120572minus1 (1(1120572120573120572

minus1)) 119878120573minus1 (1(3120573minus1)V(1120573minus1)) sdot 119908)

otimes 1(2120572)

sdot V(0)

=120572

(119878120573minus1 (V(1120573minus1)) sdot 119908) otimes V

(0)

= 119888119881119882

(V otimes 119908)

(33)

Secondly we prove that 119888119881119882

is119867-linear For all ℎ isin 119867120572120573

we compute

119888119881119882

(ℎ sdot (V otimes 119908))

=

120572

(119878120573minus1 ((ℎ(1120572)

sdot V)(1120573minus1)) ℎ(2120573)

sdot 119908) otimes (ℎ(1120572)

sdot V)(0)

=120572

(119878120573minus1 (ℎ(3120573minus1)V(1120573minus1)119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) ℎ(4120573)

sdot 119908)

otimes ℎ(2120572)

sdot V(0)

=120572

(119878120573minus1 (V(1120573minus1)119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) 120576119904

120573(ℎ(31)

) sdot 119908)

otimes ℎ(2120572)

sdot V(0)

=120572

(119878120573minus1 (V(1120573minus1)119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1)))

times 119878120573minus1 (1(2120573minus1)) sdot 119908) otimes ℎ

(2120572)1(1120572)

sdot V(0)

Abstract and Applied Analysis 7

=120572

(119878120573minus1 (1(3120573minus1)V(1120573minus1)119878minus1

120593120572minus1 (1(1120572120573120572

minus1))

times 119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) sdot 119908) otimes ℎ

(2120572)1(2120572)

sdot V(0)

=120572

(119878120573minus1 (V(1120573minus1)119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) sdot 119908)

otimes ℎ(2120572)

sdot V(0)

=120572

(120593120572minus1 (ℎ(1120572120573120572

minus1)) 119878120573minus1 (V(1120573minus1)) sdot 119908) otimes ℎ

(2120572)sdot V(0)

= ℎ(1120572120573120572

minus1)⊳120572

(119878120573minus1 (V(1120573minus1)) sdot 119908) otimes ℎ

(2120572)sdot V

= ℎ sdot 119888119881119882

(V otimes 119908)

(34)as required

Finally we check that 119888119881119882

is satisfing the 119867-colinearcondition In fact

120588

119881

119882otimes119881

120582∘ 119888119881119882

(V otimes 119908)

=

120572

((119878120573minus1 (V(1120573minus1)) sdot 119908)

(0)

) otimes V(0)(0)

otimes V(0)(1120582)

(119878120573minus1 (V(1120573minus1)) sdot 119908)

(1120582)

=

120572

(119878120573minus1(V(1120573minus1))(2120573)

sdot 119908(0)) otimes V

(0)(0)

otimes V(0)(1120582)

119878120573minus1(V(1120573minus1))(3120582)

119908(1120582)

119878minus1

120593120573minus1

(119878120573minus1(V(1120573minus1))(1120573120582minus1120573minus1)

)

=120572

(119878120573minus1 (V(3120573minus1)) sdot 119908(0)) otimes V(0)

otimes V(1120582)

119878120582minus1 (V(2120582minus1))

times 119908(1120582)

119878minus1

120593120573minus1119878120573120582120573minus1 (V(4120573120582120573

minus1))

=120572

(119878120573minus1 (V(2120573minus1)) sdot 119908(0)) otimes V(0)

otimes 120576119905

120582(V(11)

)

times 119908(1120582)

120593120573minus1 (V(3120573120582120573

minus1))

=120572

(119878120573minus1 (1(2120573minus1)V(1120573minus1)) sdot 119908(0)) otimes V(0)

otimes 119878120582minus1 (1(1120582minus1)) 119908(1120582)

120593120573minus1 (V(2120573120582120573

minus1))

=120572

(119878120573minus1 (1(2120573minus1)V(1120573minus1)) sdot 119908(0)) otimes V(0)

otimes 119878120582minus1 (1(1120582minus1)) 119908(1120582)

120593120573minus1119878minus1

119878120573120582120573minus1 (1(3120573120582120573

minus1)) 120593120573minus1 (V(2120573120582120573

minus1))

=120572

(119878120573minus1 (V(1120573minus1)) 119878120573minus1(1)(2120573)

sdot 119908(0))

otimes V(0)

otimes 119878120573minus1(1)(3120582)

119908(1120582)

120593120573minus1119878minus1

(119878120573minus1(1)(1120573120582minus1120573minus1)) 120593120573minus1 (V(2120573120582120573

minus1))

=120572

(119878120573minus1 (V(1120573minus1)) sdot 119908(0)) otimes V(0)

otimes 119908(1120582)

120593120573minus1 (V(2120573120582120573

minus1))

= (119888119881119882

otimes id119867120582

) (V(0)

otimes 119908(0)

otimes 119908(1120582)

120593120573minus1

times (V(1120573120582120573

minus1)))

= (119888119881119882

otimes id119867120582

) ∘ 120588119881otimes119882

120582(V otimes 119908)

(35)

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

Lemma 7 The map 119888119881119882

defined by (31) is bijective withinverse

119888minus1

119881119882119881

119882otimes119881 997888rarr 119881 otimes119882

119888minus1

119881119882(120572

119908otimes V) = V(0)

otimes V(1120573)

sdot 119908

(36)

for all V isin 119881120572

119908 isin119881

119882

Proof Firstly we prove 119888minus1

119881119882∘ 119888119881119882

= id119881otimes119882

For all V isin 119881119908 isin 119882 we have

119888minus1

119881119882∘ 119888119881119882

(V otimes 119908)

= V(0)(0)

otimes V(0)(1120573)

119878120573minus1 (V(1120573minus1)) sdot 119908

= V(0)

otimes 120576119905

120573(V(11)

) sdot 119908 = 1(1120572)

sdot V(0)

otimes 1(2120573)

120576119905

120573(V(11)

) sdot 119908

= 1(1120572)

119878minus1

120576119905

120572minus1 (V(11)

) sdot V(0)

otimes 1(2120573)

sdot 119908

= 120576 (11015840

(21)V(11)

) 1(1120572)

11015840

(1120572)sdot V(0)

otimes 1(2120573)

sdot 119908

= 1(1120572)

sdot V otimes 1(2120573)

sdot 119908 = V otimes 119908

(37)

Secondly we check 119888119881119882

∘ 119888minus1

119881119882= id 119881

119882otimes119881as follows

119888119881119882

∘ 119888minus1

119881119882(120572

119908otimesV)

=120572

(119878120573minus1 (V(0)(1120573

minus1)) V(1120573)

sdot 119908) otimesV(0)(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) 119878120573minus1 (V(0)(1120573

minus1)) V(1120573)

sdot 119908)

otimes 1(2120572)

sdot V(0)(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) 120576119904

120573(V(11)

) sdot 119908) otimes1(2120572)

sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)120576119904

120572120573120572minus1120593120572(V(11)

)) sdot 119908) otimes1(2120572)

sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes1

(2120572)

times 119878120572minus1120576119904

120572(120593120572(V(11)

)) sdot V(0)

8 Abstract and Applied Analysis

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes 1

(2120572)120576119905

120572120593120572119878 (V(11)

) sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes1

(2120572)

times 120576 (120593120572minus1 (11015840

(11)) 119878 (V(11)

)) 11015840

(2120572)sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes 1

(2120572)120576119878 (V(11)

) sdot V(0)

= 1(1120572120573120572

minus1)⊳120572

119908otimes1(2120572)

sdot V

=120572

119908otimes V(38)

This completes the proof

Define WYD(119867) = ∐120572isin120587

WYD120572(119867) the disjoint

union of the categories WYD120572(119867) for all 120572 isin 120587 If we

endow WYD(119867) with tensor product as in Proposition 3then WYD(119867) becomes a monoidal category The unit is119867119879

= 119867119905

120572= 120576119905

120572(1198671)120572isin120587

The group homomorphism 120595 119866 rarr aut(WYD(119867))

120573 rarr 120595120573is given on components as

120595120573 WYD

120572(119867) 997888rarr WYD

120573120572120573minus1 (119867) (39)

where the functor 120595120573acts as follows given a morphism 119891

(119881 120588119881

) rarr (119882 120588119882

) for any V isin 119881 we set ( 120573119891)( 120573V) =

120573

(119891(V))The braiding in WYD(119867) is given by the family 119888

119881119882

as shown in Proposition 6 Then we have the followingtheorem

Theorem 8 For a weak crossed Hopf group coalgebra 119867WYD(119867) is a braided 119879-category over group 120587

Example 9 Let 119867 be a weak Hopf algebra 119866 a finite groupand 119896(119866) the dual Hopf algebra of the group algebra 119896119866

Then have the weak Hopf group coalgebra 119896(119866) otimes119867 themultiplication in 119896(119866) otimes 119867 is given by

(119901120572otimes ℎ) (119901

120573otimes 119892) = 119901

120572119901120573otimes ℎ119892 (40)

for all 119901120572 119901120573

isin 119896(119866) ℎ 119892 isin 119867 and the comultiplicationcounit and antipode are given by

Δ119906V (119901120572 otimes ℎ) = sum

119906V=120572(119901119906otimes ℎ1) otimes (119901V otimes ℎ

2)

120576 (119901120572otimes ℎ) = 120575

1205721120576 (ℎ)

119878 (119901120572otimes ℎ) = 119901

120572minus1 otimes 119878 (ℎ)

(41)

Moreover 119896(119866) otimes 119867 is a weak crossed Hopf groupcoalgebra with the following crossing

Φ120573(119901120572otimes ℎ) = 119901

120573minus1120572120573

otimes ℎ (42)

ByTheorem 8WYD(119896(119866)otimes119867) is a braided 119879-category

4 Braided 119879-Categories over Weak LongDimodule Categories

In this section we introduce the notion of a (left-right)weak 120572-Long dimodule over a weak crossed Hopf groupcoalgebra119867 and prove that the category

119867WL119867 is a braided

119879-subcategory of Yetter-Drinfelrsquod category WYD(119867 otimes 119867)

when 119867 is a quasitriangular and coquasitriangular weakcrossed Hopf group coalgebra

Definition 10 Let119867 be a weak crossed Hopf group coalgebraover 120587 For a fixed element 120572 isin 120587 a (left-right) weak 120572-Longdimodule is a couple 119881 = (119881 120588

119881

= 120588119881

120582120582isin120587

) where 119881 is aleft 119867

120572-module and for any 120582 isin 120587 120588119881

120582 119881 rarr 119881 otimes 119867

120582is

a 119896-linear morphism such that

(1) 119881 is coassociative in the sense that for any 1205821 1205822isin 120587

we have

(id119881otimes Δ1205821 1205822

) ∘ 120588119881

12058211205822

= (120588119881

1205821

otimes id1198671205822

) ∘ 120588119881

1205822

(43)

(2) 119881 is counitary in the sense that

(id119881otimes 120576) ∘ 120588

119881

1= id119881 (44)

(3) 119881 satisfies the following compatible condition

120588119881

120582(119909 sdot V) = 119909 sdot V

(0)otimes V(1120582)

(45)

where 119909 isin 119867120572and V isin 119881

Now we can form the category119867WL119867120572of (left-right)

weak 120572-Long dimodules where the composition of mor-phisms of weak 120572-Long dimodules is the standard compo-sition of the underlying linear maps

Let119867WL119867 = ∐

120572isin120587 119867WL119867120572 the disjoint union of the

categories119867WL119867120572for all 120572 isin 120587

Proposition 11 Thecategory119867WL119867 is amonoidal category

Moreover for any 120572 120573 isin 119866 let 119881 isin119867WL119867120572

and let119882isin119867WL119867120573 Set

119881otimes119882 = V otimes 119908 isin 119881 otimes119882 | V otimes 119908

= 1(1120572)

sdot V otimes 1(2120573)

sdot 119908

= 120576 (119908(11)

120593120573minus1 (V(11)

)) V(0)

otimes 119908(0)

(46)

Then 119881 otimes119882isin119867WL119867120572with the following structures

119909 sdot (V otimes 119908) = 119909(1120572)

sdot V otimes 119909(2120573)

sdot 119908

120588119881otimes119882

120582(V otimes 119908) = V

(0)otimes 119908(0)

otimes 119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

(47)

for all 119909 isin 119867120572120573 V otimes 119908 isin 119881otimes119882

Proof It is straightforward

Abstract and Applied Analysis 9

Let (119867 120590 119877) be a coquasitriangular and quasitriangularweak crossed Hopf group coalgebra with crossing 120593 Define afamily of vector spaces 119867 otimes 119867 = (119867 otimes 119867)

120572= 1198671otimes 119867120572120572isin120587

where the 119867 on the left we consider its coquasitriangularstructure and for the right one we consider its quasitriangularstructureThen119867otimes119867 is aweak crossedHopf group coalgebrawith the natural tensor product and the crossing Φ = id otimes

120593120572120572isin120587

Theorem 12 Let (119867 120590 119877) be a weak crossed Hopf groupcoalgebra with coquasitriangular structure 120590 and quasitrian-gular structure 119877 Then the category

119867WL119867 is a braided

119879-subcategory of Yetter-Drinfelrsquod category WYD(119867 otimes 119867)

under the following action and coaction given by

120575119881

120582(V) = 119886

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1) = V

[0]otimes V[1120582]

(ℎ otimes 119909) ⊳120572V = 120590 (ℎ V

(11)) 119909 sdot V(0)

(48)

where ℎ otimes 119909 isin (119867 otimes 119867)120572 ℎ isin 119867

1 119909 isin 119867

120572 V isin 119881 and

119881isin119867WL119867120572

The braiding on119867WL119867 120591

119881119882 119881 otimes 119882 rarr

119881

119882otimes119881 isgiven by

120591119881119882

(V otimes 119908) = 120590 (119878 (V(11)

) 119908(11)

)120572

(119887120573sdot 119908(0)) otimes 119886120572sdot V(0)

(49)

for all 119881isin119867WL119867120572119882isin

119867WL119867120573

Proof Obviously119881 is a left (119867otimes119867)120572-moduleThen we show

that119881 satisfies the conditions inDefinition 1 First we need tocheck that119881 is coassociative In fact for all V isin 119881 isin

119867WL119867120572

and 1205821 1205822isin 120587

(id119881otimes Δ1205821 1205822

) ∘ 120575119881

12058211205822(V)

= 119886120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1

2120582minus1

1(2120582minus1

1))

otimes V(21)

otimes 119878minus1

(119887120582minus1

2120582minus1

1(1120582minus1

2))

= 1198861205721198861015840

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1

1

)

otimes V(21)

otimes 119878minus1

(1198871015840

120582minus1

2

)

= 119886120572sdot (1198861015840

120572sdot V(0))(0)

otimes (1198861015840

120572sdot V(0))(11)

otimes 119878minus1

(119887120582minus1

1

) otimes V(11)

otimes 119878minus1

(1198871015840

120582minus1

2

)

= (120575119881

1205821

otimes id(119867otimes119867)1205822

) (1198861015840

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(1198871015840

120582minus1

2

))

= (120575119881

1205821

otimes id(119867otimes119867)1205822

) ∘ 120575119881

1205822(V)

(50)

Next one directly shows that counitary condition (17) holdsas follows

(id119881otimes 120576) ∘ 120575

119881

1(V) = 119886

120572sdot V(0)120576 (119898(11)

) 120576119878minus1

(1198871)

= 119886120572sdot V120576 (119887

1) = 1120572sdot V = V

(51)

Then we have to prove that crossed condition (18) issatisfied For all ℎ isin 119867

1 119909 isin 119867

120572 and V isin 119881isin

119867WL119867120572 we

have(ℎ otimes 119909)

(2120572)sdot V[0]

otimes (ℎ otimes 119909)(3120582)

times V[1120582]

119878minus1

Φ120572minus1 ((ℎ otimes 119909)

(1120572120582minus1120572minus1))

= 120590 (ℎ(21)

(119886120572sdot V(0))(11)

) 119909(2120572)

sdot (119886120572sdot V(0))(0)

otimes ℎ(31)

V(11)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(119887120582minus1) 119878minus1

120595120572minus1 (119909(1120572120582minus1120572minus1))

= 120590 (ℎ(21)

V(11)

) 119909(2120572)

119886120572sdot V(0)

otimes ℎ(31)

V(21)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(119887120582minus1)

119878minus1

120595120572minus1 (119909(1120572120582minus1120572minus1))

= 120590 (ℎ(31)

V(21)

) 119909(2120572)

119886120572sdot V(0)

otimes V(11)

ℎ(21)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(120595120572minus1 (119909(1120572120582minus1120572minus1)) 119887120582minus1)

= 120590 (ℎ(21)

V(21)

) 119886120572119909(1120572)

sdot V(0)

otimes V(11)

119878minus1

120576119905

1(ℎ(11)

)

otimes 119909(3120582)

119878minus1

(119887120582minus1 (119909(2120582minus1)))

= 120590 (ℎ(21)

V(21)

) 119886120572119909(1120572)

sdot V(0)

otimes V(11)

119878minus1

120576119905

1(ℎ(11)

)

otimes 119878minus1

120576119905

120582minus1 (119909(21)

) 119878minus1

(119887120582minus1)

= 120590 (11015840

(21)ℎ V(21)

) 1198861205721(1120572)

119909 sdot V(0)

otimes V(11)

11015840

(11)

otimes 119878minus1

(1(2120582minus1)) 119878minus1

(119887120582minus1)

= 120590 (11015840

(21) V(21)

) 120590 (ℎ V(31)

) 1198861205721(1120572)

119909 sdot V(0)

otimes V(11)

11015840

(11)otimes 119878minus1

(119887120582minus11(2120582minus1))

= 120576 (V(21)

1(21)

) 120590 (ℎ V(31)

) 119886120572119909 sdot V(0)

otimes V(11)

1(11)

otimes 119878minus1

(119887120582minus1)

= 120590 (ℎ V(21)

) 119886120572119909 sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1)

= 120590 (ℎ V(11)

) 119886120572sdot (119909 sdot V

(0))(0)

otimes (119909 sdot V(0))(11)

otimes 119878minus1

(119887120582minus1)

= 120575119881

120582((ℎ otimes 119909) ⊳

120572V)

(52)

Finally it follows from Proposition 6 the braiding onWYD(119867 otimes 119867) that the braiding on

119867WL119867 is as the

following

120591119881119882

(V otimes 119908) =120572

(119878120573minus1 (V[1120573minus1])) ⊳120573119908otimesV[0]

= 120590 (119878 (V(11)

) 119908(11)

)120572

(119887120573sdot 119908(0)) otimes119886120572sdot V(0)

(53)

for all 119881isin119867WL119867120572119882isin119867WL119867120573 V isin 119881 and 119908 isin 119882

This completes the proof

10 Abstract and Applied Analysis

Acknowledgments

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

References

[1] V G Turaev ldquoHomotopy field theory in dimension 3 andcrossed group categoriesrdquo httparxivorgabsmath0005291

[2] P J Freyd and D N Yetter ldquoBraided compact closed categorieswith applications to low-dimensional topologyrdquo Advances inMathematics vol 77 no 2 pp 156ndash182 1989

[3] A Virelizier Algebres de Hopf graduees et fibres plats sur les3-varietes [PhD thesis] Universite Louis Pasteur StrasbourgFrance 2001

[4] A Virelizier ldquoInvolutory Hopf group-coalgebras and flat bun-dles over 3-manifoldsrdquo Fundamenta Mathematicae vol 188 pp241ndash270 2005

[5] A J Kirillov ldquoOn G-equivariant modular categoriesrdquohttparxivorgabsmath0401119

[6] S Wang ldquoNew Turaev braided group categories and groupSchur-Weyl dualityrdquo Applied Categorical Structures vol 21 no2 pp 141ndash166 2013

[7] T Yang and S Wang ldquoConstructing new braided 119879-categoriesover regular multiplier Hopf algebrasrdquo Communications inAlgebra vol 39 no 9 pp 3073ndash3089 2011

[8] M Zunino ldquoYetter-Drinfelrsquod modules for crossed structuresrdquoJournal of Pure and Applied Algebra vol 193 no 1-3 pp 313ndash343 2004

[9] A Van Daele ldquoMultiplier Hopf algebrasrdquo Transactions of theAmerican Mathematical Society vol 342 no 2 pp 917ndash9321994

[10] A Van Daele and S H Wang ldquoWeak multiplier Hopf algebrasI The main theoryrdquo Journal fur die reine und angewandteMathematik 2013

[11] A Van Daele and S Wang ldquoNew braided crossed categoriesandDrinfelrsquod quantumdouble for weakHopf group coalgebrasrdquoCommunications in Algebra vol 36 no 6 pp 2341ndash2386 2008

[12] G Bohm F Nill and K Szlachanyi ldquoWeak Hopf algebras IIntegral theory and 119862

lowast-structurerdquo Journal of Algebra vol 221no 2 pp 385ndash438 1999

[13] X Zhou and S Wang ldquoThe duality theorem for weak Hopfalgebra (co) actionsrdquo Communications in Algebra vol 38 no12 pp 4613ndash4632 2010

[14] A Virelizier ldquoHopf group-coalgebrasrdquo Journal of Pure andApplied Algebra vol 171 no 1 pp 75ndash122 2002

[15] M E Sweedler Hopf Algebras Mathematics Lecture NoteSeries W A Benjamin New York NY USA 1969

[16] X Zhou and T Yang ldquoKegelrsquos theorem over weak Hopf groupcoalgebrasrdquo Journal of Mathematics vol 33 no 2 pp 228ndash2362013

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

(1) 119877120572120573

Δ120572120573

(ℎ) = Δcop120573minus1120572minus1(ℎ)119877

120572120573 for all ℎ isin 119867

120572120573 120572 120573 isin

120587(2) (id

119867120572otimes Δ120573120574

)(119877120572120573120574

) = (119877120572120574

)11205733

(119877120572120573

)12120574

for all 120572 120573120574 isin 120587

(3) (Δ120572120573

otimesid119867120574) (119877120573minus1120572minus1120574) = (119877

120572minus1120574)1120573minus13(119877120573minus1120574)120572minus123 for

all 120572 120573 120574 isin 120587

whereΔ120572120573

= (120593120573otimesid119867120573minus1)∘Δ120573minus1120572minus1120573120573minus1 Δcop120572120573

= 119905119867120572minus1 119867120573minus1∘(120593120573otimes

id119867120573minus1) ∘Δ120573minus1120572minus1120573120573minus1 for all 120572 120573 isin 120587 and such that there exists

a family of119877 = 119877120572120573

isin Δ120572120573

(1120572120573)(119867120572otimes119867120573)Δ

cop120573minus1120572minus1(1120572120573)with

119877120572120573

119877120572120573

= Δcop120573minus1120572minus1 (1120572120573) 119877

120572120573119877120572120573

= Δ120572120573

(1120572120573)

(120593120573otimes 120593120573) (119877120572120574

) = 119877120573120572120573minus1120573120574120573minus1

(8)

for all 120572 120573 120574 isin 120587 In this paper we denote 119877120572120573

= 119886120572otimes 119887120573

Recall from [16] that a coquasitriangular weak Hopfgroup coalgebra (119867 120590) consists of a weak Hopf groupcoalgebra 119867 = 119867

120572 119898120572 1120572 Δ 120576 119878

120572isin120587and a map 120590 119867

1otimes

1198671rarr 119896 satisfying

120590 (ℎ(11)

119892(11)

) ℎ(2120572)

119892(2120572)

= 119892(1120572)

ℎ(1120572)

120590 (ℎ(21)

119892(21)

)

120590 (119886 119887119888) = 120590 (119886(11)

119888) 120590 (119886(21)

119887)

120590 (119886119887 119888) = 120590 (119886 119888(11)

) 120590 (119887 119888(21)

)

120576 (119886(11)

119887(11)

) 120590 (119887(21)

119886(21)

) 120576 (119887(31)

119886(31)

) = 120590 (119887 119886)

(9)

and there exists 120590minus1 1198671otimes 1198671rarr 119896 such that

120590 (119886(11)

119887(11)

) 120590minus1

(119886(21)

119887(21)

) = 120576 (119887119886)

120590minus1

(119886(11)

119887(11)

) 120590 (119886(21)

119887(21)

) = 120576 (119886119887)

120576 (119886(11)

119887(11)

) 120590minus1

(119886(21)

119887(21)

) 120576 (119887(31)

119886(31)

) = 120590minus1

(119886 119887)

(10)

for all ℎ 119892 isin 119867120572 119886 119887 119888 isin 119867

1 where 120590

minus1 is called a weakinverse of 120590

22 Braided 119879-Categories We recall that a monoidal cate-goryC is called a crossed category over group 120587 if it consistsof the following data

(1) A family of subcategories C120572120572isin120587

such that C is adisjoint union of this family and such that for any119880 isin C

120572and 119881 isin C

120573 119880 otimes 119881 isin C

120572120573 Here the

subcategoryC120572is called the 120572th component ofC

(2) A group homomorphism 120595 120587 rarr aut(C) 120573 997891rarr

120595120573 the conjugation (where aut(C) is the group of

invertible strict tensor functors fromC to itself) suchthat 120595

120573(C120572) = C

120573120572120573minus1 for any 120572 120573 isin 120587 Here the

functors 120595120573are called conjugation isomorphisms

We will use the Turaevrsquos left index notation in [1] for anyobject 119880 isin C

120572 119881119882 isin C

120573and any morphism 119891 119881 rarr 119882

inC120573 we set119880

119881 = 120595120572(119881) isin C

120572120573120572minus1 119880

119891 = 120595120572(119891)

119880

119881 997888rarr119880

119882

(11)

Recall form [1] that a braided 119879-category is a crossedcategory C endowed with braiding that is a family ofisomorphisms

119888 = 119888119880119881

isin C (119880 otimes 119881 (119880

119881) otimes 119880) 119880119881isinC

(12)

satisfying the following conditions

(1) for any morphism 119891 isin C120572(119880 1198801015840

) with 120572 isin 120587 119892 isin

C(119881 1198811015840

) we have

( (120572

119892) otimes 119891) ∘ 119888119880119881

= 11988811988010158401198811015840 ∘ (119891 otimes 119892) (13)

(2) for all 119880119881119882 isin C we have

119888119880otimes119881119882

= 119886 119880otimes119881119882119880119881

∘ (119888119880119881119882

otimes id119881) ∘ 119886minus1

119880119881119882119881

∘ (id119880otimes 119888119881119882

) ∘ 119886119880119881119882

119888119880119881otimes119882

= 119886minus1

119880119881119880119882119880

∘ (id119880119881

otimes 119888119880119882

) ∘ 119886 119880119881119880119882

∘ (119888119880119881

otimes id119882) ∘ 119886minus1

119880119881119882

(14)

(3) for any 119880119881 isin C 120572 isin 120587

120595120572(119888119880119881

) = 119888120595120572(119880)120595120572(119881)

(15)

3 Yetter-Drinfelrsquod Categories for WeakCrossed Hopf Group Coalgebras

In this section we first introduce the definition of weak120572-Yetter-Drinfelrsquod modules over a weak crossed Hopf groupcoalgebra 119867 and then use it to construct a class of braided119879-categories

Definition 1 Let 119867 be a weak crossed Hopf group coalgebraover group 120587 and let 120572 be a fixed element in 120587 A (left-right) weak 120572-Yetter-Drinfelrsquod module or simply a WYD

120572-

module is a couple 119881 = (119881 120588119881

= 120588119881

120582120582isin120587

) where 119881 is a left119867120572-module and for any 120582 isin 120587 120588119881

120582 119881 rarr 119881otimes119867

120582is a 119896-linear

morphism such that

(1) 119881 is coassociative in the sense that for any 1205821 1205822isin 120587

we have

(id119881otimes Δ1205821 1205822

) ∘ 120588119881

12058211205822

= (120588119881

1205821

otimes id1198671205822

) ∘ 120588119881

1205822

(16)

(2) 119881 is counitary in the sense that

(id119881otimes 120576) ∘ 120588

119881

1= id119881 (17)

(3) 119881 is crossed in the sense that for any 120582 isin 120587 ℎ isin 119867120572

120588119881

120582(ℎ sdot V) = ℎ

(2120572)sdot V(0)

otimes ℎ(3120582)

V(1120582)

119878minus1

120593120572minus1 (ℎ(1120572120582minus1120572minus1))

(18)

where 120588119881120582(V) = V

(0)otimes V(1120582)

4 Abstract and Applied Analysis

Given two WYD120572-modules (119881 120588

119881

) and (119882 120588119882

) amorphism 119891 (119881 120588

119881

) rarr (119882 120588119882

) of this two WYD120572-

modules is an 119867120572-linear map 119891 119881 119882 such that for any

120582 isin 120587

120588119882

120582∘ 119891 = (119891 otimes id

119867120582) ∘ 120588119881

120582 (19)

Then we can form the categoryWYD120572(119867) ofWYD

120572-

modules where the composition of morphisms of WYD120572-

modules is the standard composition of the underlying linearmaps

Proposition 2 Equation (18) is equivalent to the followingequations

ℎ(1120572)

sdot V(0)

otimes ℎ(2120582)

V(1120582)

= (ℎ(2120572)

sdot V)(0)

otimes (ℎ(2120572)

sdot V)(1120582)

120593120572minus1 (ℎ(1120572120582120572

minus1))

(20)

120588119881

120582(V) = V

(0)otimes V(1120582)

isin 119881otimes119905120572120582119867120582= Δ120572120582

(1120572120582) sdot (119881 otimes 119867

120582)

(21)

for any V isin 119881 ℎ isin 119867120572120582

Proof Assume that (20) and (21) hold for all ℎ isin 119867120572120582 V isin 119881

We compute

ℎ(2120572)

sdot V(0)

otimes ℎ(3120582)

V(1120582)

119878minus1

120593120572minus1 (ℎ(1120572120582minus1120572minus1))

= (ℎ(3120572)

sdot V)(0)

otimes (ℎ(3120572)

sdot V)(1120582)

times 120593120572minus1 (ℎ(2120572120582120572

minus1)) 119878minus1

120593120572minus1 (ℎ(1120572120582minus1120572minus1))

= (ℎ(3120572)

sdot V)(0)

otimes (ℎ(3120572)

sdot V)(1120582)

times 120593120572minus1 (ℎ(2120572120582120572

minus1)119878minus1

(ℎ(1120572120582minus1120572minus1)))

= (ℎ(2120572)

sdot V)(0)

otimes (ℎ(2120572)

sdot V)(1120582)

times 120593120572minus1119878minus1

(120576119905

120572120582minus1120572minus1 (ℎ(11)

))

= (11015840

(2120572)ℎ(2120572)

sdot V)(0)

otimes (11015840

(2120572)ℎ(2120572)

sdot V)(1120582)

times 120593120572minus1 (1(1120572120582120572

minus1)) times 120576 (1

(21)11015840

(11)ℎ(11)

)

= (1(2120572)

ℎ sdot V)(0)

otimes (1(2120572)

ℎ sdot V)(1120582)

120593120572minus1 (1(1120572120582120572

minus1))

= 1(1120572)

sdot (ℎ sdot V)(0)

otimes 1(2120582)

(ℎ sdot V)(1120582)

= (ℎ sdot V)(0)

otimes (ℎ sdot V)(1120582)

(22)

as requiredConversely suppose that119881 is crossed in the sense of (18)

We first note that

V(0)

otimes V(1120582)

= 1(2120572)

sdot V(0)

otimes 1(3120582)

V(1120582)

119878minus1

120593120572minus1 (1(1120572120582minus1120572minus1))

= 11015840

(1120572)1(2120572)

sdot V(0)

otimes 11015840

(2120582)V(1120582)

119878minus1

times 120593120572minus1 (1(1120572120582minus1120572minus1))

= 11015840

(1120572)sdot (1(2120572)

sdot V(0))

otimes 11015840

(2120582)[V(1120582)

119878minus1

120593120572minus1 (1(1120572120582minus1120572minus1))]

isin 119881otimes119905120572120582119867120582

(23)

To show that (21) is satisfied for all ℎ isin 119867120572120582 we do the

following calculations

(ℎ(2120572)

sdot V)(0)

otimes (ℎ(2120572)

sdot V)(1120582)

120593120572minus1 (ℎ(1120572120582120572

minus1))

= ℎ(3120572)

sdot V(0)

otimes ℎ(4120582)

V(1120582)

119878minus1

120593120572minus1 (ℎ(2120572120582minus1120572minus1))

times 120593120572minus1 (ℎ(1120572120582120572

minus1))

= ℎ(2120572)

sdot V(0)

otimes ℎ(3120582)

V(1120582)

120593120572minus1119878minus1

(120576119904

120572120582minus1120572minus1 (ℎ(11)

))

= ℎ(2120572)

11015840

(2120572)sdot V(0)

otimes ℎ(3120582)

V(1120582)

120593120572minus1119878minus1

(1(1120572120582minus1120572minus1))

times 120576 (ℎ(11)

11015840

(11)1(21)

)

= ℎ(1120572)

1(2120572)

sdot V(0)

otimes ℎ(2120582)

V(1120582)

120593120572minus1119878minus1

(1(1120572120582minus1120572minus1))

= ℎ(1120572)

11015840

(1120572)1(2120572)

sdot V(0)

otimes ℎ(2120582)

11015840

(2120582)V(1120582)

119878minus1

times 120593120572minus1 (1(1120572120582minus1120572minus1))

= ℎ(1120572)

1(2120572)

sdot V(0)

otimes ℎ(2120582)

1(3120582)

V(1120582)

119878minus1

times 120593120572minus1 (1(1120572120582minus1120572minus1))

= ℎ(1120572)

sdot V(0)

otimes ℎ(2120582)

V(1120582)

(24)

This completes the proof

Proposition 3 If (119881 120588119881

) isin WYD120572(119867) (119882 120588

119882

) isin

WYD120573(119867) then 119881otimes

119905120572120573119882 = Δ

120572120573(1120572120573) sdot (119881 otimes 119882) isin

WYD120572120573(119867) with the action and coaction structures as

follows

ℎ sdot (V otimes 119908) = ℎ(1120572)

sdot V otimes ℎ(2120573)

sdot 119908

120588119881otimes119905120572120573119882

120582(V otimes 119908) = V

(0)otimes 119908(0)

otimes 119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

(25)

for all ℎ isin 119867120572120573 120582 isin 120587 V otimes 119908 isin 119881otimes

119905120572120573119882

Proof It is easy to prove that 119881otimes119905120572120573119882 is a left 119867

120572120573-module

and the proof of coassociativity of 119881otimes119905120572120573119882 is similar to the

Hopf group coalgebra case For all V otimes 119908 isin 119881otimes119905120572120573119882 we have

(id119881otimes119905120572120573119882

otimes 120576) ∘ 120588119881otimes119905120572120573119882

1(V otimes 119908)

= 120576 (119908(11)

120593120573minus1(11015840

)(21)

)

times 120576 (120593120573minus1(11015840

)(11)

120593120573minus1 (V(11)

)) V(0)

otimes 119908(0)

Abstract and Applied Analysis 5

= 120576 ((1(2120573)

sdot 119908)(11)

120593120573minus1 (1(11)

) 120593120573minus1 (11015840

(21)))

times 120576 (120593120573minus1 (11015840

(11)) 120593120573minus1 (V(11)

)) V(0)

otimes (1(2120573)

sdot 119908)(0)

= 120576 (1(51)

119908(11)

119878minus1

120593120573minus1 (1(31)

) 120593120573minus1 (1(21)

))

times 120576 (1(11)

V(11)

) V(0)

otimes 1(4120573)

sdot 119908(0)

= 120576 (1(41)

119908(11)

120593120573minus1119878minus1

(120576119904

1(1(21)

))) 120576 (1(11)

V(11)

) V(0)

otimes 1(3120573)

sdot 119908(0)

= 120576 (1(41)

119908(11)

120576119905

1119878minus1

120593120573minus1 (1(21)

)) 120576 (1(11)

V(11)

) V(0)

otimes 1(3120573)

sdot 119908(0)

= 120576 ((1(2120573)

sdot 119908)(11)

) 120576 (1(11)

V(11)

) V(0)

otimes (1(2120573)

sdot 119908)(0)

= 120576 ((1(2120572)

sdot V)(11)

120593120572minus1 (1(11)

)) (1(2120572)

sdot V)(0)

otimes 1(3120573)

sdot 119908

= 120576 (1(31)

V(11)

120593120572minus1119878minus1

120576119904

1(1(11)

)) 1(2120572)

sdot V(0)

otimes 1(4120573)

sdot 119908

= 120576 ((1(1120572)

sdot V)(11)

) (1(1120572)

sdot V)(0)

otimes 1(2120573)

sdot 119908

= V otimes 119908

(26)

This shows that119881otimes119905120572120573119882 is satisfing counitary condition (17)

Then we check the equivalent form of crossed conditions(20) and (21) In fact for all ℎ isin 119867

120572120573120582 V otimes 119908 isin 119881otimes

119905120572120573119882 we

have

(ℎ(2120572120573)

sdot (V otimes 119908))(0)

otimes (ℎ(2120572120573)

sdot (V otimes 119908))(1120582)

120593(120572120573)minus1 (ℎ(1120572120573120582120573

minus1120572minus1))

= (ℎ(2120572)

sdot V)(0)

otimes (ℎ(3120573)

sdot 119908)(0)

otimes (ℎ(3120573)

sdot 119908)(1120582)

120593120573minus1 ((ℎ(2120572)

sdot V)(1120573120582120573

minus1))

120593120573minus1120572minus1 (ℎ(1120572120573120582120573

minus1120572minus1))

= ℎ(3120572)

sdot V(0)

otimes ℎ(6120573)

sdot 119908(0)

otimes ℎ(7120582)

119908(1120582)

119878minus1

120593120573minus1 (ℎ(5120573120582minus1120573minus1))

120593120573minus1 (ℎ(4120573120582120573

minus1)V(1120573120582120573

minus1)) 120593120573minus1120593120572minus1119878minus1

times (ℎ(2120572120573120582

minus1120573minus1120572minus1)) 120593120573minus1120572minus1 (ℎ(1120572120573120582120573

minus1120572minus1))

= ℎ(3120572)

sdot V(0)

otimes ℎ(6120573)

sdot 119908(0)

otimes ℎ(7120582)

119908(1120582)

119878minus1

120593120573minus1 (ℎ(5120573120582minus1120573minus1))

120593120573minus1 (ℎ(4120573120582120573

minus1)V(1120573120582120573

minus1)) 120593120573minus1120572minus1119878minus1

times (119878 (ℎ(1120572120573120582120573

minus1120572minus1)) ℎ(2120572120573120582

minus1120573minus1120572minus1))

= ℎ(1120572)

1(2120572)

sdot V(0)

otimes ℎ(4120573)

sdot 119908(0)

otimes ℎ(5120582)

119908(1120582)

119878minus1

120593120573minus1 (ℎ(3120573120582minus1120573minus1))

120593120573minus1 (ℎ(2120573120582120573

minus1)V(1120573120582120573

minus1)) 120593120573minus1120572minus1119878minus1

(1(1120572120573120582

minus1120573minus1120572minus1))

= ℎ(1120572)

1(2120572)

sdot V(0)

otimes ℎ(4120573)

sdot 119908(0)

otimes ℎ(5120582)

119908(1120582)

119878minus1

120593120573minus1 (ℎ(3120573120582minus1120573minus1))

120593120573minus1 (ℎ(2120573120582120573

minus1)1(3120573120582120573

minus1)V(1120573120582120573

minus1)119878minus1

times120593120572minus1 (1(1120572120573120582

minus1120573minus1120572minus1)))

= ℎ(1120572)

sdot V(0)

otimes ℎ(3120573)

sdot 119908(0)

otimes ℎ(4120582)

119908(1120582)

times 120593120573minus1119878minus1

(120576119904

120573120582minus1120573minus1 (ℎ(21)

)) 120593120573minus1 (V(1120573120582120573

minus1))

= ℎ(1120572)

sdot V(0)

otimes ℎ(2120573)

11015840

(1120573)1(2120573)

sdot 119908(0)

otimes ℎ(3120582)

11015840

(2120582)119908(1120582)

119878minus1

120593120573minus1 (1(1120573120582minus1120573minus1))

120593120573minus1 (V(1120573120582120573

minus1))

= ℎ(1120572)

sdot V(0)

otimes ℎ(2120573)

sdot 119908(0)

otimes ℎ(3120582)

119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

= ℎ(1120572120573)

sdot (V otimes 119908)(0)

otimes ℎ(2120582)

(V otimes 119908)(1120582)

1(1120572120573)

sdot (V otimes 119908)(0)

otimes 1(2120582)

(V otimes 119908)(1120582)

= 1(1120572120573)

sdot (V(0)

otimes 119908(0)) otimes 1(2120582)

119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

= 1(1120572)

sdot V(0)

otimes 1(2120573)

sdot 119908(0)

otimes 1(3120582)

119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

= 1(1120572)

sdot V(0)

otimes 1(2120573)

11015840

(1120573)sdot 119908(0)

otimes 11015840

(2120582)119908(1120582)

times 120593120573minus1 (V(1120573120582120573

minus1))

= 1(1120572)

sdot V(0)

otimes 1(2120573)

sdot 119908(0)

otimes 119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

= V(0)

otimes 119908(0)

otimes 119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

= (V otimes 119908)(0)

otimes (V otimes 119908)(1120582)

(27)

This finishes the proof

Proposition 4 Let (119881 120588119881) isin WYD120572(119867) and let 120573 isin 120587 Set

120573

119881 = 119881 as vector space with action and coaction structuresdefined by

ℎ ⊳120573V =120573

(120593120573minus1 (ℎ) sdot V) forallℎ isin 119867

120573120572120573minus1 120573V isin120573

119881

120588120573119881

120582(120573V) =

120573

(V(0)) otimes 120593120573(V(1120573minus1120582120573)

)

= V⟨0⟩

otimes V⟨1120582⟩

forall120573V isin120573

119881

(28)

Then120573

119881 isin WYD120573120572120573minus1(119867)

6 Abstract and Applied Analysis

Proof Obviously 120573119881 is a left119867120573120572120573minus1-module and conditions

(16) and (17) are straightforward Then it remains to showthat conditions (20) and (21) hold For all 120573V isin

120573

119881 we have

1(1120573120572120573

minus1)⊳ V⟨0⟩

otimes 1(2120582)

V⟨1120582⟩

=120573

(120593120573minus1 (1(1120573120572120573

minus1)) sdot V(0)) otimes1(2120582)

120593120573(V(1120573minus1120582120573)

)

=120573

(1(1120572)

sdot V(0)) otimes 120593120573(1(2120582)

) 120593120573(V(1120573minus1120582120573)

)

= V⟨0⟩

otimes V⟨1120582⟩

(29)

Next for all ℎ isin 119867120573120572120573minus1120582 120573V isin

120573

119881 we get

(ℎ(2120573120572120573

minus1)⊳120573V)⟨0⟩

otimes (ℎ(2120573120572120573

minus1)⊳120573V)⟨1120582⟩

times 120593120573120572minus1120573minus1 (ℎ(1120573120572120573

minus1120582120573120572minus1120573minus1))

=

120573

((120593120573minus1 (ℎ(2120573120572120573

minus1)) sdot V)(0)

)

otimes 120593120573((120593120573minus1 (ℎ(2120573120572120573

minus1)) sdot V)(1120573minus1120582120573)

)

120593120573120572minus1120573minus1 (ℎ(1120573120572120573

minus1120582120573120572minus1120573minus1))

=

120573

((120593120573minus1(ℎ)(2120572)

sdot V)(0)

)

otimes 120593120573((120593120573minus1(ℎ)(2120572)

sdot V)(1120573minus1120582120573)

times 120593120572minus1 (120593120573minus1(ℎ)(1120572120573minus1120582120573120572minus1)))

=120573

(120593120573minus1(ℎ)(1120572)

sdot V(0))

otimes 120593120573(120593120573minus1(ℎ)(2120573minus1120582120573)

V(1120573minus1120582120573)

)

=120573

(120593120573minus1 (ℎ(1120573120572120573

minus1)) sdot V(0))

otimes 120593120573(120593120573minus1 (ℎ(2120582)

) V(1120573minus1120582120573)

)

= ℎ(1120573120572120573

minus1)⊳120573

(V(0))

otimes 120593120573(120593120573minus1 (ℎ(2120582)

) V(1120573minus1120582120573)

)

= ℎ(1120573120572120573

minus1)⊳120573V⟨0⟩

otimes ℎ(2120582)

V⟨1120582⟩

(30)

This completes the proof of the proposition

Remark 5 Let (119881 120588119881

) isin WYD120572(119867) and let (119882 120588

119882

) isin

WYD120573(119867) then we have 119904119905119881 =

119904

(119905

119881) as an object inWYD

119904119905120572119905minus1119904minus1(119867) and 119904(119881otimes

119905120572120573119882) =

119904

119881otimes119905119904120572120573119904minus1

119904

119882 as anobject inWYD

119904120572120573119904minus1(119867)

Proposition 6 Let (119881 120588119881

) isin WYD120572(119867) (119882 120588

119882

) isin

WYD120573(119867) Set 119881119882 =

120572

119882 as an object inWYD120572120573120572minus1(119867)

Define the map

119888119881119882

119881 otimes119882 997888rarr119881

119882otimes119881

119888119881119882

(V otimes 119908) =120572

(119878120573minus1 (V(1120573minus1)) sdot 119908) otimesV

(0)

(31)

Then 119888119881119882

is 119867-linear 119867-colinear and satisfies the followingconditions

119888119881otimes119882119880

= (119888119881

119882

119880otimes id119882) ∘ (id

119881otimes 119888119882119880

)

119888119881119882otimes119880

= (id119881119882

otimes 119888119881119880

) ∘ (119888119881119882

otimes id119880)

(32)

Furthermore 119888120574119881120574119882

= 119888119881119882

for all 120574 isin 120587

Proof Firstly we need to show that 119888119881119882

is well definedIndeed we have

119888119881sdot119882

(1(1120572)

sdot V otimes 1(2120573)

sdot 119908)

=

120572

(119878120573minus1 ((1(1120572)

sdot V)(1120573minus1)) 1(2120573)

sdot 119908) otimes(1(1120572)

sdot V)(0)

=120572

(119878120573minus1 (V(1120573minus1)119878minus1

120593120572minus1 (1(1120572120573120572

minus1)))

times 119878120573minus1 (1(3120573minus1)) 1(4120573)

sdot 119908) otimes 1(2120572)

sdot V(0)

=120572

(119878120573minus1119878minus1

120593120572minus1 (1(1120572120573120572

minus1))

times 119878120573minus1 (V(1120573minus1)) 120576119904

120573(1(31)

) sdot 119908) otimes 1(2120572)

sdot V(0)

=120572

(119878120573minus1119878minus1

120593120572minus1 (1(1120572120573120572

minus1)) 119878120573minus1 (1(3120573minus1)V(1120573minus1)) sdot 119908)

otimes 1(2120572)

sdot V(0)

=120572

(119878120573minus1 (V(1120573minus1)) sdot 119908) otimes V

(0)

= 119888119881119882

(V otimes 119908)

(33)

Secondly we prove that 119888119881119882

is119867-linear For all ℎ isin 119867120572120573

we compute

119888119881119882

(ℎ sdot (V otimes 119908))

=

120572

(119878120573minus1 ((ℎ(1120572)

sdot V)(1120573minus1)) ℎ(2120573)

sdot 119908) otimes (ℎ(1120572)

sdot V)(0)

=120572

(119878120573minus1 (ℎ(3120573minus1)V(1120573minus1)119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) ℎ(4120573)

sdot 119908)

otimes ℎ(2120572)

sdot V(0)

=120572

(119878120573minus1 (V(1120573minus1)119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) 120576119904

120573(ℎ(31)

) sdot 119908)

otimes ℎ(2120572)

sdot V(0)

=120572

(119878120573minus1 (V(1120573minus1)119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1)))

times 119878120573minus1 (1(2120573minus1)) sdot 119908) otimes ℎ

(2120572)1(1120572)

sdot V(0)

Abstract and Applied Analysis 7

=120572

(119878120573minus1 (1(3120573minus1)V(1120573minus1)119878minus1

120593120572minus1 (1(1120572120573120572

minus1))

times 119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) sdot 119908) otimes ℎ

(2120572)1(2120572)

sdot V(0)

=120572

(119878120573minus1 (V(1120573minus1)119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) sdot 119908)

otimes ℎ(2120572)

sdot V(0)

=120572

(120593120572minus1 (ℎ(1120572120573120572

minus1)) 119878120573minus1 (V(1120573minus1)) sdot 119908) otimes ℎ

(2120572)sdot V(0)

= ℎ(1120572120573120572

minus1)⊳120572

(119878120573minus1 (V(1120573minus1)) sdot 119908) otimes ℎ

(2120572)sdot V

= ℎ sdot 119888119881119882

(V otimes 119908)

(34)as required

Finally we check that 119888119881119882

is satisfing the 119867-colinearcondition In fact

120588

119881

119882otimes119881

120582∘ 119888119881119882

(V otimes 119908)

=

120572

((119878120573minus1 (V(1120573minus1)) sdot 119908)

(0)

) otimes V(0)(0)

otimes V(0)(1120582)

(119878120573minus1 (V(1120573minus1)) sdot 119908)

(1120582)

=

120572

(119878120573minus1(V(1120573minus1))(2120573)

sdot 119908(0)) otimes V

(0)(0)

otimes V(0)(1120582)

119878120573minus1(V(1120573minus1))(3120582)

119908(1120582)

119878minus1

120593120573minus1

(119878120573minus1(V(1120573minus1))(1120573120582minus1120573minus1)

)

=120572

(119878120573minus1 (V(3120573minus1)) sdot 119908(0)) otimes V(0)

otimes V(1120582)

119878120582minus1 (V(2120582minus1))

times 119908(1120582)

119878minus1

120593120573minus1119878120573120582120573minus1 (V(4120573120582120573

minus1))

=120572

(119878120573minus1 (V(2120573minus1)) sdot 119908(0)) otimes V(0)

otimes 120576119905

120582(V(11)

)

times 119908(1120582)

120593120573minus1 (V(3120573120582120573

minus1))

=120572

(119878120573minus1 (1(2120573minus1)V(1120573minus1)) sdot 119908(0)) otimes V(0)

otimes 119878120582minus1 (1(1120582minus1)) 119908(1120582)

120593120573minus1 (V(2120573120582120573

minus1))

=120572

(119878120573minus1 (1(2120573minus1)V(1120573minus1)) sdot 119908(0)) otimes V(0)

otimes 119878120582minus1 (1(1120582minus1)) 119908(1120582)

120593120573minus1119878minus1

119878120573120582120573minus1 (1(3120573120582120573

minus1)) 120593120573minus1 (V(2120573120582120573

minus1))

=120572

(119878120573minus1 (V(1120573minus1)) 119878120573minus1(1)(2120573)

sdot 119908(0))

otimes V(0)

otimes 119878120573minus1(1)(3120582)

119908(1120582)

120593120573minus1119878minus1

(119878120573minus1(1)(1120573120582minus1120573minus1)) 120593120573minus1 (V(2120573120582120573

minus1))

=120572

(119878120573minus1 (V(1120573minus1)) sdot 119908(0)) otimes V(0)

otimes 119908(1120582)

120593120573minus1 (V(2120573120582120573

minus1))

= (119888119881119882

otimes id119867120582

) (V(0)

otimes 119908(0)

otimes 119908(1120582)

120593120573minus1

times (V(1120573120582120573

minus1)))

= (119888119881119882

otimes id119867120582

) ∘ 120588119881otimes119882

120582(V otimes 119908)

(35)

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

Lemma 7 The map 119888119881119882

defined by (31) is bijective withinverse

119888minus1

119881119882119881

119882otimes119881 997888rarr 119881 otimes119882

119888minus1

119881119882(120572

119908otimes V) = V(0)

otimes V(1120573)

sdot 119908

(36)

for all V isin 119881120572

119908 isin119881

119882

Proof Firstly we prove 119888minus1

119881119882∘ 119888119881119882

= id119881otimes119882

For all V isin 119881119908 isin 119882 we have

119888minus1

119881119882∘ 119888119881119882

(V otimes 119908)

= V(0)(0)

otimes V(0)(1120573)

119878120573minus1 (V(1120573minus1)) sdot 119908

= V(0)

otimes 120576119905

120573(V(11)

) sdot 119908 = 1(1120572)

sdot V(0)

otimes 1(2120573)

120576119905

120573(V(11)

) sdot 119908

= 1(1120572)

119878minus1

120576119905

120572minus1 (V(11)

) sdot V(0)

otimes 1(2120573)

sdot 119908

= 120576 (11015840

(21)V(11)

) 1(1120572)

11015840

(1120572)sdot V(0)

otimes 1(2120573)

sdot 119908

= 1(1120572)

sdot V otimes 1(2120573)

sdot 119908 = V otimes 119908

(37)

Secondly we check 119888119881119882

∘ 119888minus1

119881119882= id 119881

119882otimes119881as follows

119888119881119882

∘ 119888minus1

119881119882(120572

119908otimesV)

=120572

(119878120573minus1 (V(0)(1120573

minus1)) V(1120573)

sdot 119908) otimesV(0)(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) 119878120573minus1 (V(0)(1120573

minus1)) V(1120573)

sdot 119908)

otimes 1(2120572)

sdot V(0)(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) 120576119904

120573(V(11)

) sdot 119908) otimes1(2120572)

sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)120576119904

120572120573120572minus1120593120572(V(11)

)) sdot 119908) otimes1(2120572)

sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes1

(2120572)

times 119878120572minus1120576119904

120572(120593120572(V(11)

)) sdot V(0)

8 Abstract and Applied Analysis

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes 1

(2120572)120576119905

120572120593120572119878 (V(11)

) sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes1

(2120572)

times 120576 (120593120572minus1 (11015840

(11)) 119878 (V(11)

)) 11015840

(2120572)sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes 1

(2120572)120576119878 (V(11)

) sdot V(0)

= 1(1120572120573120572

minus1)⊳120572

119908otimes1(2120572)

sdot V

=120572

119908otimes V(38)

This completes the proof

Define WYD(119867) = ∐120572isin120587

WYD120572(119867) the disjoint

union of the categories WYD120572(119867) for all 120572 isin 120587 If we

endow WYD(119867) with tensor product as in Proposition 3then WYD(119867) becomes a monoidal category The unit is119867119879

= 119867119905

120572= 120576119905

120572(1198671)120572isin120587

The group homomorphism 120595 119866 rarr aut(WYD(119867))

120573 rarr 120595120573is given on components as

120595120573 WYD

120572(119867) 997888rarr WYD

120573120572120573minus1 (119867) (39)

where the functor 120595120573acts as follows given a morphism 119891

(119881 120588119881

) rarr (119882 120588119882

) for any V isin 119881 we set ( 120573119891)( 120573V) =

120573

(119891(V))The braiding in WYD(119867) is given by the family 119888

119881119882

as shown in Proposition 6 Then we have the followingtheorem

Theorem 8 For a weak crossed Hopf group coalgebra 119867WYD(119867) is a braided 119879-category over group 120587

Example 9 Let 119867 be a weak Hopf algebra 119866 a finite groupand 119896(119866) the dual Hopf algebra of the group algebra 119896119866

Then have the weak Hopf group coalgebra 119896(119866) otimes119867 themultiplication in 119896(119866) otimes 119867 is given by

(119901120572otimes ℎ) (119901

120573otimes 119892) = 119901

120572119901120573otimes ℎ119892 (40)

for all 119901120572 119901120573

isin 119896(119866) ℎ 119892 isin 119867 and the comultiplicationcounit and antipode are given by

Δ119906V (119901120572 otimes ℎ) = sum

119906V=120572(119901119906otimes ℎ1) otimes (119901V otimes ℎ

2)

120576 (119901120572otimes ℎ) = 120575

1205721120576 (ℎ)

119878 (119901120572otimes ℎ) = 119901

120572minus1 otimes 119878 (ℎ)

(41)

Moreover 119896(119866) otimes 119867 is a weak crossed Hopf groupcoalgebra with the following crossing

Φ120573(119901120572otimes ℎ) = 119901

120573minus1120572120573

otimes ℎ (42)

ByTheorem 8WYD(119896(119866)otimes119867) is a braided 119879-category

4 Braided 119879-Categories over Weak LongDimodule Categories

In this section we introduce the notion of a (left-right)weak 120572-Long dimodule over a weak crossed Hopf groupcoalgebra119867 and prove that the category

119867WL119867 is a braided

119879-subcategory of Yetter-Drinfelrsquod category WYD(119867 otimes 119867)

when 119867 is a quasitriangular and coquasitriangular weakcrossed Hopf group coalgebra

Definition 10 Let119867 be a weak crossed Hopf group coalgebraover 120587 For a fixed element 120572 isin 120587 a (left-right) weak 120572-Longdimodule is a couple 119881 = (119881 120588

119881

= 120588119881

120582120582isin120587

) where 119881 is aleft 119867

120572-module and for any 120582 isin 120587 120588119881

120582 119881 rarr 119881 otimes 119867

120582is

a 119896-linear morphism such that

(1) 119881 is coassociative in the sense that for any 1205821 1205822isin 120587

we have

(id119881otimes Δ1205821 1205822

) ∘ 120588119881

12058211205822

= (120588119881

1205821

otimes id1198671205822

) ∘ 120588119881

1205822

(43)

(2) 119881 is counitary in the sense that

(id119881otimes 120576) ∘ 120588

119881

1= id119881 (44)

(3) 119881 satisfies the following compatible condition

120588119881

120582(119909 sdot V) = 119909 sdot V

(0)otimes V(1120582)

(45)

where 119909 isin 119867120572and V isin 119881

Now we can form the category119867WL119867120572of (left-right)

weak 120572-Long dimodules where the composition of mor-phisms of weak 120572-Long dimodules is the standard compo-sition of the underlying linear maps

Let119867WL119867 = ∐

120572isin120587 119867WL119867120572 the disjoint union of the

categories119867WL119867120572for all 120572 isin 120587

Proposition 11 Thecategory119867WL119867 is amonoidal category

Moreover for any 120572 120573 isin 119866 let 119881 isin119867WL119867120572

and let119882isin119867WL119867120573 Set

119881otimes119882 = V otimes 119908 isin 119881 otimes119882 | V otimes 119908

= 1(1120572)

sdot V otimes 1(2120573)

sdot 119908

= 120576 (119908(11)

120593120573minus1 (V(11)

)) V(0)

otimes 119908(0)

(46)

Then 119881 otimes119882isin119867WL119867120572with the following structures

119909 sdot (V otimes 119908) = 119909(1120572)

sdot V otimes 119909(2120573)

sdot 119908

120588119881otimes119882

120582(V otimes 119908) = V

(0)otimes 119908(0)

otimes 119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

(47)

for all 119909 isin 119867120572120573 V otimes 119908 isin 119881otimes119882

Proof It is straightforward

Abstract and Applied Analysis 9

Let (119867 120590 119877) be a coquasitriangular and quasitriangularweak crossed Hopf group coalgebra with crossing 120593 Define afamily of vector spaces 119867 otimes 119867 = (119867 otimes 119867)

120572= 1198671otimes 119867120572120572isin120587

where the 119867 on the left we consider its coquasitriangularstructure and for the right one we consider its quasitriangularstructureThen119867otimes119867 is aweak crossedHopf group coalgebrawith the natural tensor product and the crossing Φ = id otimes

120593120572120572isin120587

Theorem 12 Let (119867 120590 119877) be a weak crossed Hopf groupcoalgebra with coquasitriangular structure 120590 and quasitrian-gular structure 119877 Then the category

119867WL119867 is a braided

119879-subcategory of Yetter-Drinfelrsquod category WYD(119867 otimes 119867)

under the following action and coaction given by

120575119881

120582(V) = 119886

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1) = V

[0]otimes V[1120582]

(ℎ otimes 119909) ⊳120572V = 120590 (ℎ V

(11)) 119909 sdot V(0)

(48)

where ℎ otimes 119909 isin (119867 otimes 119867)120572 ℎ isin 119867

1 119909 isin 119867

120572 V isin 119881 and

119881isin119867WL119867120572

The braiding on119867WL119867 120591

119881119882 119881 otimes 119882 rarr

119881

119882otimes119881 isgiven by

120591119881119882

(V otimes 119908) = 120590 (119878 (V(11)

) 119908(11)

)120572

(119887120573sdot 119908(0)) otimes 119886120572sdot V(0)

(49)

for all 119881isin119867WL119867120572119882isin

119867WL119867120573

Proof Obviously119881 is a left (119867otimes119867)120572-moduleThen we show

that119881 satisfies the conditions inDefinition 1 First we need tocheck that119881 is coassociative In fact for all V isin 119881 isin

119867WL119867120572

and 1205821 1205822isin 120587

(id119881otimes Δ1205821 1205822

) ∘ 120575119881

12058211205822(V)

= 119886120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1

2120582minus1

1(2120582minus1

1))

otimes V(21)

otimes 119878minus1

(119887120582minus1

2120582minus1

1(1120582minus1

2))

= 1198861205721198861015840

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1

1

)

otimes V(21)

otimes 119878minus1

(1198871015840

120582minus1

2

)

= 119886120572sdot (1198861015840

120572sdot V(0))(0)

otimes (1198861015840

120572sdot V(0))(11)

otimes 119878minus1

(119887120582minus1

1

) otimes V(11)

otimes 119878minus1

(1198871015840

120582minus1

2

)

= (120575119881

1205821

otimes id(119867otimes119867)1205822

) (1198861015840

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(1198871015840

120582minus1

2

))

= (120575119881

1205821

otimes id(119867otimes119867)1205822

) ∘ 120575119881

1205822(V)

(50)

Next one directly shows that counitary condition (17) holdsas follows

(id119881otimes 120576) ∘ 120575

119881

1(V) = 119886

120572sdot V(0)120576 (119898(11)

) 120576119878minus1

(1198871)

= 119886120572sdot V120576 (119887

1) = 1120572sdot V = V

(51)

Then we have to prove that crossed condition (18) issatisfied For all ℎ isin 119867

1 119909 isin 119867

120572 and V isin 119881isin

119867WL119867120572 we

have(ℎ otimes 119909)

(2120572)sdot V[0]

otimes (ℎ otimes 119909)(3120582)

times V[1120582]

119878minus1

Φ120572minus1 ((ℎ otimes 119909)

(1120572120582minus1120572minus1))

= 120590 (ℎ(21)

(119886120572sdot V(0))(11)

) 119909(2120572)

sdot (119886120572sdot V(0))(0)

otimes ℎ(31)

V(11)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(119887120582minus1) 119878minus1

120595120572minus1 (119909(1120572120582minus1120572minus1))

= 120590 (ℎ(21)

V(11)

) 119909(2120572)

119886120572sdot V(0)

otimes ℎ(31)

V(21)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(119887120582minus1)

119878minus1

120595120572minus1 (119909(1120572120582minus1120572minus1))

= 120590 (ℎ(31)

V(21)

) 119909(2120572)

119886120572sdot V(0)

otimes V(11)

ℎ(21)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(120595120572minus1 (119909(1120572120582minus1120572minus1)) 119887120582minus1)

= 120590 (ℎ(21)

V(21)

) 119886120572119909(1120572)

sdot V(0)

otimes V(11)

119878minus1

120576119905

1(ℎ(11)

)

otimes 119909(3120582)

119878minus1

(119887120582minus1 (119909(2120582minus1)))

= 120590 (ℎ(21)

V(21)

) 119886120572119909(1120572)

sdot V(0)

otimes V(11)

119878minus1

120576119905

1(ℎ(11)

)

otimes 119878minus1

120576119905

120582minus1 (119909(21)

) 119878minus1

(119887120582minus1)

= 120590 (11015840

(21)ℎ V(21)

) 1198861205721(1120572)

119909 sdot V(0)

otimes V(11)

11015840

(11)

otimes 119878minus1

(1(2120582minus1)) 119878minus1

(119887120582minus1)

= 120590 (11015840

(21) V(21)

) 120590 (ℎ V(31)

) 1198861205721(1120572)

119909 sdot V(0)

otimes V(11)

11015840

(11)otimes 119878minus1

(119887120582minus11(2120582minus1))

= 120576 (V(21)

1(21)

) 120590 (ℎ V(31)

) 119886120572119909 sdot V(0)

otimes V(11)

1(11)

otimes 119878minus1

(119887120582minus1)

= 120590 (ℎ V(21)

) 119886120572119909 sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1)

= 120590 (ℎ V(11)

) 119886120572sdot (119909 sdot V

(0))(0)

otimes (119909 sdot V(0))(11)

otimes 119878minus1

(119887120582minus1)

= 120575119881

120582((ℎ otimes 119909) ⊳

120572V)

(52)

Finally it follows from Proposition 6 the braiding onWYD(119867 otimes 119867) that the braiding on

119867WL119867 is as the

following

120591119881119882

(V otimes 119908) =120572

(119878120573minus1 (V[1120573minus1])) ⊳120573119908otimesV[0]

= 120590 (119878 (V(11)

) 119908(11)

)120572

(119887120573sdot 119908(0)) otimes119886120572sdot V(0)

(53)

for all 119881isin119867WL119867120572119882isin119867WL119867120573 V isin 119881 and 119908 isin 119882

This completes the proof

10 Abstract and Applied Analysis

Acknowledgments

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

References

[1] V G Turaev ldquoHomotopy field theory in dimension 3 andcrossed group categoriesrdquo httparxivorgabsmath0005291

[2] P J Freyd and D N Yetter ldquoBraided compact closed categorieswith applications to low-dimensional topologyrdquo Advances inMathematics vol 77 no 2 pp 156ndash182 1989

[3] A Virelizier Algebres de Hopf graduees et fibres plats sur les3-varietes [PhD thesis] Universite Louis Pasteur StrasbourgFrance 2001

[4] A Virelizier ldquoInvolutory Hopf group-coalgebras and flat bun-dles over 3-manifoldsrdquo Fundamenta Mathematicae vol 188 pp241ndash270 2005

[5] A J Kirillov ldquoOn G-equivariant modular categoriesrdquohttparxivorgabsmath0401119

[6] S Wang ldquoNew Turaev braided group categories and groupSchur-Weyl dualityrdquo Applied Categorical Structures vol 21 no2 pp 141ndash166 2013

[7] T Yang and S Wang ldquoConstructing new braided 119879-categoriesover regular multiplier Hopf algebrasrdquo Communications inAlgebra vol 39 no 9 pp 3073ndash3089 2011

[8] M Zunino ldquoYetter-Drinfelrsquod modules for crossed structuresrdquoJournal of Pure and Applied Algebra vol 193 no 1-3 pp 313ndash343 2004

[9] A Van Daele ldquoMultiplier Hopf algebrasrdquo Transactions of theAmerican Mathematical Society vol 342 no 2 pp 917ndash9321994

[10] A Van Daele and S H Wang ldquoWeak multiplier Hopf algebrasI The main theoryrdquo Journal fur die reine und angewandteMathematik 2013

[11] A Van Daele and S Wang ldquoNew braided crossed categoriesandDrinfelrsquod quantumdouble for weakHopf group coalgebrasrdquoCommunications in Algebra vol 36 no 6 pp 2341ndash2386 2008

[12] G Bohm F Nill and K Szlachanyi ldquoWeak Hopf algebras IIntegral theory and 119862

lowast-structurerdquo Journal of Algebra vol 221no 2 pp 385ndash438 1999

[13] X Zhou and S Wang ldquoThe duality theorem for weak Hopfalgebra (co) actionsrdquo Communications in Algebra vol 38 no12 pp 4613ndash4632 2010

[14] A Virelizier ldquoHopf group-coalgebrasrdquo Journal of Pure andApplied Algebra vol 171 no 1 pp 75ndash122 2002

[15] M E Sweedler Hopf Algebras Mathematics Lecture NoteSeries W A Benjamin New York NY USA 1969

[16] X Zhou and T Yang ldquoKegelrsquos theorem over weak Hopf groupcoalgebrasrdquo Journal of Mathematics vol 33 no 2 pp 228ndash2362013

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Given two WYD120572-modules (119881 120588

119881

) and (119882 120588119882

) amorphism 119891 (119881 120588

119881

) rarr (119882 120588119882

) of this two WYD120572-

modules is an 119867120572-linear map 119891 119881 119882 such that for any

120582 isin 120587

120588119882

120582∘ 119891 = (119891 otimes id

119867120582) ∘ 120588119881

120582 (19)

Then we can form the categoryWYD120572(119867) ofWYD

120572-

modules where the composition of morphisms of WYD120572-

modules is the standard composition of the underlying linearmaps

Proposition 2 Equation (18) is equivalent to the followingequations

ℎ(1120572)

sdot V(0)

otimes ℎ(2120582)

V(1120582)

= (ℎ(2120572)

sdot V)(0)

otimes (ℎ(2120572)

sdot V)(1120582)

120593120572minus1 (ℎ(1120572120582120572

minus1))

(20)

120588119881

120582(V) = V

(0)otimes V(1120582)

isin 119881otimes119905120572120582119867120582= Δ120572120582

(1120572120582) sdot (119881 otimes 119867

120582)

(21)

for any V isin 119881 ℎ isin 119867120572120582

Proof Assume that (20) and (21) hold for all ℎ isin 119867120572120582 V isin 119881

We compute

ℎ(2120572)

sdot V(0)

otimes ℎ(3120582)

V(1120582)

119878minus1

120593120572minus1 (ℎ(1120572120582minus1120572minus1))

= (ℎ(3120572)

sdot V)(0)

otimes (ℎ(3120572)

sdot V)(1120582)

times 120593120572minus1 (ℎ(2120572120582120572

minus1)) 119878minus1

120593120572minus1 (ℎ(1120572120582minus1120572minus1))

= (ℎ(3120572)

sdot V)(0)

otimes (ℎ(3120572)

sdot V)(1120582)

times 120593120572minus1 (ℎ(2120572120582120572

minus1)119878minus1

(ℎ(1120572120582minus1120572minus1)))

= (ℎ(2120572)

sdot V)(0)

otimes (ℎ(2120572)

sdot V)(1120582)

times 120593120572minus1119878minus1

(120576119905

120572120582minus1120572minus1 (ℎ(11)

))

= (11015840

(2120572)ℎ(2120572)

sdot V)(0)

otimes (11015840

(2120572)ℎ(2120572)

sdot V)(1120582)

times 120593120572minus1 (1(1120572120582120572

minus1)) times 120576 (1

(21)11015840

(11)ℎ(11)

)

= (1(2120572)

ℎ sdot V)(0)

otimes (1(2120572)

ℎ sdot V)(1120582)

120593120572minus1 (1(1120572120582120572

minus1))

= 1(1120572)

sdot (ℎ sdot V)(0)

otimes 1(2120582)

(ℎ sdot V)(1120582)

= (ℎ sdot V)(0)

otimes (ℎ sdot V)(1120582)

(22)

as requiredConversely suppose that119881 is crossed in the sense of (18)

We first note that

V(0)

otimes V(1120582)

= 1(2120572)

sdot V(0)

otimes 1(3120582)

V(1120582)

119878minus1

120593120572minus1 (1(1120572120582minus1120572minus1))

= 11015840

(1120572)1(2120572)

sdot V(0)

otimes 11015840

(2120582)V(1120582)

119878minus1

times 120593120572minus1 (1(1120572120582minus1120572minus1))

= 11015840

(1120572)sdot (1(2120572)

sdot V(0))

otimes 11015840

(2120582)[V(1120582)

119878minus1

120593120572minus1 (1(1120572120582minus1120572minus1))]

isin 119881otimes119905120572120582119867120582

(23)

To show that (21) is satisfied for all ℎ isin 119867120572120582 we do the

following calculations

(ℎ(2120572)

sdot V)(0)

otimes (ℎ(2120572)

sdot V)(1120582)

120593120572minus1 (ℎ(1120572120582120572

minus1))

= ℎ(3120572)

sdot V(0)

otimes ℎ(4120582)

V(1120582)

119878minus1

120593120572minus1 (ℎ(2120572120582minus1120572minus1))

times 120593120572minus1 (ℎ(1120572120582120572

minus1))

= ℎ(2120572)

sdot V(0)

otimes ℎ(3120582)

V(1120582)

120593120572minus1119878minus1

(120576119904

120572120582minus1120572minus1 (ℎ(11)

))

= ℎ(2120572)

11015840

(2120572)sdot V(0)

otimes ℎ(3120582)

V(1120582)

120593120572minus1119878minus1

(1(1120572120582minus1120572minus1))

times 120576 (ℎ(11)

11015840

(11)1(21)

)

= ℎ(1120572)

1(2120572)

sdot V(0)

otimes ℎ(2120582)

V(1120582)

120593120572minus1119878minus1

(1(1120572120582minus1120572minus1))

= ℎ(1120572)

11015840

(1120572)1(2120572)

sdot V(0)

otimes ℎ(2120582)

11015840

(2120582)V(1120582)

119878minus1

times 120593120572minus1 (1(1120572120582minus1120572minus1))

= ℎ(1120572)

1(2120572)

sdot V(0)

otimes ℎ(2120582)

1(3120582)

V(1120582)

119878minus1

times 120593120572minus1 (1(1120572120582minus1120572minus1))

= ℎ(1120572)

sdot V(0)

otimes ℎ(2120582)

V(1120582)

(24)

This completes the proof

Proposition 3 If (119881 120588119881

) isin WYD120572(119867) (119882 120588

119882

) isin

WYD120573(119867) then 119881otimes

119905120572120573119882 = Δ

120572120573(1120572120573) sdot (119881 otimes 119882) isin

WYD120572120573(119867) with the action and coaction structures as

follows

ℎ sdot (V otimes 119908) = ℎ(1120572)

sdot V otimes ℎ(2120573)

sdot 119908

120588119881otimes119905120572120573119882

120582(V otimes 119908) = V

(0)otimes 119908(0)

otimes 119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

(25)

for all ℎ isin 119867120572120573 120582 isin 120587 V otimes 119908 isin 119881otimes

119905120572120573119882

Proof It is easy to prove that 119881otimes119905120572120573119882 is a left 119867

120572120573-module

and the proof of coassociativity of 119881otimes119905120572120573119882 is similar to the

Hopf group coalgebra case For all V otimes 119908 isin 119881otimes119905120572120573119882 we have

(id119881otimes119905120572120573119882

otimes 120576) ∘ 120588119881otimes119905120572120573119882

1(V otimes 119908)

= 120576 (119908(11)

120593120573minus1(11015840

)(21)

)

times 120576 (120593120573minus1(11015840

)(11)

120593120573minus1 (V(11)

)) V(0)

otimes 119908(0)

Abstract and Applied Analysis 5

= 120576 ((1(2120573)

sdot 119908)(11)

120593120573minus1 (1(11)

) 120593120573minus1 (11015840

(21)))

times 120576 (120593120573minus1 (11015840

(11)) 120593120573minus1 (V(11)

)) V(0)

otimes (1(2120573)

sdot 119908)(0)

= 120576 (1(51)

119908(11)

119878minus1

120593120573minus1 (1(31)

) 120593120573minus1 (1(21)

))

times 120576 (1(11)

V(11)

) V(0)

otimes 1(4120573)

sdot 119908(0)

= 120576 (1(41)

119908(11)

120593120573minus1119878minus1

(120576119904

1(1(21)

))) 120576 (1(11)

V(11)

) V(0)

otimes 1(3120573)

sdot 119908(0)

= 120576 (1(41)

119908(11)

120576119905

1119878minus1

120593120573minus1 (1(21)

)) 120576 (1(11)

V(11)

) V(0)

otimes 1(3120573)

sdot 119908(0)

= 120576 ((1(2120573)

sdot 119908)(11)

) 120576 (1(11)

V(11)

) V(0)

otimes (1(2120573)

sdot 119908)(0)

= 120576 ((1(2120572)

sdot V)(11)

120593120572minus1 (1(11)

)) (1(2120572)

sdot V)(0)

otimes 1(3120573)

sdot 119908

= 120576 (1(31)

V(11)

120593120572minus1119878minus1

120576119904

1(1(11)

)) 1(2120572)

sdot V(0)

otimes 1(4120573)

sdot 119908

= 120576 ((1(1120572)

sdot V)(11)

) (1(1120572)

sdot V)(0)

otimes 1(2120573)

sdot 119908

= V otimes 119908

(26)

This shows that119881otimes119905120572120573119882 is satisfing counitary condition (17)

Then we check the equivalent form of crossed conditions(20) and (21) In fact for all ℎ isin 119867

120572120573120582 V otimes 119908 isin 119881otimes

119905120572120573119882 we

have

(ℎ(2120572120573)

sdot (V otimes 119908))(0)

otimes (ℎ(2120572120573)

sdot (V otimes 119908))(1120582)

120593(120572120573)minus1 (ℎ(1120572120573120582120573

minus1120572minus1))

= (ℎ(2120572)

sdot V)(0)

otimes (ℎ(3120573)

sdot 119908)(0)

otimes (ℎ(3120573)

sdot 119908)(1120582)

120593120573minus1 ((ℎ(2120572)

sdot V)(1120573120582120573

minus1))

120593120573minus1120572minus1 (ℎ(1120572120573120582120573

minus1120572minus1))

= ℎ(3120572)

sdot V(0)

otimes ℎ(6120573)

sdot 119908(0)

otimes ℎ(7120582)

119908(1120582)

119878minus1

120593120573minus1 (ℎ(5120573120582minus1120573minus1))

120593120573minus1 (ℎ(4120573120582120573

minus1)V(1120573120582120573

minus1)) 120593120573minus1120593120572minus1119878minus1

times (ℎ(2120572120573120582

minus1120573minus1120572minus1)) 120593120573minus1120572minus1 (ℎ(1120572120573120582120573

minus1120572minus1))

= ℎ(3120572)

sdot V(0)

otimes ℎ(6120573)

sdot 119908(0)

otimes ℎ(7120582)

119908(1120582)

119878minus1

120593120573minus1 (ℎ(5120573120582minus1120573minus1))

120593120573minus1 (ℎ(4120573120582120573

minus1)V(1120573120582120573

minus1)) 120593120573minus1120572minus1119878minus1

times (119878 (ℎ(1120572120573120582120573

minus1120572minus1)) ℎ(2120572120573120582

minus1120573minus1120572minus1))

= ℎ(1120572)

1(2120572)

sdot V(0)

otimes ℎ(4120573)

sdot 119908(0)

otimes ℎ(5120582)

119908(1120582)

119878minus1

120593120573minus1 (ℎ(3120573120582minus1120573minus1))

120593120573minus1 (ℎ(2120573120582120573

minus1)V(1120573120582120573

minus1)) 120593120573minus1120572minus1119878minus1

(1(1120572120573120582

minus1120573minus1120572minus1))

= ℎ(1120572)

1(2120572)

sdot V(0)

otimes ℎ(4120573)

sdot 119908(0)

otimes ℎ(5120582)

119908(1120582)

119878minus1

120593120573minus1 (ℎ(3120573120582minus1120573minus1))

120593120573minus1 (ℎ(2120573120582120573

minus1)1(3120573120582120573

minus1)V(1120573120582120573

minus1)119878minus1

times120593120572minus1 (1(1120572120573120582

minus1120573minus1120572minus1)))

= ℎ(1120572)

sdot V(0)

otimes ℎ(3120573)

sdot 119908(0)

otimes ℎ(4120582)

119908(1120582)

times 120593120573minus1119878minus1

(120576119904

120573120582minus1120573minus1 (ℎ(21)

)) 120593120573minus1 (V(1120573120582120573

minus1))

= ℎ(1120572)

sdot V(0)

otimes ℎ(2120573)

11015840

(1120573)1(2120573)

sdot 119908(0)

otimes ℎ(3120582)

11015840

(2120582)119908(1120582)

119878minus1

120593120573minus1 (1(1120573120582minus1120573minus1))

120593120573minus1 (V(1120573120582120573

minus1))

= ℎ(1120572)

sdot V(0)

otimes ℎ(2120573)

sdot 119908(0)

otimes ℎ(3120582)

119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

= ℎ(1120572120573)

sdot (V otimes 119908)(0)

otimes ℎ(2120582)

(V otimes 119908)(1120582)

1(1120572120573)

sdot (V otimes 119908)(0)

otimes 1(2120582)

(V otimes 119908)(1120582)

= 1(1120572120573)

sdot (V(0)

otimes 119908(0)) otimes 1(2120582)

119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

= 1(1120572)

sdot V(0)

otimes 1(2120573)

sdot 119908(0)

otimes 1(3120582)

119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

= 1(1120572)

sdot V(0)

otimes 1(2120573)

11015840

(1120573)sdot 119908(0)

otimes 11015840

(2120582)119908(1120582)

times 120593120573minus1 (V(1120573120582120573

minus1))

= 1(1120572)

sdot V(0)

otimes 1(2120573)

sdot 119908(0)

otimes 119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

= V(0)

otimes 119908(0)

otimes 119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

= (V otimes 119908)(0)

otimes (V otimes 119908)(1120582)

(27)

This finishes the proof

Proposition 4 Let (119881 120588119881) isin WYD120572(119867) and let 120573 isin 120587 Set

120573

119881 = 119881 as vector space with action and coaction structuresdefined by

ℎ ⊳120573V =120573

(120593120573minus1 (ℎ) sdot V) forallℎ isin 119867

120573120572120573minus1 120573V isin120573

119881

120588120573119881

120582(120573V) =

120573

(V(0)) otimes 120593120573(V(1120573minus1120582120573)

)

= V⟨0⟩

otimes V⟨1120582⟩

forall120573V isin120573

119881

(28)

Then120573

119881 isin WYD120573120572120573minus1(119867)

6 Abstract and Applied Analysis

Proof Obviously 120573119881 is a left119867120573120572120573minus1-module and conditions

(16) and (17) are straightforward Then it remains to showthat conditions (20) and (21) hold For all 120573V isin

120573

119881 we have

1(1120573120572120573

minus1)⊳ V⟨0⟩

otimes 1(2120582)

V⟨1120582⟩

=120573

(120593120573minus1 (1(1120573120572120573

minus1)) sdot V(0)) otimes1(2120582)

120593120573(V(1120573minus1120582120573)

)

=120573

(1(1120572)

sdot V(0)) otimes 120593120573(1(2120582)

) 120593120573(V(1120573minus1120582120573)

)

= V⟨0⟩

otimes V⟨1120582⟩

(29)

Next for all ℎ isin 119867120573120572120573minus1120582 120573V isin

120573

119881 we get

(ℎ(2120573120572120573

minus1)⊳120573V)⟨0⟩

otimes (ℎ(2120573120572120573

minus1)⊳120573V)⟨1120582⟩

times 120593120573120572minus1120573minus1 (ℎ(1120573120572120573

minus1120582120573120572minus1120573minus1))

=

120573

((120593120573minus1 (ℎ(2120573120572120573

minus1)) sdot V)(0)

)

otimes 120593120573((120593120573minus1 (ℎ(2120573120572120573

minus1)) sdot V)(1120573minus1120582120573)

)

120593120573120572minus1120573minus1 (ℎ(1120573120572120573

minus1120582120573120572minus1120573minus1))

=

120573

((120593120573minus1(ℎ)(2120572)

sdot V)(0)

)

otimes 120593120573((120593120573minus1(ℎ)(2120572)

sdot V)(1120573minus1120582120573)

times 120593120572minus1 (120593120573minus1(ℎ)(1120572120573minus1120582120573120572minus1)))

=120573

(120593120573minus1(ℎ)(1120572)

sdot V(0))

otimes 120593120573(120593120573minus1(ℎ)(2120573minus1120582120573)

V(1120573minus1120582120573)

)

=120573

(120593120573minus1 (ℎ(1120573120572120573

minus1)) sdot V(0))

otimes 120593120573(120593120573minus1 (ℎ(2120582)

) V(1120573minus1120582120573)

)

= ℎ(1120573120572120573

minus1)⊳120573

(V(0))

otimes 120593120573(120593120573minus1 (ℎ(2120582)

) V(1120573minus1120582120573)

)

= ℎ(1120573120572120573

minus1)⊳120573V⟨0⟩

otimes ℎ(2120582)

V⟨1120582⟩

(30)

This completes the proof of the proposition

Remark 5 Let (119881 120588119881

) isin WYD120572(119867) and let (119882 120588

119882

) isin

WYD120573(119867) then we have 119904119905119881 =

119904

(119905

119881) as an object inWYD

119904119905120572119905minus1119904minus1(119867) and 119904(119881otimes

119905120572120573119882) =

119904

119881otimes119905119904120572120573119904minus1

119904

119882 as anobject inWYD

119904120572120573119904minus1(119867)

Proposition 6 Let (119881 120588119881

) isin WYD120572(119867) (119882 120588

119882

) isin

WYD120573(119867) Set 119881119882 =

120572

119882 as an object inWYD120572120573120572minus1(119867)

Define the map

119888119881119882

119881 otimes119882 997888rarr119881

119882otimes119881

119888119881119882

(V otimes 119908) =120572

(119878120573minus1 (V(1120573minus1)) sdot 119908) otimesV

(0)

(31)

Then 119888119881119882

is 119867-linear 119867-colinear and satisfies the followingconditions

119888119881otimes119882119880

= (119888119881

119882

119880otimes id119882) ∘ (id

119881otimes 119888119882119880

)

119888119881119882otimes119880

= (id119881119882

otimes 119888119881119880

) ∘ (119888119881119882

otimes id119880)

(32)

Furthermore 119888120574119881120574119882

= 119888119881119882

for all 120574 isin 120587

Proof Firstly we need to show that 119888119881119882

is well definedIndeed we have

119888119881sdot119882

(1(1120572)

sdot V otimes 1(2120573)

sdot 119908)

=

120572

(119878120573minus1 ((1(1120572)

sdot V)(1120573minus1)) 1(2120573)

sdot 119908) otimes(1(1120572)

sdot V)(0)

=120572

(119878120573minus1 (V(1120573minus1)119878minus1

120593120572minus1 (1(1120572120573120572

minus1)))

times 119878120573minus1 (1(3120573minus1)) 1(4120573)

sdot 119908) otimes 1(2120572)

sdot V(0)

=120572

(119878120573minus1119878minus1

120593120572minus1 (1(1120572120573120572

minus1))

times 119878120573minus1 (V(1120573minus1)) 120576119904

120573(1(31)

) sdot 119908) otimes 1(2120572)

sdot V(0)

=120572

(119878120573minus1119878minus1

120593120572minus1 (1(1120572120573120572

minus1)) 119878120573minus1 (1(3120573minus1)V(1120573minus1)) sdot 119908)

otimes 1(2120572)

sdot V(0)

=120572

(119878120573minus1 (V(1120573minus1)) sdot 119908) otimes V

(0)

= 119888119881119882

(V otimes 119908)

(33)

Secondly we prove that 119888119881119882

is119867-linear For all ℎ isin 119867120572120573

we compute

119888119881119882

(ℎ sdot (V otimes 119908))

=

120572

(119878120573minus1 ((ℎ(1120572)

sdot V)(1120573minus1)) ℎ(2120573)

sdot 119908) otimes (ℎ(1120572)

sdot V)(0)

=120572

(119878120573minus1 (ℎ(3120573minus1)V(1120573minus1)119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) ℎ(4120573)

sdot 119908)

otimes ℎ(2120572)

sdot V(0)

=120572

(119878120573minus1 (V(1120573minus1)119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) 120576119904

120573(ℎ(31)

) sdot 119908)

otimes ℎ(2120572)

sdot V(0)

=120572

(119878120573minus1 (V(1120573minus1)119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1)))

times 119878120573minus1 (1(2120573minus1)) sdot 119908) otimes ℎ

(2120572)1(1120572)

sdot V(0)

Abstract and Applied Analysis 7

=120572

(119878120573minus1 (1(3120573minus1)V(1120573minus1)119878minus1

120593120572minus1 (1(1120572120573120572

minus1))

times 119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) sdot 119908) otimes ℎ

(2120572)1(2120572)

sdot V(0)

=120572

(119878120573minus1 (V(1120573minus1)119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) sdot 119908)

otimes ℎ(2120572)

sdot V(0)

=120572

(120593120572minus1 (ℎ(1120572120573120572

minus1)) 119878120573minus1 (V(1120573minus1)) sdot 119908) otimes ℎ

(2120572)sdot V(0)

= ℎ(1120572120573120572

minus1)⊳120572

(119878120573minus1 (V(1120573minus1)) sdot 119908) otimes ℎ

(2120572)sdot V

= ℎ sdot 119888119881119882

(V otimes 119908)

(34)as required

Finally we check that 119888119881119882

is satisfing the 119867-colinearcondition In fact

120588

119881

119882otimes119881

120582∘ 119888119881119882

(V otimes 119908)

=

120572

((119878120573minus1 (V(1120573minus1)) sdot 119908)

(0)

) otimes V(0)(0)

otimes V(0)(1120582)

(119878120573minus1 (V(1120573minus1)) sdot 119908)

(1120582)

=

120572

(119878120573minus1(V(1120573minus1))(2120573)

sdot 119908(0)) otimes V

(0)(0)

otimes V(0)(1120582)

119878120573minus1(V(1120573minus1))(3120582)

119908(1120582)

119878minus1

120593120573minus1

(119878120573minus1(V(1120573minus1))(1120573120582minus1120573minus1)

)

=120572

(119878120573minus1 (V(3120573minus1)) sdot 119908(0)) otimes V(0)

otimes V(1120582)

119878120582minus1 (V(2120582minus1))

times 119908(1120582)

119878minus1

120593120573minus1119878120573120582120573minus1 (V(4120573120582120573

minus1))

=120572

(119878120573minus1 (V(2120573minus1)) sdot 119908(0)) otimes V(0)

otimes 120576119905

120582(V(11)

)

times 119908(1120582)

120593120573minus1 (V(3120573120582120573

minus1))

=120572

(119878120573minus1 (1(2120573minus1)V(1120573minus1)) sdot 119908(0)) otimes V(0)

otimes 119878120582minus1 (1(1120582minus1)) 119908(1120582)

120593120573minus1 (V(2120573120582120573

minus1))

=120572

(119878120573minus1 (1(2120573minus1)V(1120573minus1)) sdot 119908(0)) otimes V(0)

otimes 119878120582minus1 (1(1120582minus1)) 119908(1120582)

120593120573minus1119878minus1

119878120573120582120573minus1 (1(3120573120582120573

minus1)) 120593120573minus1 (V(2120573120582120573

minus1))

=120572

(119878120573minus1 (V(1120573minus1)) 119878120573minus1(1)(2120573)

sdot 119908(0))

otimes V(0)

otimes 119878120573minus1(1)(3120582)

119908(1120582)

120593120573minus1119878minus1

(119878120573minus1(1)(1120573120582minus1120573minus1)) 120593120573minus1 (V(2120573120582120573

minus1))

=120572

(119878120573minus1 (V(1120573minus1)) sdot 119908(0)) otimes V(0)

otimes 119908(1120582)

120593120573minus1 (V(2120573120582120573

minus1))

= (119888119881119882

otimes id119867120582

) (V(0)

otimes 119908(0)

otimes 119908(1120582)

120593120573minus1

times (V(1120573120582120573

minus1)))

= (119888119881119882

otimes id119867120582

) ∘ 120588119881otimes119882

120582(V otimes 119908)

(35)

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

Lemma 7 The map 119888119881119882

defined by (31) is bijective withinverse

119888minus1

119881119882119881

119882otimes119881 997888rarr 119881 otimes119882

119888minus1

119881119882(120572

119908otimes V) = V(0)

otimes V(1120573)

sdot 119908

(36)

for all V isin 119881120572

119908 isin119881

119882

Proof Firstly we prove 119888minus1

119881119882∘ 119888119881119882

= id119881otimes119882

For all V isin 119881119908 isin 119882 we have

119888minus1

119881119882∘ 119888119881119882

(V otimes 119908)

= V(0)(0)

otimes V(0)(1120573)

119878120573minus1 (V(1120573minus1)) sdot 119908

= V(0)

otimes 120576119905

120573(V(11)

) sdot 119908 = 1(1120572)

sdot V(0)

otimes 1(2120573)

120576119905

120573(V(11)

) sdot 119908

= 1(1120572)

119878minus1

120576119905

120572minus1 (V(11)

) sdot V(0)

otimes 1(2120573)

sdot 119908

= 120576 (11015840

(21)V(11)

) 1(1120572)

11015840

(1120572)sdot V(0)

otimes 1(2120573)

sdot 119908

= 1(1120572)

sdot V otimes 1(2120573)

sdot 119908 = V otimes 119908

(37)

Secondly we check 119888119881119882

∘ 119888minus1

119881119882= id 119881

119882otimes119881as follows

119888119881119882

∘ 119888minus1

119881119882(120572

119908otimesV)

=120572

(119878120573minus1 (V(0)(1120573

minus1)) V(1120573)

sdot 119908) otimesV(0)(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) 119878120573minus1 (V(0)(1120573

minus1)) V(1120573)

sdot 119908)

otimes 1(2120572)

sdot V(0)(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) 120576119904

120573(V(11)

) sdot 119908) otimes1(2120572)

sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)120576119904

120572120573120572minus1120593120572(V(11)

)) sdot 119908) otimes1(2120572)

sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes1

(2120572)

times 119878120572minus1120576119904

120572(120593120572(V(11)

)) sdot V(0)

8 Abstract and Applied Analysis

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes 1

(2120572)120576119905

120572120593120572119878 (V(11)

) sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes1

(2120572)

times 120576 (120593120572minus1 (11015840

(11)) 119878 (V(11)

)) 11015840

(2120572)sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes 1

(2120572)120576119878 (V(11)

) sdot V(0)

= 1(1120572120573120572

minus1)⊳120572

119908otimes1(2120572)

sdot V

=120572

119908otimes V(38)

This completes the proof

Define WYD(119867) = ∐120572isin120587

WYD120572(119867) the disjoint

union of the categories WYD120572(119867) for all 120572 isin 120587 If we

endow WYD(119867) with tensor product as in Proposition 3then WYD(119867) becomes a monoidal category The unit is119867119879

= 119867119905

120572= 120576119905

120572(1198671)120572isin120587

The group homomorphism 120595 119866 rarr aut(WYD(119867))

120573 rarr 120595120573is given on components as

120595120573 WYD

120572(119867) 997888rarr WYD

120573120572120573minus1 (119867) (39)

where the functor 120595120573acts as follows given a morphism 119891

(119881 120588119881

) rarr (119882 120588119882

) for any V isin 119881 we set ( 120573119891)( 120573V) =

120573

(119891(V))The braiding in WYD(119867) is given by the family 119888

119881119882

as shown in Proposition 6 Then we have the followingtheorem

Theorem 8 For a weak crossed Hopf group coalgebra 119867WYD(119867) is a braided 119879-category over group 120587

Example 9 Let 119867 be a weak Hopf algebra 119866 a finite groupand 119896(119866) the dual Hopf algebra of the group algebra 119896119866

Then have the weak Hopf group coalgebra 119896(119866) otimes119867 themultiplication in 119896(119866) otimes 119867 is given by

(119901120572otimes ℎ) (119901

120573otimes 119892) = 119901

120572119901120573otimes ℎ119892 (40)

for all 119901120572 119901120573

isin 119896(119866) ℎ 119892 isin 119867 and the comultiplicationcounit and antipode are given by

Δ119906V (119901120572 otimes ℎ) = sum

119906V=120572(119901119906otimes ℎ1) otimes (119901V otimes ℎ

2)

120576 (119901120572otimes ℎ) = 120575

1205721120576 (ℎ)

119878 (119901120572otimes ℎ) = 119901

120572minus1 otimes 119878 (ℎ)

(41)

Moreover 119896(119866) otimes 119867 is a weak crossed Hopf groupcoalgebra with the following crossing

Φ120573(119901120572otimes ℎ) = 119901

120573minus1120572120573

otimes ℎ (42)

ByTheorem 8WYD(119896(119866)otimes119867) is a braided 119879-category

4 Braided 119879-Categories over Weak LongDimodule Categories

In this section we introduce the notion of a (left-right)weak 120572-Long dimodule over a weak crossed Hopf groupcoalgebra119867 and prove that the category

119867WL119867 is a braided

119879-subcategory of Yetter-Drinfelrsquod category WYD(119867 otimes 119867)

when 119867 is a quasitriangular and coquasitriangular weakcrossed Hopf group coalgebra

Definition 10 Let119867 be a weak crossed Hopf group coalgebraover 120587 For a fixed element 120572 isin 120587 a (left-right) weak 120572-Longdimodule is a couple 119881 = (119881 120588

119881

= 120588119881

120582120582isin120587

) where 119881 is aleft 119867

120572-module and for any 120582 isin 120587 120588119881

120582 119881 rarr 119881 otimes 119867

120582is

a 119896-linear morphism such that

(1) 119881 is coassociative in the sense that for any 1205821 1205822isin 120587

we have

(id119881otimes Δ1205821 1205822

) ∘ 120588119881

12058211205822

= (120588119881

1205821

otimes id1198671205822

) ∘ 120588119881

1205822

(43)

(2) 119881 is counitary in the sense that

(id119881otimes 120576) ∘ 120588

119881

1= id119881 (44)

(3) 119881 satisfies the following compatible condition

120588119881

120582(119909 sdot V) = 119909 sdot V

(0)otimes V(1120582)

(45)

where 119909 isin 119867120572and V isin 119881

Now we can form the category119867WL119867120572of (left-right)

weak 120572-Long dimodules where the composition of mor-phisms of weak 120572-Long dimodules is the standard compo-sition of the underlying linear maps

Let119867WL119867 = ∐

120572isin120587 119867WL119867120572 the disjoint union of the

categories119867WL119867120572for all 120572 isin 120587

Proposition 11 Thecategory119867WL119867 is amonoidal category

Moreover for any 120572 120573 isin 119866 let 119881 isin119867WL119867120572

and let119882isin119867WL119867120573 Set

119881otimes119882 = V otimes 119908 isin 119881 otimes119882 | V otimes 119908

= 1(1120572)

sdot V otimes 1(2120573)

sdot 119908

= 120576 (119908(11)

120593120573minus1 (V(11)

)) V(0)

otimes 119908(0)

(46)

Then 119881 otimes119882isin119867WL119867120572with the following structures

119909 sdot (V otimes 119908) = 119909(1120572)

sdot V otimes 119909(2120573)

sdot 119908

120588119881otimes119882

120582(V otimes 119908) = V

(0)otimes 119908(0)

otimes 119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

(47)

for all 119909 isin 119867120572120573 V otimes 119908 isin 119881otimes119882

Proof It is straightforward

Abstract and Applied Analysis 9

Let (119867 120590 119877) be a coquasitriangular and quasitriangularweak crossed Hopf group coalgebra with crossing 120593 Define afamily of vector spaces 119867 otimes 119867 = (119867 otimes 119867)

120572= 1198671otimes 119867120572120572isin120587

where the 119867 on the left we consider its coquasitriangularstructure and for the right one we consider its quasitriangularstructureThen119867otimes119867 is aweak crossedHopf group coalgebrawith the natural tensor product and the crossing Φ = id otimes

120593120572120572isin120587

Theorem 12 Let (119867 120590 119877) be a weak crossed Hopf groupcoalgebra with coquasitriangular structure 120590 and quasitrian-gular structure 119877 Then the category

119867WL119867 is a braided

119879-subcategory of Yetter-Drinfelrsquod category WYD(119867 otimes 119867)

under the following action and coaction given by

120575119881

120582(V) = 119886

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1) = V

[0]otimes V[1120582]

(ℎ otimes 119909) ⊳120572V = 120590 (ℎ V

(11)) 119909 sdot V(0)

(48)

where ℎ otimes 119909 isin (119867 otimes 119867)120572 ℎ isin 119867

1 119909 isin 119867

120572 V isin 119881 and

119881isin119867WL119867120572

The braiding on119867WL119867 120591

119881119882 119881 otimes 119882 rarr

119881

119882otimes119881 isgiven by

120591119881119882

(V otimes 119908) = 120590 (119878 (V(11)

) 119908(11)

)120572

(119887120573sdot 119908(0)) otimes 119886120572sdot V(0)

(49)

for all 119881isin119867WL119867120572119882isin

119867WL119867120573

Proof Obviously119881 is a left (119867otimes119867)120572-moduleThen we show

that119881 satisfies the conditions inDefinition 1 First we need tocheck that119881 is coassociative In fact for all V isin 119881 isin

119867WL119867120572

and 1205821 1205822isin 120587

(id119881otimes Δ1205821 1205822

) ∘ 120575119881

12058211205822(V)

= 119886120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1

2120582minus1

1(2120582minus1

1))

otimes V(21)

otimes 119878minus1

(119887120582minus1

2120582minus1

1(1120582minus1

2))

= 1198861205721198861015840

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1

1

)

otimes V(21)

otimes 119878minus1

(1198871015840

120582minus1

2

)

= 119886120572sdot (1198861015840

120572sdot V(0))(0)

otimes (1198861015840

120572sdot V(0))(11)

otimes 119878minus1

(119887120582minus1

1

) otimes V(11)

otimes 119878minus1

(1198871015840

120582minus1

2

)

= (120575119881

1205821

otimes id(119867otimes119867)1205822

) (1198861015840

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(1198871015840

120582minus1

2

))

= (120575119881

1205821

otimes id(119867otimes119867)1205822

) ∘ 120575119881

1205822(V)

(50)

Next one directly shows that counitary condition (17) holdsas follows

(id119881otimes 120576) ∘ 120575

119881

1(V) = 119886

120572sdot V(0)120576 (119898(11)

) 120576119878minus1

(1198871)

= 119886120572sdot V120576 (119887

1) = 1120572sdot V = V

(51)

Then we have to prove that crossed condition (18) issatisfied For all ℎ isin 119867

1 119909 isin 119867

120572 and V isin 119881isin

119867WL119867120572 we

have(ℎ otimes 119909)

(2120572)sdot V[0]

otimes (ℎ otimes 119909)(3120582)

times V[1120582]

119878minus1

Φ120572minus1 ((ℎ otimes 119909)

(1120572120582minus1120572minus1))

= 120590 (ℎ(21)

(119886120572sdot V(0))(11)

) 119909(2120572)

sdot (119886120572sdot V(0))(0)

otimes ℎ(31)

V(11)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(119887120582minus1) 119878minus1

120595120572minus1 (119909(1120572120582minus1120572minus1))

= 120590 (ℎ(21)

V(11)

) 119909(2120572)

119886120572sdot V(0)

otimes ℎ(31)

V(21)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(119887120582minus1)

119878minus1

120595120572minus1 (119909(1120572120582minus1120572minus1))

= 120590 (ℎ(31)

V(21)

) 119909(2120572)

119886120572sdot V(0)

otimes V(11)

ℎ(21)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(120595120572minus1 (119909(1120572120582minus1120572minus1)) 119887120582minus1)

= 120590 (ℎ(21)

V(21)

) 119886120572119909(1120572)

sdot V(0)

otimes V(11)

119878minus1

120576119905

1(ℎ(11)

)

otimes 119909(3120582)

119878minus1

(119887120582minus1 (119909(2120582minus1)))

= 120590 (ℎ(21)

V(21)

) 119886120572119909(1120572)

sdot V(0)

otimes V(11)

119878minus1

120576119905

1(ℎ(11)

)

otimes 119878minus1

120576119905

120582minus1 (119909(21)

) 119878minus1

(119887120582minus1)

= 120590 (11015840

(21)ℎ V(21)

) 1198861205721(1120572)

119909 sdot V(0)

otimes V(11)

11015840

(11)

otimes 119878minus1

(1(2120582minus1)) 119878minus1

(119887120582minus1)

= 120590 (11015840

(21) V(21)

) 120590 (ℎ V(31)

) 1198861205721(1120572)

119909 sdot V(0)

otimes V(11)

11015840

(11)otimes 119878minus1

(119887120582minus11(2120582minus1))

= 120576 (V(21)

1(21)

) 120590 (ℎ V(31)

) 119886120572119909 sdot V(0)

otimes V(11)

1(11)

otimes 119878minus1

(119887120582minus1)

= 120590 (ℎ V(21)

) 119886120572119909 sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1)

= 120590 (ℎ V(11)

) 119886120572sdot (119909 sdot V

(0))(0)

otimes (119909 sdot V(0))(11)

otimes 119878minus1

(119887120582minus1)

= 120575119881

120582((ℎ otimes 119909) ⊳

120572V)

(52)

Finally it follows from Proposition 6 the braiding onWYD(119867 otimes 119867) that the braiding on

119867WL119867 is as the

following

120591119881119882

(V otimes 119908) =120572

(119878120573minus1 (V[1120573minus1])) ⊳120573119908otimesV[0]

= 120590 (119878 (V(11)

) 119908(11)

)120572

(119887120573sdot 119908(0)) otimes119886120572sdot V(0)

(53)

for all 119881isin119867WL119867120572119882isin119867WL119867120573 V isin 119881 and 119908 isin 119882

This completes the proof

10 Abstract and Applied Analysis

Acknowledgments

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

References

[1] V G Turaev ldquoHomotopy field theory in dimension 3 andcrossed group categoriesrdquo httparxivorgabsmath0005291

[2] P J Freyd and D N Yetter ldquoBraided compact closed categorieswith applications to low-dimensional topologyrdquo Advances inMathematics vol 77 no 2 pp 156ndash182 1989

[3] A Virelizier Algebres de Hopf graduees et fibres plats sur les3-varietes [PhD thesis] Universite Louis Pasteur StrasbourgFrance 2001

[4] A Virelizier ldquoInvolutory Hopf group-coalgebras and flat bun-dles over 3-manifoldsrdquo Fundamenta Mathematicae vol 188 pp241ndash270 2005

[5] A J Kirillov ldquoOn G-equivariant modular categoriesrdquohttparxivorgabsmath0401119

[6] S Wang ldquoNew Turaev braided group categories and groupSchur-Weyl dualityrdquo Applied Categorical Structures vol 21 no2 pp 141ndash166 2013

[7] T Yang and S Wang ldquoConstructing new braided 119879-categoriesover regular multiplier Hopf algebrasrdquo Communications inAlgebra vol 39 no 9 pp 3073ndash3089 2011

[8] M Zunino ldquoYetter-Drinfelrsquod modules for crossed structuresrdquoJournal of Pure and Applied Algebra vol 193 no 1-3 pp 313ndash343 2004

[9] A Van Daele ldquoMultiplier Hopf algebrasrdquo Transactions of theAmerican Mathematical Society vol 342 no 2 pp 917ndash9321994

[10] A Van Daele and S H Wang ldquoWeak multiplier Hopf algebrasI The main theoryrdquo Journal fur die reine und angewandteMathematik 2013

[11] A Van Daele and S Wang ldquoNew braided crossed categoriesandDrinfelrsquod quantumdouble for weakHopf group coalgebrasrdquoCommunications in Algebra vol 36 no 6 pp 2341ndash2386 2008

[12] G Bohm F Nill and K Szlachanyi ldquoWeak Hopf algebras IIntegral theory and 119862

lowast-structurerdquo Journal of Algebra vol 221no 2 pp 385ndash438 1999

[13] X Zhou and S Wang ldquoThe duality theorem for weak Hopfalgebra (co) actionsrdquo Communications in Algebra vol 38 no12 pp 4613ndash4632 2010

[14] A Virelizier ldquoHopf group-coalgebrasrdquo Journal of Pure andApplied Algebra vol 171 no 1 pp 75ndash122 2002

[15] M E Sweedler Hopf Algebras Mathematics Lecture NoteSeries W A Benjamin New York NY USA 1969

[16] X Zhou and T Yang ldquoKegelrsquos theorem over weak Hopf groupcoalgebrasrdquo Journal of Mathematics vol 33 no 2 pp 228ndash2362013

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Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

= 120576 ((1(2120573)

sdot 119908)(11)

120593120573minus1 (1(11)

) 120593120573minus1 (11015840

(21)))

times 120576 (120593120573minus1 (11015840

(11)) 120593120573minus1 (V(11)

)) V(0)

otimes (1(2120573)

sdot 119908)(0)

= 120576 (1(51)

119908(11)

119878minus1

120593120573minus1 (1(31)

) 120593120573minus1 (1(21)

))

times 120576 (1(11)

V(11)

) V(0)

otimes 1(4120573)

sdot 119908(0)

= 120576 (1(41)

119908(11)

120593120573minus1119878minus1

(120576119904

1(1(21)

))) 120576 (1(11)

V(11)

) V(0)

otimes 1(3120573)

sdot 119908(0)

= 120576 (1(41)

119908(11)

120576119905

1119878minus1

120593120573minus1 (1(21)

)) 120576 (1(11)

V(11)

) V(0)

otimes 1(3120573)

sdot 119908(0)

= 120576 ((1(2120573)

sdot 119908)(11)

) 120576 (1(11)

V(11)

) V(0)

otimes (1(2120573)

sdot 119908)(0)

= 120576 ((1(2120572)

sdot V)(11)

120593120572minus1 (1(11)

)) (1(2120572)

sdot V)(0)

otimes 1(3120573)

sdot 119908

= 120576 (1(31)

V(11)

120593120572minus1119878minus1

120576119904

1(1(11)

)) 1(2120572)

sdot V(0)

otimes 1(4120573)

sdot 119908

= 120576 ((1(1120572)

sdot V)(11)

) (1(1120572)

sdot V)(0)

otimes 1(2120573)

sdot 119908

= V otimes 119908

(26)

This shows that119881otimes119905120572120573119882 is satisfing counitary condition (17)

Then we check the equivalent form of crossed conditions(20) and (21) In fact for all ℎ isin 119867

120572120573120582 V otimes 119908 isin 119881otimes

119905120572120573119882 we

have

(ℎ(2120572120573)

sdot (V otimes 119908))(0)

otimes (ℎ(2120572120573)

sdot (V otimes 119908))(1120582)

120593(120572120573)minus1 (ℎ(1120572120573120582120573

minus1120572minus1))

= (ℎ(2120572)

sdot V)(0)

otimes (ℎ(3120573)

sdot 119908)(0)

otimes (ℎ(3120573)

sdot 119908)(1120582)

120593120573minus1 ((ℎ(2120572)

sdot V)(1120573120582120573

minus1))

120593120573minus1120572minus1 (ℎ(1120572120573120582120573

minus1120572minus1))

= ℎ(3120572)

sdot V(0)

otimes ℎ(6120573)

sdot 119908(0)

otimes ℎ(7120582)

119908(1120582)

119878minus1

120593120573minus1 (ℎ(5120573120582minus1120573minus1))

120593120573minus1 (ℎ(4120573120582120573

minus1)V(1120573120582120573

minus1)) 120593120573minus1120593120572minus1119878minus1

times (ℎ(2120572120573120582

minus1120573minus1120572minus1)) 120593120573minus1120572minus1 (ℎ(1120572120573120582120573

minus1120572minus1))

= ℎ(3120572)

sdot V(0)

otimes ℎ(6120573)

sdot 119908(0)

otimes ℎ(7120582)

119908(1120582)

119878minus1

120593120573minus1 (ℎ(5120573120582minus1120573minus1))

120593120573minus1 (ℎ(4120573120582120573

minus1)V(1120573120582120573

minus1)) 120593120573minus1120572minus1119878minus1

times (119878 (ℎ(1120572120573120582120573

minus1120572minus1)) ℎ(2120572120573120582

minus1120573minus1120572minus1))

= ℎ(1120572)

1(2120572)

sdot V(0)

otimes ℎ(4120573)

sdot 119908(0)

otimes ℎ(5120582)

119908(1120582)

119878minus1

120593120573minus1 (ℎ(3120573120582minus1120573minus1))

120593120573minus1 (ℎ(2120573120582120573

minus1)V(1120573120582120573

minus1)) 120593120573minus1120572minus1119878minus1

(1(1120572120573120582

minus1120573minus1120572minus1))

= ℎ(1120572)

1(2120572)

sdot V(0)

otimes ℎ(4120573)

sdot 119908(0)

otimes ℎ(5120582)

119908(1120582)

119878minus1

120593120573minus1 (ℎ(3120573120582minus1120573minus1))

120593120573minus1 (ℎ(2120573120582120573

minus1)1(3120573120582120573

minus1)V(1120573120582120573

minus1)119878minus1

times120593120572minus1 (1(1120572120573120582

minus1120573minus1120572minus1)))

= ℎ(1120572)

sdot V(0)

otimes ℎ(3120573)

sdot 119908(0)

otimes ℎ(4120582)

119908(1120582)

times 120593120573minus1119878minus1

(120576119904

120573120582minus1120573minus1 (ℎ(21)

)) 120593120573minus1 (V(1120573120582120573

minus1))

= ℎ(1120572)

sdot V(0)

otimes ℎ(2120573)

11015840

(1120573)1(2120573)

sdot 119908(0)

otimes ℎ(3120582)

11015840

(2120582)119908(1120582)

119878minus1

120593120573minus1 (1(1120573120582minus1120573minus1))

120593120573minus1 (V(1120573120582120573

minus1))

= ℎ(1120572)

sdot V(0)

otimes ℎ(2120573)

sdot 119908(0)

otimes ℎ(3120582)

119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

= ℎ(1120572120573)

sdot (V otimes 119908)(0)

otimes ℎ(2120582)

(V otimes 119908)(1120582)

1(1120572120573)

sdot (V otimes 119908)(0)

otimes 1(2120582)

(V otimes 119908)(1120582)

= 1(1120572120573)

sdot (V(0)

otimes 119908(0)) otimes 1(2120582)

119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

= 1(1120572)

sdot V(0)

otimes 1(2120573)

sdot 119908(0)

otimes 1(3120582)

119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

= 1(1120572)

sdot V(0)

otimes 1(2120573)

11015840

(1120573)sdot 119908(0)

otimes 11015840

(2120582)119908(1120582)

times 120593120573minus1 (V(1120573120582120573

minus1))

= 1(1120572)

sdot V(0)

otimes 1(2120573)

sdot 119908(0)

otimes 119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

= V(0)

otimes 119908(0)

otimes 119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

= (V otimes 119908)(0)

otimes (V otimes 119908)(1120582)

(27)

This finishes the proof

Proposition 4 Let (119881 120588119881) isin WYD120572(119867) and let 120573 isin 120587 Set

120573

119881 = 119881 as vector space with action and coaction structuresdefined by

ℎ ⊳120573V =120573

(120593120573minus1 (ℎ) sdot V) forallℎ isin 119867

120573120572120573minus1 120573V isin120573

119881

120588120573119881

120582(120573V) =

120573

(V(0)) otimes 120593120573(V(1120573minus1120582120573)

)

= V⟨0⟩

otimes V⟨1120582⟩

forall120573V isin120573

119881

(28)

Then120573

119881 isin WYD120573120572120573minus1(119867)

6 Abstract and Applied Analysis

Proof Obviously 120573119881 is a left119867120573120572120573minus1-module and conditions

(16) and (17) are straightforward Then it remains to showthat conditions (20) and (21) hold For all 120573V isin

120573

119881 we have

1(1120573120572120573

minus1)⊳ V⟨0⟩

otimes 1(2120582)

V⟨1120582⟩

=120573

(120593120573minus1 (1(1120573120572120573

minus1)) sdot V(0)) otimes1(2120582)

120593120573(V(1120573minus1120582120573)

)

=120573

(1(1120572)

sdot V(0)) otimes 120593120573(1(2120582)

) 120593120573(V(1120573minus1120582120573)

)

= V⟨0⟩

otimes V⟨1120582⟩

(29)

Next for all ℎ isin 119867120573120572120573minus1120582 120573V isin

120573

119881 we get

(ℎ(2120573120572120573

minus1)⊳120573V)⟨0⟩

otimes (ℎ(2120573120572120573

minus1)⊳120573V)⟨1120582⟩

times 120593120573120572minus1120573minus1 (ℎ(1120573120572120573

minus1120582120573120572minus1120573minus1))

=

120573

((120593120573minus1 (ℎ(2120573120572120573

minus1)) sdot V)(0)

)

otimes 120593120573((120593120573minus1 (ℎ(2120573120572120573

minus1)) sdot V)(1120573minus1120582120573)

)

120593120573120572minus1120573minus1 (ℎ(1120573120572120573

minus1120582120573120572minus1120573minus1))

=

120573

((120593120573minus1(ℎ)(2120572)

sdot V)(0)

)

otimes 120593120573((120593120573minus1(ℎ)(2120572)

sdot V)(1120573minus1120582120573)

times 120593120572minus1 (120593120573minus1(ℎ)(1120572120573minus1120582120573120572minus1)))

=120573

(120593120573minus1(ℎ)(1120572)

sdot V(0))

otimes 120593120573(120593120573minus1(ℎ)(2120573minus1120582120573)

V(1120573minus1120582120573)

)

=120573

(120593120573minus1 (ℎ(1120573120572120573

minus1)) sdot V(0))

otimes 120593120573(120593120573minus1 (ℎ(2120582)

) V(1120573minus1120582120573)

)

= ℎ(1120573120572120573

minus1)⊳120573

(V(0))

otimes 120593120573(120593120573minus1 (ℎ(2120582)

) V(1120573minus1120582120573)

)

= ℎ(1120573120572120573

minus1)⊳120573V⟨0⟩

otimes ℎ(2120582)

V⟨1120582⟩

(30)

This completes the proof of the proposition

Remark 5 Let (119881 120588119881

) isin WYD120572(119867) and let (119882 120588

119882

) isin

WYD120573(119867) then we have 119904119905119881 =

119904

(119905

119881) as an object inWYD

119904119905120572119905minus1119904minus1(119867) and 119904(119881otimes

119905120572120573119882) =

119904

119881otimes119905119904120572120573119904minus1

119904

119882 as anobject inWYD

119904120572120573119904minus1(119867)

Proposition 6 Let (119881 120588119881

) isin WYD120572(119867) (119882 120588

119882

) isin

WYD120573(119867) Set 119881119882 =

120572

119882 as an object inWYD120572120573120572minus1(119867)

Define the map

119888119881119882

119881 otimes119882 997888rarr119881

119882otimes119881

119888119881119882

(V otimes 119908) =120572

(119878120573minus1 (V(1120573minus1)) sdot 119908) otimesV

(0)

(31)

Then 119888119881119882

is 119867-linear 119867-colinear and satisfies the followingconditions

119888119881otimes119882119880

= (119888119881

119882

119880otimes id119882) ∘ (id

119881otimes 119888119882119880

)

119888119881119882otimes119880

= (id119881119882

otimes 119888119881119880

) ∘ (119888119881119882

otimes id119880)

(32)

Furthermore 119888120574119881120574119882

= 119888119881119882

for all 120574 isin 120587

Proof Firstly we need to show that 119888119881119882

is well definedIndeed we have

119888119881sdot119882

(1(1120572)

sdot V otimes 1(2120573)

sdot 119908)

=

120572

(119878120573minus1 ((1(1120572)

sdot V)(1120573minus1)) 1(2120573)

sdot 119908) otimes(1(1120572)

sdot V)(0)

=120572

(119878120573minus1 (V(1120573minus1)119878minus1

120593120572minus1 (1(1120572120573120572

minus1)))

times 119878120573minus1 (1(3120573minus1)) 1(4120573)

sdot 119908) otimes 1(2120572)

sdot V(0)

=120572

(119878120573minus1119878minus1

120593120572minus1 (1(1120572120573120572

minus1))

times 119878120573minus1 (V(1120573minus1)) 120576119904

120573(1(31)

) sdot 119908) otimes 1(2120572)

sdot V(0)

=120572

(119878120573minus1119878minus1

120593120572minus1 (1(1120572120573120572

minus1)) 119878120573minus1 (1(3120573minus1)V(1120573minus1)) sdot 119908)

otimes 1(2120572)

sdot V(0)

=120572

(119878120573minus1 (V(1120573minus1)) sdot 119908) otimes V

(0)

= 119888119881119882

(V otimes 119908)

(33)

Secondly we prove that 119888119881119882

is119867-linear For all ℎ isin 119867120572120573

we compute

119888119881119882

(ℎ sdot (V otimes 119908))

=

120572

(119878120573minus1 ((ℎ(1120572)

sdot V)(1120573minus1)) ℎ(2120573)

sdot 119908) otimes (ℎ(1120572)

sdot V)(0)

=120572

(119878120573minus1 (ℎ(3120573minus1)V(1120573minus1)119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) ℎ(4120573)

sdot 119908)

otimes ℎ(2120572)

sdot V(0)

=120572

(119878120573minus1 (V(1120573minus1)119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) 120576119904

120573(ℎ(31)

) sdot 119908)

otimes ℎ(2120572)

sdot V(0)

=120572

(119878120573minus1 (V(1120573minus1)119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1)))

times 119878120573minus1 (1(2120573minus1)) sdot 119908) otimes ℎ

(2120572)1(1120572)

sdot V(0)

Abstract and Applied Analysis 7

=120572

(119878120573minus1 (1(3120573minus1)V(1120573minus1)119878minus1

120593120572minus1 (1(1120572120573120572

minus1))

times 119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) sdot 119908) otimes ℎ

(2120572)1(2120572)

sdot V(0)

=120572

(119878120573minus1 (V(1120573minus1)119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) sdot 119908)

otimes ℎ(2120572)

sdot V(0)

=120572

(120593120572minus1 (ℎ(1120572120573120572

minus1)) 119878120573minus1 (V(1120573minus1)) sdot 119908) otimes ℎ

(2120572)sdot V(0)

= ℎ(1120572120573120572

minus1)⊳120572

(119878120573minus1 (V(1120573minus1)) sdot 119908) otimes ℎ

(2120572)sdot V

= ℎ sdot 119888119881119882

(V otimes 119908)

(34)as required

Finally we check that 119888119881119882

is satisfing the 119867-colinearcondition In fact

120588

119881

119882otimes119881

120582∘ 119888119881119882

(V otimes 119908)

=

120572

((119878120573minus1 (V(1120573minus1)) sdot 119908)

(0)

) otimes V(0)(0)

otimes V(0)(1120582)

(119878120573minus1 (V(1120573minus1)) sdot 119908)

(1120582)

=

120572

(119878120573minus1(V(1120573minus1))(2120573)

sdot 119908(0)) otimes V

(0)(0)

otimes V(0)(1120582)

119878120573minus1(V(1120573minus1))(3120582)

119908(1120582)

119878minus1

120593120573minus1

(119878120573minus1(V(1120573minus1))(1120573120582minus1120573minus1)

)

=120572

(119878120573minus1 (V(3120573minus1)) sdot 119908(0)) otimes V(0)

otimes V(1120582)

119878120582minus1 (V(2120582minus1))

times 119908(1120582)

119878minus1

120593120573minus1119878120573120582120573minus1 (V(4120573120582120573

minus1))

=120572

(119878120573minus1 (V(2120573minus1)) sdot 119908(0)) otimes V(0)

otimes 120576119905

120582(V(11)

)

times 119908(1120582)

120593120573minus1 (V(3120573120582120573

minus1))

=120572

(119878120573minus1 (1(2120573minus1)V(1120573minus1)) sdot 119908(0)) otimes V(0)

otimes 119878120582minus1 (1(1120582minus1)) 119908(1120582)

120593120573minus1 (V(2120573120582120573

minus1))

=120572

(119878120573minus1 (1(2120573minus1)V(1120573minus1)) sdot 119908(0)) otimes V(0)

otimes 119878120582minus1 (1(1120582minus1)) 119908(1120582)

120593120573minus1119878minus1

119878120573120582120573minus1 (1(3120573120582120573

minus1)) 120593120573minus1 (V(2120573120582120573

minus1))

=120572

(119878120573minus1 (V(1120573minus1)) 119878120573minus1(1)(2120573)

sdot 119908(0))

otimes V(0)

otimes 119878120573minus1(1)(3120582)

119908(1120582)

120593120573minus1119878minus1

(119878120573minus1(1)(1120573120582minus1120573minus1)) 120593120573minus1 (V(2120573120582120573

minus1))

=120572

(119878120573minus1 (V(1120573minus1)) sdot 119908(0)) otimes V(0)

otimes 119908(1120582)

120593120573minus1 (V(2120573120582120573

minus1))

= (119888119881119882

otimes id119867120582

) (V(0)

otimes 119908(0)

otimes 119908(1120582)

120593120573minus1

times (V(1120573120582120573

minus1)))

= (119888119881119882

otimes id119867120582

) ∘ 120588119881otimes119882

120582(V otimes 119908)

(35)

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

Lemma 7 The map 119888119881119882

defined by (31) is bijective withinverse

119888minus1

119881119882119881

119882otimes119881 997888rarr 119881 otimes119882

119888minus1

119881119882(120572

119908otimes V) = V(0)

otimes V(1120573)

sdot 119908

(36)

for all V isin 119881120572

119908 isin119881

119882

Proof Firstly we prove 119888minus1

119881119882∘ 119888119881119882

= id119881otimes119882

For all V isin 119881119908 isin 119882 we have

119888minus1

119881119882∘ 119888119881119882

(V otimes 119908)

= V(0)(0)

otimes V(0)(1120573)

119878120573minus1 (V(1120573minus1)) sdot 119908

= V(0)

otimes 120576119905

120573(V(11)

) sdot 119908 = 1(1120572)

sdot V(0)

otimes 1(2120573)

120576119905

120573(V(11)

) sdot 119908

= 1(1120572)

119878minus1

120576119905

120572minus1 (V(11)

) sdot V(0)

otimes 1(2120573)

sdot 119908

= 120576 (11015840

(21)V(11)

) 1(1120572)

11015840

(1120572)sdot V(0)

otimes 1(2120573)

sdot 119908

= 1(1120572)

sdot V otimes 1(2120573)

sdot 119908 = V otimes 119908

(37)

Secondly we check 119888119881119882

∘ 119888minus1

119881119882= id 119881

119882otimes119881as follows

119888119881119882

∘ 119888minus1

119881119882(120572

119908otimesV)

=120572

(119878120573minus1 (V(0)(1120573

minus1)) V(1120573)

sdot 119908) otimesV(0)(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) 119878120573minus1 (V(0)(1120573

minus1)) V(1120573)

sdot 119908)

otimes 1(2120572)

sdot V(0)(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) 120576119904

120573(V(11)

) sdot 119908) otimes1(2120572)

sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)120576119904

120572120573120572minus1120593120572(V(11)

)) sdot 119908) otimes1(2120572)

sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes1

(2120572)

times 119878120572minus1120576119904

120572(120593120572(V(11)

)) sdot V(0)

8 Abstract and Applied Analysis

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes 1

(2120572)120576119905

120572120593120572119878 (V(11)

) sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes1

(2120572)

times 120576 (120593120572minus1 (11015840

(11)) 119878 (V(11)

)) 11015840

(2120572)sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes 1

(2120572)120576119878 (V(11)

) sdot V(0)

= 1(1120572120573120572

minus1)⊳120572

119908otimes1(2120572)

sdot V

=120572

119908otimes V(38)

This completes the proof

Define WYD(119867) = ∐120572isin120587

WYD120572(119867) the disjoint

union of the categories WYD120572(119867) for all 120572 isin 120587 If we

endow WYD(119867) with tensor product as in Proposition 3then WYD(119867) becomes a monoidal category The unit is119867119879

= 119867119905

120572= 120576119905

120572(1198671)120572isin120587

The group homomorphism 120595 119866 rarr aut(WYD(119867))

120573 rarr 120595120573is given on components as

120595120573 WYD

120572(119867) 997888rarr WYD

120573120572120573minus1 (119867) (39)

where the functor 120595120573acts as follows given a morphism 119891

(119881 120588119881

) rarr (119882 120588119882

) for any V isin 119881 we set ( 120573119891)( 120573V) =

120573

(119891(V))The braiding in WYD(119867) is given by the family 119888

119881119882

as shown in Proposition 6 Then we have the followingtheorem

Theorem 8 For a weak crossed Hopf group coalgebra 119867WYD(119867) is a braided 119879-category over group 120587

Example 9 Let 119867 be a weak Hopf algebra 119866 a finite groupand 119896(119866) the dual Hopf algebra of the group algebra 119896119866

Then have the weak Hopf group coalgebra 119896(119866) otimes119867 themultiplication in 119896(119866) otimes 119867 is given by

(119901120572otimes ℎ) (119901

120573otimes 119892) = 119901

120572119901120573otimes ℎ119892 (40)

for all 119901120572 119901120573

isin 119896(119866) ℎ 119892 isin 119867 and the comultiplicationcounit and antipode are given by

Δ119906V (119901120572 otimes ℎ) = sum

119906V=120572(119901119906otimes ℎ1) otimes (119901V otimes ℎ

2)

120576 (119901120572otimes ℎ) = 120575

1205721120576 (ℎ)

119878 (119901120572otimes ℎ) = 119901

120572minus1 otimes 119878 (ℎ)

(41)

Moreover 119896(119866) otimes 119867 is a weak crossed Hopf groupcoalgebra with the following crossing

Φ120573(119901120572otimes ℎ) = 119901

120573minus1120572120573

otimes ℎ (42)

ByTheorem 8WYD(119896(119866)otimes119867) is a braided 119879-category

4 Braided 119879-Categories over Weak LongDimodule Categories

In this section we introduce the notion of a (left-right)weak 120572-Long dimodule over a weak crossed Hopf groupcoalgebra119867 and prove that the category

119867WL119867 is a braided

119879-subcategory of Yetter-Drinfelrsquod category WYD(119867 otimes 119867)

when 119867 is a quasitriangular and coquasitriangular weakcrossed Hopf group coalgebra

Definition 10 Let119867 be a weak crossed Hopf group coalgebraover 120587 For a fixed element 120572 isin 120587 a (left-right) weak 120572-Longdimodule is a couple 119881 = (119881 120588

119881

= 120588119881

120582120582isin120587

) where 119881 is aleft 119867

120572-module and for any 120582 isin 120587 120588119881

120582 119881 rarr 119881 otimes 119867

120582is

a 119896-linear morphism such that

(1) 119881 is coassociative in the sense that for any 1205821 1205822isin 120587

we have

(id119881otimes Δ1205821 1205822

) ∘ 120588119881

12058211205822

= (120588119881

1205821

otimes id1198671205822

) ∘ 120588119881

1205822

(43)

(2) 119881 is counitary in the sense that

(id119881otimes 120576) ∘ 120588

119881

1= id119881 (44)

(3) 119881 satisfies the following compatible condition

120588119881

120582(119909 sdot V) = 119909 sdot V

(0)otimes V(1120582)

(45)

where 119909 isin 119867120572and V isin 119881

Now we can form the category119867WL119867120572of (left-right)

weak 120572-Long dimodules where the composition of mor-phisms of weak 120572-Long dimodules is the standard compo-sition of the underlying linear maps

Let119867WL119867 = ∐

120572isin120587 119867WL119867120572 the disjoint union of the

categories119867WL119867120572for all 120572 isin 120587

Proposition 11 Thecategory119867WL119867 is amonoidal category

Moreover for any 120572 120573 isin 119866 let 119881 isin119867WL119867120572

and let119882isin119867WL119867120573 Set

119881otimes119882 = V otimes 119908 isin 119881 otimes119882 | V otimes 119908

= 1(1120572)

sdot V otimes 1(2120573)

sdot 119908

= 120576 (119908(11)

120593120573minus1 (V(11)

)) V(0)

otimes 119908(0)

(46)

Then 119881 otimes119882isin119867WL119867120572with the following structures

119909 sdot (V otimes 119908) = 119909(1120572)

sdot V otimes 119909(2120573)

sdot 119908

120588119881otimes119882

120582(V otimes 119908) = V

(0)otimes 119908(0)

otimes 119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

(47)

for all 119909 isin 119867120572120573 V otimes 119908 isin 119881otimes119882

Proof It is straightforward

Abstract and Applied Analysis 9

Let (119867 120590 119877) be a coquasitriangular and quasitriangularweak crossed Hopf group coalgebra with crossing 120593 Define afamily of vector spaces 119867 otimes 119867 = (119867 otimes 119867)

120572= 1198671otimes 119867120572120572isin120587

where the 119867 on the left we consider its coquasitriangularstructure and for the right one we consider its quasitriangularstructureThen119867otimes119867 is aweak crossedHopf group coalgebrawith the natural tensor product and the crossing Φ = id otimes

120593120572120572isin120587

Theorem 12 Let (119867 120590 119877) be a weak crossed Hopf groupcoalgebra with coquasitriangular structure 120590 and quasitrian-gular structure 119877 Then the category

119867WL119867 is a braided

119879-subcategory of Yetter-Drinfelrsquod category WYD(119867 otimes 119867)

under the following action and coaction given by

120575119881

120582(V) = 119886

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1) = V

[0]otimes V[1120582]

(ℎ otimes 119909) ⊳120572V = 120590 (ℎ V

(11)) 119909 sdot V(0)

(48)

where ℎ otimes 119909 isin (119867 otimes 119867)120572 ℎ isin 119867

1 119909 isin 119867

120572 V isin 119881 and

119881isin119867WL119867120572

The braiding on119867WL119867 120591

119881119882 119881 otimes 119882 rarr

119881

119882otimes119881 isgiven by

120591119881119882

(V otimes 119908) = 120590 (119878 (V(11)

) 119908(11)

)120572

(119887120573sdot 119908(0)) otimes 119886120572sdot V(0)

(49)

for all 119881isin119867WL119867120572119882isin

119867WL119867120573

Proof Obviously119881 is a left (119867otimes119867)120572-moduleThen we show

that119881 satisfies the conditions inDefinition 1 First we need tocheck that119881 is coassociative In fact for all V isin 119881 isin

119867WL119867120572

and 1205821 1205822isin 120587

(id119881otimes Δ1205821 1205822

) ∘ 120575119881

12058211205822(V)

= 119886120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1

2120582minus1

1(2120582minus1

1))

otimes V(21)

otimes 119878minus1

(119887120582minus1

2120582minus1

1(1120582minus1

2))

= 1198861205721198861015840

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1

1

)

otimes V(21)

otimes 119878minus1

(1198871015840

120582minus1

2

)

= 119886120572sdot (1198861015840

120572sdot V(0))(0)

otimes (1198861015840

120572sdot V(0))(11)

otimes 119878minus1

(119887120582minus1

1

) otimes V(11)

otimes 119878minus1

(1198871015840

120582minus1

2

)

= (120575119881

1205821

otimes id(119867otimes119867)1205822

) (1198861015840

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(1198871015840

120582minus1

2

))

= (120575119881

1205821

otimes id(119867otimes119867)1205822

) ∘ 120575119881

1205822(V)

(50)

Next one directly shows that counitary condition (17) holdsas follows

(id119881otimes 120576) ∘ 120575

119881

1(V) = 119886

120572sdot V(0)120576 (119898(11)

) 120576119878minus1

(1198871)

= 119886120572sdot V120576 (119887

1) = 1120572sdot V = V

(51)

Then we have to prove that crossed condition (18) issatisfied For all ℎ isin 119867

1 119909 isin 119867

120572 and V isin 119881isin

119867WL119867120572 we

have(ℎ otimes 119909)

(2120572)sdot V[0]

otimes (ℎ otimes 119909)(3120582)

times V[1120582]

119878minus1

Φ120572minus1 ((ℎ otimes 119909)

(1120572120582minus1120572minus1))

= 120590 (ℎ(21)

(119886120572sdot V(0))(11)

) 119909(2120572)

sdot (119886120572sdot V(0))(0)

otimes ℎ(31)

V(11)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(119887120582minus1) 119878minus1

120595120572minus1 (119909(1120572120582minus1120572minus1))

= 120590 (ℎ(21)

V(11)

) 119909(2120572)

119886120572sdot V(0)

otimes ℎ(31)

V(21)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(119887120582minus1)

119878minus1

120595120572minus1 (119909(1120572120582minus1120572minus1))

= 120590 (ℎ(31)

V(21)

) 119909(2120572)

119886120572sdot V(0)

otimes V(11)

ℎ(21)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(120595120572minus1 (119909(1120572120582minus1120572minus1)) 119887120582minus1)

= 120590 (ℎ(21)

V(21)

) 119886120572119909(1120572)

sdot V(0)

otimes V(11)

119878minus1

120576119905

1(ℎ(11)

)

otimes 119909(3120582)

119878minus1

(119887120582minus1 (119909(2120582minus1)))

= 120590 (ℎ(21)

V(21)

) 119886120572119909(1120572)

sdot V(0)

otimes V(11)

119878minus1

120576119905

1(ℎ(11)

)

otimes 119878minus1

120576119905

120582minus1 (119909(21)

) 119878minus1

(119887120582minus1)

= 120590 (11015840

(21)ℎ V(21)

) 1198861205721(1120572)

119909 sdot V(0)

otimes V(11)

11015840

(11)

otimes 119878minus1

(1(2120582minus1)) 119878minus1

(119887120582minus1)

= 120590 (11015840

(21) V(21)

) 120590 (ℎ V(31)

) 1198861205721(1120572)

119909 sdot V(0)

otimes V(11)

11015840

(11)otimes 119878minus1

(119887120582minus11(2120582minus1))

= 120576 (V(21)

1(21)

) 120590 (ℎ V(31)

) 119886120572119909 sdot V(0)

otimes V(11)

1(11)

otimes 119878minus1

(119887120582minus1)

= 120590 (ℎ V(21)

) 119886120572119909 sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1)

= 120590 (ℎ V(11)

) 119886120572sdot (119909 sdot V

(0))(0)

otimes (119909 sdot V(0))(11)

otimes 119878minus1

(119887120582minus1)

= 120575119881

120582((ℎ otimes 119909) ⊳

120572V)

(52)

Finally it follows from Proposition 6 the braiding onWYD(119867 otimes 119867) that the braiding on

119867WL119867 is as the

following

120591119881119882

(V otimes 119908) =120572

(119878120573minus1 (V[1120573minus1])) ⊳120573119908otimesV[0]

= 120590 (119878 (V(11)

) 119908(11)

)120572

(119887120573sdot 119908(0)) otimes119886120572sdot V(0)

(53)

for all 119881isin119867WL119867120572119882isin119867WL119867120573 V isin 119881 and 119908 isin 119882

This completes the proof

10 Abstract and Applied Analysis

Acknowledgments

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

References

[1] V G Turaev ldquoHomotopy field theory in dimension 3 andcrossed group categoriesrdquo httparxivorgabsmath0005291

[2] P J Freyd and D N Yetter ldquoBraided compact closed categorieswith applications to low-dimensional topologyrdquo Advances inMathematics vol 77 no 2 pp 156ndash182 1989

[3] A Virelizier Algebres de Hopf graduees et fibres plats sur les3-varietes [PhD thesis] Universite Louis Pasteur StrasbourgFrance 2001

[4] A Virelizier ldquoInvolutory Hopf group-coalgebras and flat bun-dles over 3-manifoldsrdquo Fundamenta Mathematicae vol 188 pp241ndash270 2005

[5] A J Kirillov ldquoOn G-equivariant modular categoriesrdquohttparxivorgabsmath0401119

[6] S Wang ldquoNew Turaev braided group categories and groupSchur-Weyl dualityrdquo Applied Categorical Structures vol 21 no2 pp 141ndash166 2013

[7] T Yang and S Wang ldquoConstructing new braided 119879-categoriesover regular multiplier Hopf algebrasrdquo Communications inAlgebra vol 39 no 9 pp 3073ndash3089 2011

[8] M Zunino ldquoYetter-Drinfelrsquod modules for crossed structuresrdquoJournal of Pure and Applied Algebra vol 193 no 1-3 pp 313ndash343 2004

[9] A Van Daele ldquoMultiplier Hopf algebrasrdquo Transactions of theAmerican Mathematical Society vol 342 no 2 pp 917ndash9321994

[10] A Van Daele and S H Wang ldquoWeak multiplier Hopf algebrasI The main theoryrdquo Journal fur die reine und angewandteMathematik 2013

[11] A Van Daele and S Wang ldquoNew braided crossed categoriesandDrinfelrsquod quantumdouble for weakHopf group coalgebrasrdquoCommunications in Algebra vol 36 no 6 pp 2341ndash2386 2008

[12] G Bohm F Nill and K Szlachanyi ldquoWeak Hopf algebras IIntegral theory and 119862

lowast-structurerdquo Journal of Algebra vol 221no 2 pp 385ndash438 1999

[13] X Zhou and S Wang ldquoThe duality theorem for weak Hopfalgebra (co) actionsrdquo Communications in Algebra vol 38 no12 pp 4613ndash4632 2010

[14] A Virelizier ldquoHopf group-coalgebrasrdquo Journal of Pure andApplied Algebra vol 171 no 1 pp 75ndash122 2002

[15] M E Sweedler Hopf Algebras Mathematics Lecture NoteSeries W A Benjamin New York NY USA 1969

[16] X Zhou and T Yang ldquoKegelrsquos theorem over weak Hopf groupcoalgebrasrdquo Journal of Mathematics vol 33 no 2 pp 228ndash2362013

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

Proof Obviously 120573119881 is a left119867120573120572120573minus1-module and conditions

(16) and (17) are straightforward Then it remains to showthat conditions (20) and (21) hold For all 120573V isin

120573

119881 we have

1(1120573120572120573

minus1)⊳ V⟨0⟩

otimes 1(2120582)

V⟨1120582⟩

=120573

(120593120573minus1 (1(1120573120572120573

minus1)) sdot V(0)) otimes1(2120582)

120593120573(V(1120573minus1120582120573)

)

=120573

(1(1120572)

sdot V(0)) otimes 120593120573(1(2120582)

) 120593120573(V(1120573minus1120582120573)

)

= V⟨0⟩

otimes V⟨1120582⟩

(29)

Next for all ℎ isin 119867120573120572120573minus1120582 120573V isin

120573

119881 we get

(ℎ(2120573120572120573

minus1)⊳120573V)⟨0⟩

otimes (ℎ(2120573120572120573

minus1)⊳120573V)⟨1120582⟩

times 120593120573120572minus1120573minus1 (ℎ(1120573120572120573

minus1120582120573120572minus1120573minus1))

=

120573

((120593120573minus1 (ℎ(2120573120572120573

minus1)) sdot V)(0)

)

otimes 120593120573((120593120573minus1 (ℎ(2120573120572120573

minus1)) sdot V)(1120573minus1120582120573)

)

120593120573120572minus1120573minus1 (ℎ(1120573120572120573

minus1120582120573120572minus1120573minus1))

=

120573

((120593120573minus1(ℎ)(2120572)

sdot V)(0)

)

otimes 120593120573((120593120573minus1(ℎ)(2120572)

sdot V)(1120573minus1120582120573)

times 120593120572minus1 (120593120573minus1(ℎ)(1120572120573minus1120582120573120572minus1)))

=120573

(120593120573minus1(ℎ)(1120572)

sdot V(0))

otimes 120593120573(120593120573minus1(ℎ)(2120573minus1120582120573)

V(1120573minus1120582120573)

)

=120573

(120593120573minus1 (ℎ(1120573120572120573

minus1)) sdot V(0))

otimes 120593120573(120593120573minus1 (ℎ(2120582)

) V(1120573minus1120582120573)

)

= ℎ(1120573120572120573

minus1)⊳120573

(V(0))

otimes 120593120573(120593120573minus1 (ℎ(2120582)

) V(1120573minus1120582120573)

)

= ℎ(1120573120572120573

minus1)⊳120573V⟨0⟩

otimes ℎ(2120582)

V⟨1120582⟩

(30)

This completes the proof of the proposition

Remark 5 Let (119881 120588119881

) isin WYD120572(119867) and let (119882 120588

119882

) isin

WYD120573(119867) then we have 119904119905119881 =

119904

(119905

119881) as an object inWYD

119904119905120572119905minus1119904minus1(119867) and 119904(119881otimes

119905120572120573119882) =

119904

119881otimes119905119904120572120573119904minus1

119904

119882 as anobject inWYD

119904120572120573119904minus1(119867)

Proposition 6 Let (119881 120588119881

) isin WYD120572(119867) (119882 120588

119882

) isin

WYD120573(119867) Set 119881119882 =

120572

119882 as an object inWYD120572120573120572minus1(119867)

Define the map

119888119881119882

119881 otimes119882 997888rarr119881

119882otimes119881

119888119881119882

(V otimes 119908) =120572

(119878120573minus1 (V(1120573minus1)) sdot 119908) otimesV

(0)

(31)

Then 119888119881119882

is 119867-linear 119867-colinear and satisfies the followingconditions

119888119881otimes119882119880

= (119888119881

119882

119880otimes id119882) ∘ (id

119881otimes 119888119882119880

)

119888119881119882otimes119880

= (id119881119882

otimes 119888119881119880

) ∘ (119888119881119882

otimes id119880)

(32)

Furthermore 119888120574119881120574119882

= 119888119881119882

for all 120574 isin 120587

Proof Firstly we need to show that 119888119881119882

is well definedIndeed we have

119888119881sdot119882

(1(1120572)

sdot V otimes 1(2120573)

sdot 119908)

=

120572

(119878120573minus1 ((1(1120572)

sdot V)(1120573minus1)) 1(2120573)

sdot 119908) otimes(1(1120572)

sdot V)(0)

=120572

(119878120573minus1 (V(1120573minus1)119878minus1

120593120572minus1 (1(1120572120573120572

minus1)))

times 119878120573minus1 (1(3120573minus1)) 1(4120573)

sdot 119908) otimes 1(2120572)

sdot V(0)

=120572

(119878120573minus1119878minus1

120593120572minus1 (1(1120572120573120572

minus1))

times 119878120573minus1 (V(1120573minus1)) 120576119904

120573(1(31)

) sdot 119908) otimes 1(2120572)

sdot V(0)

=120572

(119878120573minus1119878minus1

120593120572minus1 (1(1120572120573120572

minus1)) 119878120573minus1 (1(3120573minus1)V(1120573minus1)) sdot 119908)

otimes 1(2120572)

sdot V(0)

=120572

(119878120573minus1 (V(1120573minus1)) sdot 119908) otimes V

(0)

= 119888119881119882

(V otimes 119908)

(33)

Secondly we prove that 119888119881119882

is119867-linear For all ℎ isin 119867120572120573

we compute

119888119881119882

(ℎ sdot (V otimes 119908))

=

120572

(119878120573minus1 ((ℎ(1120572)

sdot V)(1120573minus1)) ℎ(2120573)

sdot 119908) otimes (ℎ(1120572)

sdot V)(0)

=120572

(119878120573minus1 (ℎ(3120573minus1)V(1120573minus1)119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) ℎ(4120573)

sdot 119908)

otimes ℎ(2120572)

sdot V(0)

=120572

(119878120573minus1 (V(1120573minus1)119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) 120576119904

120573(ℎ(31)

) sdot 119908)

otimes ℎ(2120572)

sdot V(0)

=120572

(119878120573minus1 (V(1120573minus1)119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1)))

times 119878120573minus1 (1(2120573minus1)) sdot 119908) otimes ℎ

(2120572)1(1120572)

sdot V(0)

Abstract and Applied Analysis 7

=120572

(119878120573minus1 (1(3120573minus1)V(1120573minus1)119878minus1

120593120572minus1 (1(1120572120573120572

minus1))

times 119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) sdot 119908) otimes ℎ

(2120572)1(2120572)

sdot V(0)

=120572

(119878120573minus1 (V(1120573minus1)119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) sdot 119908)

otimes ℎ(2120572)

sdot V(0)

=120572

(120593120572minus1 (ℎ(1120572120573120572

minus1)) 119878120573minus1 (V(1120573minus1)) sdot 119908) otimes ℎ

(2120572)sdot V(0)

= ℎ(1120572120573120572

minus1)⊳120572

(119878120573minus1 (V(1120573minus1)) sdot 119908) otimes ℎ

(2120572)sdot V

= ℎ sdot 119888119881119882

(V otimes 119908)

(34)as required

Finally we check that 119888119881119882

is satisfing the 119867-colinearcondition In fact

120588

119881

119882otimes119881

120582∘ 119888119881119882

(V otimes 119908)

=

120572

((119878120573minus1 (V(1120573minus1)) sdot 119908)

(0)

) otimes V(0)(0)

otimes V(0)(1120582)

(119878120573minus1 (V(1120573minus1)) sdot 119908)

(1120582)

=

120572

(119878120573minus1(V(1120573minus1))(2120573)

sdot 119908(0)) otimes V

(0)(0)

otimes V(0)(1120582)

119878120573minus1(V(1120573minus1))(3120582)

119908(1120582)

119878minus1

120593120573minus1

(119878120573minus1(V(1120573minus1))(1120573120582minus1120573minus1)

)

=120572

(119878120573minus1 (V(3120573minus1)) sdot 119908(0)) otimes V(0)

otimes V(1120582)

119878120582minus1 (V(2120582minus1))

times 119908(1120582)

119878minus1

120593120573minus1119878120573120582120573minus1 (V(4120573120582120573

minus1))

=120572

(119878120573minus1 (V(2120573minus1)) sdot 119908(0)) otimes V(0)

otimes 120576119905

120582(V(11)

)

times 119908(1120582)

120593120573minus1 (V(3120573120582120573

minus1))

=120572

(119878120573minus1 (1(2120573minus1)V(1120573minus1)) sdot 119908(0)) otimes V(0)

otimes 119878120582minus1 (1(1120582minus1)) 119908(1120582)

120593120573minus1 (V(2120573120582120573

minus1))

=120572

(119878120573minus1 (1(2120573minus1)V(1120573minus1)) sdot 119908(0)) otimes V(0)

otimes 119878120582minus1 (1(1120582minus1)) 119908(1120582)

120593120573minus1119878minus1

119878120573120582120573minus1 (1(3120573120582120573

minus1)) 120593120573minus1 (V(2120573120582120573

minus1))

=120572

(119878120573minus1 (V(1120573minus1)) 119878120573minus1(1)(2120573)

sdot 119908(0))

otimes V(0)

otimes 119878120573minus1(1)(3120582)

119908(1120582)

120593120573minus1119878minus1

(119878120573minus1(1)(1120573120582minus1120573minus1)) 120593120573minus1 (V(2120573120582120573

minus1))

=120572

(119878120573minus1 (V(1120573minus1)) sdot 119908(0)) otimes V(0)

otimes 119908(1120582)

120593120573minus1 (V(2120573120582120573

minus1))

= (119888119881119882

otimes id119867120582

) (V(0)

otimes 119908(0)

otimes 119908(1120582)

120593120573minus1

times (V(1120573120582120573

minus1)))

= (119888119881119882

otimes id119867120582

) ∘ 120588119881otimes119882

120582(V otimes 119908)

(35)

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

Lemma 7 The map 119888119881119882

defined by (31) is bijective withinverse

119888minus1

119881119882119881

119882otimes119881 997888rarr 119881 otimes119882

119888minus1

119881119882(120572

119908otimes V) = V(0)

otimes V(1120573)

sdot 119908

(36)

for all V isin 119881120572

119908 isin119881

119882

Proof Firstly we prove 119888minus1

119881119882∘ 119888119881119882

= id119881otimes119882

For all V isin 119881119908 isin 119882 we have

119888minus1

119881119882∘ 119888119881119882

(V otimes 119908)

= V(0)(0)

otimes V(0)(1120573)

119878120573minus1 (V(1120573minus1)) sdot 119908

= V(0)

otimes 120576119905

120573(V(11)

) sdot 119908 = 1(1120572)

sdot V(0)

otimes 1(2120573)

120576119905

120573(V(11)

) sdot 119908

= 1(1120572)

119878minus1

120576119905

120572minus1 (V(11)

) sdot V(0)

otimes 1(2120573)

sdot 119908

= 120576 (11015840

(21)V(11)

) 1(1120572)

11015840

(1120572)sdot V(0)

otimes 1(2120573)

sdot 119908

= 1(1120572)

sdot V otimes 1(2120573)

sdot 119908 = V otimes 119908

(37)

Secondly we check 119888119881119882

∘ 119888minus1

119881119882= id 119881

119882otimes119881as follows

119888119881119882

∘ 119888minus1

119881119882(120572

119908otimesV)

=120572

(119878120573minus1 (V(0)(1120573

minus1)) V(1120573)

sdot 119908) otimesV(0)(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) 119878120573minus1 (V(0)(1120573

minus1)) V(1120573)

sdot 119908)

otimes 1(2120572)

sdot V(0)(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) 120576119904

120573(V(11)

) sdot 119908) otimes1(2120572)

sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)120576119904

120572120573120572minus1120593120572(V(11)

)) sdot 119908) otimes1(2120572)

sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes1

(2120572)

times 119878120572minus1120576119904

120572(120593120572(V(11)

)) sdot V(0)

8 Abstract and Applied Analysis

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes 1

(2120572)120576119905

120572120593120572119878 (V(11)

) sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes1

(2120572)

times 120576 (120593120572minus1 (11015840

(11)) 119878 (V(11)

)) 11015840

(2120572)sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes 1

(2120572)120576119878 (V(11)

) sdot V(0)

= 1(1120572120573120572

minus1)⊳120572

119908otimes1(2120572)

sdot V

=120572

119908otimes V(38)

This completes the proof

Define WYD(119867) = ∐120572isin120587

WYD120572(119867) the disjoint

union of the categories WYD120572(119867) for all 120572 isin 120587 If we

endow WYD(119867) with tensor product as in Proposition 3then WYD(119867) becomes a monoidal category The unit is119867119879

= 119867119905

120572= 120576119905

120572(1198671)120572isin120587

The group homomorphism 120595 119866 rarr aut(WYD(119867))

120573 rarr 120595120573is given on components as

120595120573 WYD

120572(119867) 997888rarr WYD

120573120572120573minus1 (119867) (39)

where the functor 120595120573acts as follows given a morphism 119891

(119881 120588119881

) rarr (119882 120588119882

) for any V isin 119881 we set ( 120573119891)( 120573V) =

120573

(119891(V))The braiding in WYD(119867) is given by the family 119888

119881119882

as shown in Proposition 6 Then we have the followingtheorem

Theorem 8 For a weak crossed Hopf group coalgebra 119867WYD(119867) is a braided 119879-category over group 120587

Example 9 Let 119867 be a weak Hopf algebra 119866 a finite groupand 119896(119866) the dual Hopf algebra of the group algebra 119896119866

Then have the weak Hopf group coalgebra 119896(119866) otimes119867 themultiplication in 119896(119866) otimes 119867 is given by

(119901120572otimes ℎ) (119901

120573otimes 119892) = 119901

120572119901120573otimes ℎ119892 (40)

for all 119901120572 119901120573

isin 119896(119866) ℎ 119892 isin 119867 and the comultiplicationcounit and antipode are given by

Δ119906V (119901120572 otimes ℎ) = sum

119906V=120572(119901119906otimes ℎ1) otimes (119901V otimes ℎ

2)

120576 (119901120572otimes ℎ) = 120575

1205721120576 (ℎ)

119878 (119901120572otimes ℎ) = 119901

120572minus1 otimes 119878 (ℎ)

(41)

Moreover 119896(119866) otimes 119867 is a weak crossed Hopf groupcoalgebra with the following crossing

Φ120573(119901120572otimes ℎ) = 119901

120573minus1120572120573

otimes ℎ (42)

ByTheorem 8WYD(119896(119866)otimes119867) is a braided 119879-category

4 Braided 119879-Categories over Weak LongDimodule Categories

In this section we introduce the notion of a (left-right)weak 120572-Long dimodule over a weak crossed Hopf groupcoalgebra119867 and prove that the category

119867WL119867 is a braided

119879-subcategory of Yetter-Drinfelrsquod category WYD(119867 otimes 119867)

when 119867 is a quasitriangular and coquasitriangular weakcrossed Hopf group coalgebra

Definition 10 Let119867 be a weak crossed Hopf group coalgebraover 120587 For a fixed element 120572 isin 120587 a (left-right) weak 120572-Longdimodule is a couple 119881 = (119881 120588

119881

= 120588119881

120582120582isin120587

) where 119881 is aleft 119867

120572-module and for any 120582 isin 120587 120588119881

120582 119881 rarr 119881 otimes 119867

120582is

a 119896-linear morphism such that

(1) 119881 is coassociative in the sense that for any 1205821 1205822isin 120587

we have

(id119881otimes Δ1205821 1205822

) ∘ 120588119881

12058211205822

= (120588119881

1205821

otimes id1198671205822

) ∘ 120588119881

1205822

(43)

(2) 119881 is counitary in the sense that

(id119881otimes 120576) ∘ 120588

119881

1= id119881 (44)

(3) 119881 satisfies the following compatible condition

120588119881

120582(119909 sdot V) = 119909 sdot V

(0)otimes V(1120582)

(45)

where 119909 isin 119867120572and V isin 119881

Now we can form the category119867WL119867120572of (left-right)

weak 120572-Long dimodules where the composition of mor-phisms of weak 120572-Long dimodules is the standard compo-sition of the underlying linear maps

Let119867WL119867 = ∐

120572isin120587 119867WL119867120572 the disjoint union of the

categories119867WL119867120572for all 120572 isin 120587

Proposition 11 Thecategory119867WL119867 is amonoidal category

Moreover for any 120572 120573 isin 119866 let 119881 isin119867WL119867120572

and let119882isin119867WL119867120573 Set

119881otimes119882 = V otimes 119908 isin 119881 otimes119882 | V otimes 119908

= 1(1120572)

sdot V otimes 1(2120573)

sdot 119908

= 120576 (119908(11)

120593120573minus1 (V(11)

)) V(0)

otimes 119908(0)

(46)

Then 119881 otimes119882isin119867WL119867120572with the following structures

119909 sdot (V otimes 119908) = 119909(1120572)

sdot V otimes 119909(2120573)

sdot 119908

120588119881otimes119882

120582(V otimes 119908) = V

(0)otimes 119908(0)

otimes 119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

(47)

for all 119909 isin 119867120572120573 V otimes 119908 isin 119881otimes119882

Proof It is straightforward

Abstract and Applied Analysis 9

Let (119867 120590 119877) be a coquasitriangular and quasitriangularweak crossed Hopf group coalgebra with crossing 120593 Define afamily of vector spaces 119867 otimes 119867 = (119867 otimes 119867)

120572= 1198671otimes 119867120572120572isin120587

where the 119867 on the left we consider its coquasitriangularstructure and for the right one we consider its quasitriangularstructureThen119867otimes119867 is aweak crossedHopf group coalgebrawith the natural tensor product and the crossing Φ = id otimes

120593120572120572isin120587

Theorem 12 Let (119867 120590 119877) be a weak crossed Hopf groupcoalgebra with coquasitriangular structure 120590 and quasitrian-gular structure 119877 Then the category

119867WL119867 is a braided

119879-subcategory of Yetter-Drinfelrsquod category WYD(119867 otimes 119867)

under the following action and coaction given by

120575119881

120582(V) = 119886

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1) = V

[0]otimes V[1120582]

(ℎ otimes 119909) ⊳120572V = 120590 (ℎ V

(11)) 119909 sdot V(0)

(48)

where ℎ otimes 119909 isin (119867 otimes 119867)120572 ℎ isin 119867

1 119909 isin 119867

120572 V isin 119881 and

119881isin119867WL119867120572

The braiding on119867WL119867 120591

119881119882 119881 otimes 119882 rarr

119881

119882otimes119881 isgiven by

120591119881119882

(V otimes 119908) = 120590 (119878 (V(11)

) 119908(11)

)120572

(119887120573sdot 119908(0)) otimes 119886120572sdot V(0)

(49)

for all 119881isin119867WL119867120572119882isin

119867WL119867120573

Proof Obviously119881 is a left (119867otimes119867)120572-moduleThen we show

that119881 satisfies the conditions inDefinition 1 First we need tocheck that119881 is coassociative In fact for all V isin 119881 isin

119867WL119867120572

and 1205821 1205822isin 120587

(id119881otimes Δ1205821 1205822

) ∘ 120575119881

12058211205822(V)

= 119886120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1

2120582minus1

1(2120582minus1

1))

otimes V(21)

otimes 119878minus1

(119887120582minus1

2120582minus1

1(1120582minus1

2))

= 1198861205721198861015840

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1

1

)

otimes V(21)

otimes 119878minus1

(1198871015840

120582minus1

2

)

= 119886120572sdot (1198861015840

120572sdot V(0))(0)

otimes (1198861015840

120572sdot V(0))(11)

otimes 119878minus1

(119887120582minus1

1

) otimes V(11)

otimes 119878minus1

(1198871015840

120582minus1

2

)

= (120575119881

1205821

otimes id(119867otimes119867)1205822

) (1198861015840

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(1198871015840

120582minus1

2

))

= (120575119881

1205821

otimes id(119867otimes119867)1205822

) ∘ 120575119881

1205822(V)

(50)

Next one directly shows that counitary condition (17) holdsas follows

(id119881otimes 120576) ∘ 120575

119881

1(V) = 119886

120572sdot V(0)120576 (119898(11)

) 120576119878minus1

(1198871)

= 119886120572sdot V120576 (119887

1) = 1120572sdot V = V

(51)

Then we have to prove that crossed condition (18) issatisfied For all ℎ isin 119867

1 119909 isin 119867

120572 and V isin 119881isin

119867WL119867120572 we

have(ℎ otimes 119909)

(2120572)sdot V[0]

otimes (ℎ otimes 119909)(3120582)

times V[1120582]

119878minus1

Φ120572minus1 ((ℎ otimes 119909)

(1120572120582minus1120572minus1))

= 120590 (ℎ(21)

(119886120572sdot V(0))(11)

) 119909(2120572)

sdot (119886120572sdot V(0))(0)

otimes ℎ(31)

V(11)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(119887120582minus1) 119878minus1

120595120572minus1 (119909(1120572120582minus1120572minus1))

= 120590 (ℎ(21)

V(11)

) 119909(2120572)

119886120572sdot V(0)

otimes ℎ(31)

V(21)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(119887120582minus1)

119878minus1

120595120572minus1 (119909(1120572120582minus1120572minus1))

= 120590 (ℎ(31)

V(21)

) 119909(2120572)

119886120572sdot V(0)

otimes V(11)

ℎ(21)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(120595120572minus1 (119909(1120572120582minus1120572minus1)) 119887120582minus1)

= 120590 (ℎ(21)

V(21)

) 119886120572119909(1120572)

sdot V(0)

otimes V(11)

119878minus1

120576119905

1(ℎ(11)

)

otimes 119909(3120582)

119878minus1

(119887120582minus1 (119909(2120582minus1)))

= 120590 (ℎ(21)

V(21)

) 119886120572119909(1120572)

sdot V(0)

otimes V(11)

119878minus1

120576119905

1(ℎ(11)

)

otimes 119878minus1

120576119905

120582minus1 (119909(21)

) 119878minus1

(119887120582minus1)

= 120590 (11015840

(21)ℎ V(21)

) 1198861205721(1120572)

119909 sdot V(0)

otimes V(11)

11015840

(11)

otimes 119878minus1

(1(2120582minus1)) 119878minus1

(119887120582minus1)

= 120590 (11015840

(21) V(21)

) 120590 (ℎ V(31)

) 1198861205721(1120572)

119909 sdot V(0)

otimes V(11)

11015840

(11)otimes 119878minus1

(119887120582minus11(2120582minus1))

= 120576 (V(21)

1(21)

) 120590 (ℎ V(31)

) 119886120572119909 sdot V(0)

otimes V(11)

1(11)

otimes 119878minus1

(119887120582minus1)

= 120590 (ℎ V(21)

) 119886120572119909 sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1)

= 120590 (ℎ V(11)

) 119886120572sdot (119909 sdot V

(0))(0)

otimes (119909 sdot V(0))(11)

otimes 119878minus1

(119887120582minus1)

= 120575119881

120582((ℎ otimes 119909) ⊳

120572V)

(52)

Finally it follows from Proposition 6 the braiding onWYD(119867 otimes 119867) that the braiding on

119867WL119867 is as the

following

120591119881119882

(V otimes 119908) =120572

(119878120573minus1 (V[1120573minus1])) ⊳120573119908otimesV[0]

= 120590 (119878 (V(11)

) 119908(11)

)120572

(119887120573sdot 119908(0)) otimes119886120572sdot V(0)

(53)

for all 119881isin119867WL119867120572119882isin119867WL119867120573 V isin 119881 and 119908 isin 119882

This completes the proof

10 Abstract and Applied Analysis

Acknowledgments

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

References

[1] V G Turaev ldquoHomotopy field theory in dimension 3 andcrossed group categoriesrdquo httparxivorgabsmath0005291

[2] P J Freyd and D N Yetter ldquoBraided compact closed categorieswith applications to low-dimensional topologyrdquo Advances inMathematics vol 77 no 2 pp 156ndash182 1989

[3] A Virelizier Algebres de Hopf graduees et fibres plats sur les3-varietes [PhD thesis] Universite Louis Pasteur StrasbourgFrance 2001

[4] A Virelizier ldquoInvolutory Hopf group-coalgebras and flat bun-dles over 3-manifoldsrdquo Fundamenta Mathematicae vol 188 pp241ndash270 2005

[5] A J Kirillov ldquoOn G-equivariant modular categoriesrdquohttparxivorgabsmath0401119

[6] S Wang ldquoNew Turaev braided group categories and groupSchur-Weyl dualityrdquo Applied Categorical Structures vol 21 no2 pp 141ndash166 2013

[7] T Yang and S Wang ldquoConstructing new braided 119879-categoriesover regular multiplier Hopf algebrasrdquo Communications inAlgebra vol 39 no 9 pp 3073ndash3089 2011

[8] M Zunino ldquoYetter-Drinfelrsquod modules for crossed structuresrdquoJournal of Pure and Applied Algebra vol 193 no 1-3 pp 313ndash343 2004

[9] A Van Daele ldquoMultiplier Hopf algebrasrdquo Transactions of theAmerican Mathematical Society vol 342 no 2 pp 917ndash9321994

[10] A Van Daele and S H Wang ldquoWeak multiplier Hopf algebrasI The main theoryrdquo Journal fur die reine und angewandteMathematik 2013

[11] A Van Daele and S Wang ldquoNew braided crossed categoriesandDrinfelrsquod quantumdouble for weakHopf group coalgebrasrdquoCommunications in Algebra vol 36 no 6 pp 2341ndash2386 2008

[12] G Bohm F Nill and K Szlachanyi ldquoWeak Hopf algebras IIntegral theory and 119862

lowast-structurerdquo Journal of Algebra vol 221no 2 pp 385ndash438 1999

[13] X Zhou and S Wang ldquoThe duality theorem for weak Hopfalgebra (co) actionsrdquo Communications in Algebra vol 38 no12 pp 4613ndash4632 2010

[14] A Virelizier ldquoHopf group-coalgebrasrdquo Journal of Pure andApplied Algebra vol 171 no 1 pp 75ndash122 2002

[15] M E Sweedler Hopf Algebras Mathematics Lecture NoteSeries W A Benjamin New York NY USA 1969

[16] X Zhou and T Yang ldquoKegelrsquos theorem over weak Hopf groupcoalgebrasrdquo Journal of Mathematics vol 33 no 2 pp 228ndash2362013

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Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

=120572

(119878120573minus1 (1(3120573minus1)V(1120573minus1)119878minus1

120593120572minus1 (1(1120572120573120572

minus1))

times 119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) sdot 119908) otimes ℎ

(2120572)1(2120572)

sdot V(0)

=120572

(119878120573minus1 (V(1120573minus1)119878minus1

120593120572minus1 (ℎ(1120572120573120572

minus1))) sdot 119908)

otimes ℎ(2120572)

sdot V(0)

=120572

(120593120572minus1 (ℎ(1120572120573120572

minus1)) 119878120573minus1 (V(1120573minus1)) sdot 119908) otimes ℎ

(2120572)sdot V(0)

= ℎ(1120572120573120572

minus1)⊳120572

(119878120573minus1 (V(1120573minus1)) sdot 119908) otimes ℎ

(2120572)sdot V

= ℎ sdot 119888119881119882

(V otimes 119908)

(34)as required

Finally we check that 119888119881119882

is satisfing the 119867-colinearcondition In fact

120588

119881

119882otimes119881

120582∘ 119888119881119882

(V otimes 119908)

=

120572

((119878120573minus1 (V(1120573minus1)) sdot 119908)

(0)

) otimes V(0)(0)

otimes V(0)(1120582)

(119878120573minus1 (V(1120573minus1)) sdot 119908)

(1120582)

=

120572

(119878120573minus1(V(1120573minus1))(2120573)

sdot 119908(0)) otimes V

(0)(0)

otimes V(0)(1120582)

119878120573minus1(V(1120573minus1))(3120582)

119908(1120582)

119878minus1

120593120573minus1

(119878120573minus1(V(1120573minus1))(1120573120582minus1120573minus1)

)

=120572

(119878120573minus1 (V(3120573minus1)) sdot 119908(0)) otimes V(0)

otimes V(1120582)

119878120582minus1 (V(2120582minus1))

times 119908(1120582)

119878minus1

120593120573minus1119878120573120582120573minus1 (V(4120573120582120573

minus1))

=120572

(119878120573minus1 (V(2120573minus1)) sdot 119908(0)) otimes V(0)

otimes 120576119905

120582(V(11)

)

times 119908(1120582)

120593120573minus1 (V(3120573120582120573

minus1))

=120572

(119878120573minus1 (1(2120573minus1)V(1120573minus1)) sdot 119908(0)) otimes V(0)

otimes 119878120582minus1 (1(1120582minus1)) 119908(1120582)

120593120573minus1 (V(2120573120582120573

minus1))

=120572

(119878120573minus1 (1(2120573minus1)V(1120573minus1)) sdot 119908(0)) otimes V(0)

otimes 119878120582minus1 (1(1120582minus1)) 119908(1120582)

120593120573minus1119878minus1

119878120573120582120573minus1 (1(3120573120582120573

minus1)) 120593120573minus1 (V(2120573120582120573

minus1))

=120572

(119878120573minus1 (V(1120573minus1)) 119878120573minus1(1)(2120573)

sdot 119908(0))

otimes V(0)

otimes 119878120573minus1(1)(3120582)

119908(1120582)

120593120573minus1119878minus1

(119878120573minus1(1)(1120573120582minus1120573minus1)) 120593120573minus1 (V(2120573120582120573

minus1))

=120572

(119878120573minus1 (V(1120573minus1)) sdot 119908(0)) otimes V(0)

otimes 119908(1120582)

120593120573minus1 (V(2120573120582120573

minus1))

= (119888119881119882

otimes id119867120582

) (V(0)

otimes 119908(0)

otimes 119908(1120582)

120593120573minus1

times (V(1120573120582120573

minus1)))

= (119888119881119882

otimes id119867120582

) ∘ 120588119881otimes119882

120582(V otimes 119908)

(35)

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

Lemma 7 The map 119888119881119882

defined by (31) is bijective withinverse

119888minus1

119881119882119881

119882otimes119881 997888rarr 119881 otimes119882

119888minus1

119881119882(120572

119908otimes V) = V(0)

otimes V(1120573)

sdot 119908

(36)

for all V isin 119881120572

119908 isin119881

119882

Proof Firstly we prove 119888minus1

119881119882∘ 119888119881119882

= id119881otimes119882

For all V isin 119881119908 isin 119882 we have

119888minus1

119881119882∘ 119888119881119882

(V otimes 119908)

= V(0)(0)

otimes V(0)(1120573)

119878120573minus1 (V(1120573minus1)) sdot 119908

= V(0)

otimes 120576119905

120573(V(11)

) sdot 119908 = 1(1120572)

sdot V(0)

otimes 1(2120573)

120576119905

120573(V(11)

) sdot 119908

= 1(1120572)

119878minus1

120576119905

120572minus1 (V(11)

) sdot V(0)

otimes 1(2120573)

sdot 119908

= 120576 (11015840

(21)V(11)

) 1(1120572)

11015840

(1120572)sdot V(0)

otimes 1(2120573)

sdot 119908

= 1(1120572)

sdot V otimes 1(2120573)

sdot 119908 = V otimes 119908

(37)

Secondly we check 119888119881119882

∘ 119888minus1

119881119882= id 119881

119882otimes119881as follows

119888119881119882

∘ 119888minus1

119881119882(120572

119908otimesV)

=120572

(119878120573minus1 (V(0)(1120573

minus1)) V(1120573)

sdot 119908) otimesV(0)(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) 119878120573minus1 (V(0)(1120573

minus1)) V(1120573)

sdot 119908)

otimes 1(2120572)

sdot V(0)(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) 120576119904

120573(V(11)

) sdot 119908) otimes1(2120572)

sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)120576119904

120572120573120572minus1120593120572(V(11)

)) sdot 119908) otimes1(2120572)

sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes1

(2120572)

times 119878120572minus1120576119904

120572(120593120572(V(11)

)) sdot V(0)

8 Abstract and Applied Analysis

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes 1

(2120572)120576119905

120572120593120572119878 (V(11)

) sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes1

(2120572)

times 120576 (120593120572minus1 (11015840

(11)) 119878 (V(11)

)) 11015840

(2120572)sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes 1

(2120572)120576119878 (V(11)

) sdot V(0)

= 1(1120572120573120572

minus1)⊳120572

119908otimes1(2120572)

sdot V

=120572

119908otimes V(38)

This completes the proof

Define WYD(119867) = ∐120572isin120587

WYD120572(119867) the disjoint

union of the categories WYD120572(119867) for all 120572 isin 120587 If we

endow WYD(119867) with tensor product as in Proposition 3then WYD(119867) becomes a monoidal category The unit is119867119879

= 119867119905

120572= 120576119905

120572(1198671)120572isin120587

The group homomorphism 120595 119866 rarr aut(WYD(119867))

120573 rarr 120595120573is given on components as

120595120573 WYD

120572(119867) 997888rarr WYD

120573120572120573minus1 (119867) (39)

where the functor 120595120573acts as follows given a morphism 119891

(119881 120588119881

) rarr (119882 120588119882

) for any V isin 119881 we set ( 120573119891)( 120573V) =

120573

(119891(V))The braiding in WYD(119867) is given by the family 119888

119881119882

as shown in Proposition 6 Then we have the followingtheorem

Theorem 8 For a weak crossed Hopf group coalgebra 119867WYD(119867) is a braided 119879-category over group 120587

Example 9 Let 119867 be a weak Hopf algebra 119866 a finite groupand 119896(119866) the dual Hopf algebra of the group algebra 119896119866

Then have the weak Hopf group coalgebra 119896(119866) otimes119867 themultiplication in 119896(119866) otimes 119867 is given by

(119901120572otimes ℎ) (119901

120573otimes 119892) = 119901

120572119901120573otimes ℎ119892 (40)

for all 119901120572 119901120573

isin 119896(119866) ℎ 119892 isin 119867 and the comultiplicationcounit and antipode are given by

Δ119906V (119901120572 otimes ℎ) = sum

119906V=120572(119901119906otimes ℎ1) otimes (119901V otimes ℎ

2)

120576 (119901120572otimes ℎ) = 120575

1205721120576 (ℎ)

119878 (119901120572otimes ℎ) = 119901

120572minus1 otimes 119878 (ℎ)

(41)

Moreover 119896(119866) otimes 119867 is a weak crossed Hopf groupcoalgebra with the following crossing

Φ120573(119901120572otimes ℎ) = 119901

120573minus1120572120573

otimes ℎ (42)

ByTheorem 8WYD(119896(119866)otimes119867) is a braided 119879-category

4 Braided 119879-Categories over Weak LongDimodule Categories

In this section we introduce the notion of a (left-right)weak 120572-Long dimodule over a weak crossed Hopf groupcoalgebra119867 and prove that the category

119867WL119867 is a braided

119879-subcategory of Yetter-Drinfelrsquod category WYD(119867 otimes 119867)

when 119867 is a quasitriangular and coquasitriangular weakcrossed Hopf group coalgebra

Definition 10 Let119867 be a weak crossed Hopf group coalgebraover 120587 For a fixed element 120572 isin 120587 a (left-right) weak 120572-Longdimodule is a couple 119881 = (119881 120588

119881

= 120588119881

120582120582isin120587

) where 119881 is aleft 119867

120572-module and for any 120582 isin 120587 120588119881

120582 119881 rarr 119881 otimes 119867

120582is

a 119896-linear morphism such that

(1) 119881 is coassociative in the sense that for any 1205821 1205822isin 120587

we have

(id119881otimes Δ1205821 1205822

) ∘ 120588119881

12058211205822

= (120588119881

1205821

otimes id1198671205822

) ∘ 120588119881

1205822

(43)

(2) 119881 is counitary in the sense that

(id119881otimes 120576) ∘ 120588

119881

1= id119881 (44)

(3) 119881 satisfies the following compatible condition

120588119881

120582(119909 sdot V) = 119909 sdot V

(0)otimes V(1120582)

(45)

where 119909 isin 119867120572and V isin 119881

Now we can form the category119867WL119867120572of (left-right)

weak 120572-Long dimodules where the composition of mor-phisms of weak 120572-Long dimodules is the standard compo-sition of the underlying linear maps

Let119867WL119867 = ∐

120572isin120587 119867WL119867120572 the disjoint union of the

categories119867WL119867120572for all 120572 isin 120587

Proposition 11 Thecategory119867WL119867 is amonoidal category

Moreover for any 120572 120573 isin 119866 let 119881 isin119867WL119867120572

and let119882isin119867WL119867120573 Set

119881otimes119882 = V otimes 119908 isin 119881 otimes119882 | V otimes 119908

= 1(1120572)

sdot V otimes 1(2120573)

sdot 119908

= 120576 (119908(11)

120593120573minus1 (V(11)

)) V(0)

otimes 119908(0)

(46)

Then 119881 otimes119882isin119867WL119867120572with the following structures

119909 sdot (V otimes 119908) = 119909(1120572)

sdot V otimes 119909(2120573)

sdot 119908

120588119881otimes119882

120582(V otimes 119908) = V

(0)otimes 119908(0)

otimes 119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

(47)

for all 119909 isin 119867120572120573 V otimes 119908 isin 119881otimes119882

Proof It is straightforward

Abstract and Applied Analysis 9

Let (119867 120590 119877) be a coquasitriangular and quasitriangularweak crossed Hopf group coalgebra with crossing 120593 Define afamily of vector spaces 119867 otimes 119867 = (119867 otimes 119867)

120572= 1198671otimes 119867120572120572isin120587

where the 119867 on the left we consider its coquasitriangularstructure and for the right one we consider its quasitriangularstructureThen119867otimes119867 is aweak crossedHopf group coalgebrawith the natural tensor product and the crossing Φ = id otimes

120593120572120572isin120587

Theorem 12 Let (119867 120590 119877) be a weak crossed Hopf groupcoalgebra with coquasitriangular structure 120590 and quasitrian-gular structure 119877 Then the category

119867WL119867 is a braided

119879-subcategory of Yetter-Drinfelrsquod category WYD(119867 otimes 119867)

under the following action and coaction given by

120575119881

120582(V) = 119886

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1) = V

[0]otimes V[1120582]

(ℎ otimes 119909) ⊳120572V = 120590 (ℎ V

(11)) 119909 sdot V(0)

(48)

where ℎ otimes 119909 isin (119867 otimes 119867)120572 ℎ isin 119867

1 119909 isin 119867

120572 V isin 119881 and

119881isin119867WL119867120572

The braiding on119867WL119867 120591

119881119882 119881 otimes 119882 rarr

119881

119882otimes119881 isgiven by

120591119881119882

(V otimes 119908) = 120590 (119878 (V(11)

) 119908(11)

)120572

(119887120573sdot 119908(0)) otimes 119886120572sdot V(0)

(49)

for all 119881isin119867WL119867120572119882isin

119867WL119867120573

Proof Obviously119881 is a left (119867otimes119867)120572-moduleThen we show

that119881 satisfies the conditions inDefinition 1 First we need tocheck that119881 is coassociative In fact for all V isin 119881 isin

119867WL119867120572

and 1205821 1205822isin 120587

(id119881otimes Δ1205821 1205822

) ∘ 120575119881

12058211205822(V)

= 119886120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1

2120582minus1

1(2120582minus1

1))

otimes V(21)

otimes 119878minus1

(119887120582minus1

2120582minus1

1(1120582minus1

2))

= 1198861205721198861015840

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1

1

)

otimes V(21)

otimes 119878minus1

(1198871015840

120582minus1

2

)

= 119886120572sdot (1198861015840

120572sdot V(0))(0)

otimes (1198861015840

120572sdot V(0))(11)

otimes 119878minus1

(119887120582minus1

1

) otimes V(11)

otimes 119878minus1

(1198871015840

120582minus1

2

)

= (120575119881

1205821

otimes id(119867otimes119867)1205822

) (1198861015840

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(1198871015840

120582minus1

2

))

= (120575119881

1205821

otimes id(119867otimes119867)1205822

) ∘ 120575119881

1205822(V)

(50)

Next one directly shows that counitary condition (17) holdsas follows

(id119881otimes 120576) ∘ 120575

119881

1(V) = 119886

120572sdot V(0)120576 (119898(11)

) 120576119878minus1

(1198871)

= 119886120572sdot V120576 (119887

1) = 1120572sdot V = V

(51)

Then we have to prove that crossed condition (18) issatisfied For all ℎ isin 119867

1 119909 isin 119867

120572 and V isin 119881isin

119867WL119867120572 we

have(ℎ otimes 119909)

(2120572)sdot V[0]

otimes (ℎ otimes 119909)(3120582)

times V[1120582]

119878minus1

Φ120572minus1 ((ℎ otimes 119909)

(1120572120582minus1120572minus1))

= 120590 (ℎ(21)

(119886120572sdot V(0))(11)

) 119909(2120572)

sdot (119886120572sdot V(0))(0)

otimes ℎ(31)

V(11)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(119887120582minus1) 119878minus1

120595120572minus1 (119909(1120572120582minus1120572minus1))

= 120590 (ℎ(21)

V(11)

) 119909(2120572)

119886120572sdot V(0)

otimes ℎ(31)

V(21)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(119887120582minus1)

119878minus1

120595120572minus1 (119909(1120572120582minus1120572minus1))

= 120590 (ℎ(31)

V(21)

) 119909(2120572)

119886120572sdot V(0)

otimes V(11)

ℎ(21)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(120595120572minus1 (119909(1120572120582minus1120572minus1)) 119887120582minus1)

= 120590 (ℎ(21)

V(21)

) 119886120572119909(1120572)

sdot V(0)

otimes V(11)

119878minus1

120576119905

1(ℎ(11)

)

otimes 119909(3120582)

119878minus1

(119887120582minus1 (119909(2120582minus1)))

= 120590 (ℎ(21)

V(21)

) 119886120572119909(1120572)

sdot V(0)

otimes V(11)

119878minus1

120576119905

1(ℎ(11)

)

otimes 119878minus1

120576119905

120582minus1 (119909(21)

) 119878minus1

(119887120582minus1)

= 120590 (11015840

(21)ℎ V(21)

) 1198861205721(1120572)

119909 sdot V(0)

otimes V(11)

11015840

(11)

otimes 119878minus1

(1(2120582minus1)) 119878minus1

(119887120582minus1)

= 120590 (11015840

(21) V(21)

) 120590 (ℎ V(31)

) 1198861205721(1120572)

119909 sdot V(0)

otimes V(11)

11015840

(11)otimes 119878minus1

(119887120582minus11(2120582minus1))

= 120576 (V(21)

1(21)

) 120590 (ℎ V(31)

) 119886120572119909 sdot V(0)

otimes V(11)

1(11)

otimes 119878minus1

(119887120582minus1)

= 120590 (ℎ V(21)

) 119886120572119909 sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1)

= 120590 (ℎ V(11)

) 119886120572sdot (119909 sdot V

(0))(0)

otimes (119909 sdot V(0))(11)

otimes 119878minus1

(119887120582minus1)

= 120575119881

120582((ℎ otimes 119909) ⊳

120572V)

(52)

Finally it follows from Proposition 6 the braiding onWYD(119867 otimes 119867) that the braiding on

119867WL119867 is as the

following

120591119881119882

(V otimes 119908) =120572

(119878120573minus1 (V[1120573minus1])) ⊳120573119908otimesV[0]

= 120590 (119878 (V(11)

) 119908(11)

)120572

(119887120573sdot 119908(0)) otimes119886120572sdot V(0)

(53)

for all 119881isin119867WL119867120572119882isin119867WL119867120573 V isin 119881 and 119908 isin 119882

This completes the proof

10 Abstract and Applied Analysis

Acknowledgments

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

References

[1] V G Turaev ldquoHomotopy field theory in dimension 3 andcrossed group categoriesrdquo httparxivorgabsmath0005291

[2] P J Freyd and D N Yetter ldquoBraided compact closed categorieswith applications to low-dimensional topologyrdquo Advances inMathematics vol 77 no 2 pp 156ndash182 1989

[3] A Virelizier Algebres de Hopf graduees et fibres plats sur les3-varietes [PhD thesis] Universite Louis Pasteur StrasbourgFrance 2001

[4] A Virelizier ldquoInvolutory Hopf group-coalgebras and flat bun-dles over 3-manifoldsrdquo Fundamenta Mathematicae vol 188 pp241ndash270 2005

[5] A J Kirillov ldquoOn G-equivariant modular categoriesrdquohttparxivorgabsmath0401119

[6] S Wang ldquoNew Turaev braided group categories and groupSchur-Weyl dualityrdquo Applied Categorical Structures vol 21 no2 pp 141ndash166 2013

[7] T Yang and S Wang ldquoConstructing new braided 119879-categoriesover regular multiplier Hopf algebrasrdquo Communications inAlgebra vol 39 no 9 pp 3073ndash3089 2011

[8] M Zunino ldquoYetter-Drinfelrsquod modules for crossed structuresrdquoJournal of Pure and Applied Algebra vol 193 no 1-3 pp 313ndash343 2004

[9] A Van Daele ldquoMultiplier Hopf algebrasrdquo Transactions of theAmerican Mathematical Society vol 342 no 2 pp 917ndash9321994

[10] A Van Daele and S H Wang ldquoWeak multiplier Hopf algebrasI The main theoryrdquo Journal fur die reine und angewandteMathematik 2013

[11] A Van Daele and S Wang ldquoNew braided crossed categoriesandDrinfelrsquod quantumdouble for weakHopf group coalgebrasrdquoCommunications in Algebra vol 36 no 6 pp 2341ndash2386 2008

[12] G Bohm F Nill and K Szlachanyi ldquoWeak Hopf algebras IIntegral theory and 119862

lowast-structurerdquo Journal of Algebra vol 221no 2 pp 385ndash438 1999

[13] X Zhou and S Wang ldquoThe duality theorem for weak Hopfalgebra (co) actionsrdquo Communications in Algebra vol 38 no12 pp 4613ndash4632 2010

[14] A Virelizier ldquoHopf group-coalgebrasrdquo Journal of Pure andApplied Algebra vol 171 no 1 pp 75ndash122 2002

[15] M E Sweedler Hopf Algebras Mathematics Lecture NoteSeries W A Benjamin New York NY USA 1969

[16] X Zhou and T Yang ldquoKegelrsquos theorem over weak Hopf groupcoalgebrasrdquo Journal of Mathematics vol 33 no 2 pp 228ndash2362013

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes 1

(2120572)120576119905

120572120593120572119878 (V(11)

) sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes1

(2120572)

times 120576 (120593120572minus1 (11015840

(11)) 119878 (V(11)

)) 11015840

(2120572)sdot V(0)

=120572

(120593120572minus1 (1(1120572120573120572

minus1)) sdot 119908) otimes 1

(2120572)120576119878 (V(11)

) sdot V(0)

= 1(1120572120573120572

minus1)⊳120572

119908otimes1(2120572)

sdot V

=120572

119908otimes V(38)

This completes the proof

Define WYD(119867) = ∐120572isin120587

WYD120572(119867) the disjoint

union of the categories WYD120572(119867) for all 120572 isin 120587 If we

endow WYD(119867) with tensor product as in Proposition 3then WYD(119867) becomes a monoidal category The unit is119867119879

= 119867119905

120572= 120576119905

120572(1198671)120572isin120587

The group homomorphism 120595 119866 rarr aut(WYD(119867))

120573 rarr 120595120573is given on components as

120595120573 WYD

120572(119867) 997888rarr WYD

120573120572120573minus1 (119867) (39)

where the functor 120595120573acts as follows given a morphism 119891

(119881 120588119881

) rarr (119882 120588119882

) for any V isin 119881 we set ( 120573119891)( 120573V) =

120573

(119891(V))The braiding in WYD(119867) is given by the family 119888

119881119882

as shown in Proposition 6 Then we have the followingtheorem

Theorem 8 For a weak crossed Hopf group coalgebra 119867WYD(119867) is a braided 119879-category over group 120587

Example 9 Let 119867 be a weak Hopf algebra 119866 a finite groupand 119896(119866) the dual Hopf algebra of the group algebra 119896119866

Then have the weak Hopf group coalgebra 119896(119866) otimes119867 themultiplication in 119896(119866) otimes 119867 is given by

(119901120572otimes ℎ) (119901

120573otimes 119892) = 119901

120572119901120573otimes ℎ119892 (40)

for all 119901120572 119901120573

isin 119896(119866) ℎ 119892 isin 119867 and the comultiplicationcounit and antipode are given by

Δ119906V (119901120572 otimes ℎ) = sum

119906V=120572(119901119906otimes ℎ1) otimes (119901V otimes ℎ

2)

120576 (119901120572otimes ℎ) = 120575

1205721120576 (ℎ)

119878 (119901120572otimes ℎ) = 119901

120572minus1 otimes 119878 (ℎ)

(41)

Moreover 119896(119866) otimes 119867 is a weak crossed Hopf groupcoalgebra with the following crossing

Φ120573(119901120572otimes ℎ) = 119901

120573minus1120572120573

otimes ℎ (42)

ByTheorem 8WYD(119896(119866)otimes119867) is a braided 119879-category

4 Braided 119879-Categories over Weak LongDimodule Categories

In this section we introduce the notion of a (left-right)weak 120572-Long dimodule over a weak crossed Hopf groupcoalgebra119867 and prove that the category

119867WL119867 is a braided

119879-subcategory of Yetter-Drinfelrsquod category WYD(119867 otimes 119867)

when 119867 is a quasitriangular and coquasitriangular weakcrossed Hopf group coalgebra

Definition 10 Let119867 be a weak crossed Hopf group coalgebraover 120587 For a fixed element 120572 isin 120587 a (left-right) weak 120572-Longdimodule is a couple 119881 = (119881 120588

119881

= 120588119881

120582120582isin120587

) where 119881 is aleft 119867

120572-module and for any 120582 isin 120587 120588119881

120582 119881 rarr 119881 otimes 119867

120582is

a 119896-linear morphism such that

(1) 119881 is coassociative in the sense that for any 1205821 1205822isin 120587

we have

(id119881otimes Δ1205821 1205822

) ∘ 120588119881

12058211205822

= (120588119881

1205821

otimes id1198671205822

) ∘ 120588119881

1205822

(43)

(2) 119881 is counitary in the sense that

(id119881otimes 120576) ∘ 120588

119881

1= id119881 (44)

(3) 119881 satisfies the following compatible condition

120588119881

120582(119909 sdot V) = 119909 sdot V

(0)otimes V(1120582)

(45)

where 119909 isin 119867120572and V isin 119881

Now we can form the category119867WL119867120572of (left-right)

weak 120572-Long dimodules where the composition of mor-phisms of weak 120572-Long dimodules is the standard compo-sition of the underlying linear maps

Let119867WL119867 = ∐

120572isin120587 119867WL119867120572 the disjoint union of the

categories119867WL119867120572for all 120572 isin 120587

Proposition 11 Thecategory119867WL119867 is amonoidal category

Moreover for any 120572 120573 isin 119866 let 119881 isin119867WL119867120572

and let119882isin119867WL119867120573 Set

119881otimes119882 = V otimes 119908 isin 119881 otimes119882 | V otimes 119908

= 1(1120572)

sdot V otimes 1(2120573)

sdot 119908

= 120576 (119908(11)

120593120573minus1 (V(11)

)) V(0)

otimes 119908(0)

(46)

Then 119881 otimes119882isin119867WL119867120572with the following structures

119909 sdot (V otimes 119908) = 119909(1120572)

sdot V otimes 119909(2120573)

sdot 119908

120588119881otimes119882

120582(V otimes 119908) = V

(0)otimes 119908(0)

otimes 119908(1120582)

120593120573minus1 (V(1120573120582120573

minus1))

(47)

for all 119909 isin 119867120572120573 V otimes 119908 isin 119881otimes119882

Proof It is straightforward

Abstract and Applied Analysis 9

Let (119867 120590 119877) be a coquasitriangular and quasitriangularweak crossed Hopf group coalgebra with crossing 120593 Define afamily of vector spaces 119867 otimes 119867 = (119867 otimes 119867)

120572= 1198671otimes 119867120572120572isin120587

where the 119867 on the left we consider its coquasitriangularstructure and for the right one we consider its quasitriangularstructureThen119867otimes119867 is aweak crossedHopf group coalgebrawith the natural tensor product and the crossing Φ = id otimes

120593120572120572isin120587

Theorem 12 Let (119867 120590 119877) be a weak crossed Hopf groupcoalgebra with coquasitriangular structure 120590 and quasitrian-gular structure 119877 Then the category

119867WL119867 is a braided

119879-subcategory of Yetter-Drinfelrsquod category WYD(119867 otimes 119867)

under the following action and coaction given by

120575119881

120582(V) = 119886

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1) = V

[0]otimes V[1120582]

(ℎ otimes 119909) ⊳120572V = 120590 (ℎ V

(11)) 119909 sdot V(0)

(48)

where ℎ otimes 119909 isin (119867 otimes 119867)120572 ℎ isin 119867

1 119909 isin 119867

120572 V isin 119881 and

119881isin119867WL119867120572

The braiding on119867WL119867 120591

119881119882 119881 otimes 119882 rarr

119881

119882otimes119881 isgiven by

120591119881119882

(V otimes 119908) = 120590 (119878 (V(11)

) 119908(11)

)120572

(119887120573sdot 119908(0)) otimes 119886120572sdot V(0)

(49)

for all 119881isin119867WL119867120572119882isin

119867WL119867120573

Proof Obviously119881 is a left (119867otimes119867)120572-moduleThen we show

that119881 satisfies the conditions inDefinition 1 First we need tocheck that119881 is coassociative In fact for all V isin 119881 isin

119867WL119867120572

and 1205821 1205822isin 120587

(id119881otimes Δ1205821 1205822

) ∘ 120575119881

12058211205822(V)

= 119886120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1

2120582minus1

1(2120582minus1

1))

otimes V(21)

otimes 119878minus1

(119887120582minus1

2120582minus1

1(1120582minus1

2))

= 1198861205721198861015840

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1

1

)

otimes V(21)

otimes 119878minus1

(1198871015840

120582minus1

2

)

= 119886120572sdot (1198861015840

120572sdot V(0))(0)

otimes (1198861015840

120572sdot V(0))(11)

otimes 119878minus1

(119887120582minus1

1

) otimes V(11)

otimes 119878minus1

(1198871015840

120582minus1

2

)

= (120575119881

1205821

otimes id(119867otimes119867)1205822

) (1198861015840

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(1198871015840

120582minus1

2

))

= (120575119881

1205821

otimes id(119867otimes119867)1205822

) ∘ 120575119881

1205822(V)

(50)

Next one directly shows that counitary condition (17) holdsas follows

(id119881otimes 120576) ∘ 120575

119881

1(V) = 119886

120572sdot V(0)120576 (119898(11)

) 120576119878minus1

(1198871)

= 119886120572sdot V120576 (119887

1) = 1120572sdot V = V

(51)

Then we have to prove that crossed condition (18) issatisfied For all ℎ isin 119867

1 119909 isin 119867

120572 and V isin 119881isin

119867WL119867120572 we

have(ℎ otimes 119909)

(2120572)sdot V[0]

otimes (ℎ otimes 119909)(3120582)

times V[1120582]

119878minus1

Φ120572minus1 ((ℎ otimes 119909)

(1120572120582minus1120572minus1))

= 120590 (ℎ(21)

(119886120572sdot V(0))(11)

) 119909(2120572)

sdot (119886120572sdot V(0))(0)

otimes ℎ(31)

V(11)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(119887120582minus1) 119878minus1

120595120572minus1 (119909(1120572120582minus1120572minus1))

= 120590 (ℎ(21)

V(11)

) 119909(2120572)

119886120572sdot V(0)

otimes ℎ(31)

V(21)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(119887120582minus1)

119878minus1

120595120572minus1 (119909(1120572120582minus1120572minus1))

= 120590 (ℎ(31)

V(21)

) 119909(2120572)

119886120572sdot V(0)

otimes V(11)

ℎ(21)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(120595120572minus1 (119909(1120572120582minus1120572minus1)) 119887120582minus1)

= 120590 (ℎ(21)

V(21)

) 119886120572119909(1120572)

sdot V(0)

otimes V(11)

119878minus1

120576119905

1(ℎ(11)

)

otimes 119909(3120582)

119878minus1

(119887120582minus1 (119909(2120582minus1)))

= 120590 (ℎ(21)

V(21)

) 119886120572119909(1120572)

sdot V(0)

otimes V(11)

119878minus1

120576119905

1(ℎ(11)

)

otimes 119878minus1

120576119905

120582minus1 (119909(21)

) 119878minus1

(119887120582minus1)

= 120590 (11015840

(21)ℎ V(21)

) 1198861205721(1120572)

119909 sdot V(0)

otimes V(11)

11015840

(11)

otimes 119878minus1

(1(2120582minus1)) 119878minus1

(119887120582minus1)

= 120590 (11015840

(21) V(21)

) 120590 (ℎ V(31)

) 1198861205721(1120572)

119909 sdot V(0)

otimes V(11)

11015840

(11)otimes 119878minus1

(119887120582minus11(2120582minus1))

= 120576 (V(21)

1(21)

) 120590 (ℎ V(31)

) 119886120572119909 sdot V(0)

otimes V(11)

1(11)

otimes 119878minus1

(119887120582minus1)

= 120590 (ℎ V(21)

) 119886120572119909 sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1)

= 120590 (ℎ V(11)

) 119886120572sdot (119909 sdot V

(0))(0)

otimes (119909 sdot V(0))(11)

otimes 119878minus1

(119887120582minus1)

= 120575119881

120582((ℎ otimes 119909) ⊳

120572V)

(52)

Finally it follows from Proposition 6 the braiding onWYD(119867 otimes 119867) that the braiding on

119867WL119867 is as the

following

120591119881119882

(V otimes 119908) =120572

(119878120573minus1 (V[1120573minus1])) ⊳120573119908otimesV[0]

= 120590 (119878 (V(11)

) 119908(11)

)120572

(119887120573sdot 119908(0)) otimes119886120572sdot V(0)

(53)

for all 119881isin119867WL119867120572119882isin119867WL119867120573 V isin 119881 and 119908 isin 119882

This completes the proof

10 Abstract and Applied Analysis

Acknowledgments

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

References

[1] V G Turaev ldquoHomotopy field theory in dimension 3 andcrossed group categoriesrdquo httparxivorgabsmath0005291

[2] P J Freyd and D N Yetter ldquoBraided compact closed categorieswith applications to low-dimensional topologyrdquo Advances inMathematics vol 77 no 2 pp 156ndash182 1989

[3] A Virelizier Algebres de Hopf graduees et fibres plats sur les3-varietes [PhD thesis] Universite Louis Pasteur StrasbourgFrance 2001

[4] A Virelizier ldquoInvolutory Hopf group-coalgebras and flat bun-dles over 3-manifoldsrdquo Fundamenta Mathematicae vol 188 pp241ndash270 2005

[5] A J Kirillov ldquoOn G-equivariant modular categoriesrdquohttparxivorgabsmath0401119

[6] S Wang ldquoNew Turaev braided group categories and groupSchur-Weyl dualityrdquo Applied Categorical Structures vol 21 no2 pp 141ndash166 2013

[7] T Yang and S Wang ldquoConstructing new braided 119879-categoriesover regular multiplier Hopf algebrasrdquo Communications inAlgebra vol 39 no 9 pp 3073ndash3089 2011

[8] M Zunino ldquoYetter-Drinfelrsquod modules for crossed structuresrdquoJournal of Pure and Applied Algebra vol 193 no 1-3 pp 313ndash343 2004

[9] A Van Daele ldquoMultiplier Hopf algebrasrdquo Transactions of theAmerican Mathematical Society vol 342 no 2 pp 917ndash9321994

[10] A Van Daele and S H Wang ldquoWeak multiplier Hopf algebrasI The main theoryrdquo Journal fur die reine und angewandteMathematik 2013

[11] A Van Daele and S Wang ldquoNew braided crossed categoriesandDrinfelrsquod quantumdouble for weakHopf group coalgebrasrdquoCommunications in Algebra vol 36 no 6 pp 2341ndash2386 2008

[12] G Bohm F Nill and K Szlachanyi ldquoWeak Hopf algebras IIntegral theory and 119862

lowast-structurerdquo Journal of Algebra vol 221no 2 pp 385ndash438 1999

[13] X Zhou and S Wang ldquoThe duality theorem for weak Hopfalgebra (co) actionsrdquo Communications in Algebra vol 38 no12 pp 4613ndash4632 2010

[14] A Virelizier ldquoHopf group-coalgebrasrdquo Journal of Pure andApplied Algebra vol 171 no 1 pp 75ndash122 2002

[15] M E Sweedler Hopf Algebras Mathematics Lecture NoteSeries W A Benjamin New York NY USA 1969

[16] X Zhou and T Yang ldquoKegelrsquos theorem over weak Hopf groupcoalgebrasrdquo Journal of Mathematics vol 33 no 2 pp 228ndash2362013

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

Let (119867 120590 119877) be a coquasitriangular and quasitriangularweak crossed Hopf group coalgebra with crossing 120593 Define afamily of vector spaces 119867 otimes 119867 = (119867 otimes 119867)

120572= 1198671otimes 119867120572120572isin120587

where the 119867 on the left we consider its coquasitriangularstructure and for the right one we consider its quasitriangularstructureThen119867otimes119867 is aweak crossedHopf group coalgebrawith the natural tensor product and the crossing Φ = id otimes

120593120572120572isin120587

Theorem 12 Let (119867 120590 119877) be a weak crossed Hopf groupcoalgebra with coquasitriangular structure 120590 and quasitrian-gular structure 119877 Then the category

119867WL119867 is a braided

119879-subcategory of Yetter-Drinfelrsquod category WYD(119867 otimes 119867)

under the following action and coaction given by

120575119881

120582(V) = 119886

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1) = V

[0]otimes V[1120582]

(ℎ otimes 119909) ⊳120572V = 120590 (ℎ V

(11)) 119909 sdot V(0)

(48)

where ℎ otimes 119909 isin (119867 otimes 119867)120572 ℎ isin 119867

1 119909 isin 119867

120572 V isin 119881 and

119881isin119867WL119867120572

The braiding on119867WL119867 120591

119881119882 119881 otimes 119882 rarr

119881

119882otimes119881 isgiven by

120591119881119882

(V otimes 119908) = 120590 (119878 (V(11)

) 119908(11)

)120572

(119887120573sdot 119908(0)) otimes 119886120572sdot V(0)

(49)

for all 119881isin119867WL119867120572119882isin

119867WL119867120573

Proof Obviously119881 is a left (119867otimes119867)120572-moduleThen we show

that119881 satisfies the conditions inDefinition 1 First we need tocheck that119881 is coassociative In fact for all V isin 119881 isin

119867WL119867120572

and 1205821 1205822isin 120587

(id119881otimes Δ1205821 1205822

) ∘ 120575119881

12058211205822(V)

= 119886120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1

2120582minus1

1(2120582minus1

1))

otimes V(21)

otimes 119878minus1

(119887120582minus1

2120582minus1

1(1120582minus1

2))

= 1198861205721198861015840

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1

1

)

otimes V(21)

otimes 119878minus1

(1198871015840

120582minus1

2

)

= 119886120572sdot (1198861015840

120572sdot V(0))(0)

otimes (1198861015840

120572sdot V(0))(11)

otimes 119878minus1

(119887120582minus1

1

) otimes V(11)

otimes 119878minus1

(1198871015840

120582minus1

2

)

= (120575119881

1205821

otimes id(119867otimes119867)1205822

) (1198861015840

120572sdot V(0)

otimes V(11)

otimes 119878minus1

(1198871015840

120582minus1

2

))

= (120575119881

1205821

otimes id(119867otimes119867)1205822

) ∘ 120575119881

1205822(V)

(50)

Next one directly shows that counitary condition (17) holdsas follows

(id119881otimes 120576) ∘ 120575

119881

1(V) = 119886

120572sdot V(0)120576 (119898(11)

) 120576119878minus1

(1198871)

= 119886120572sdot V120576 (119887

1) = 1120572sdot V = V

(51)

Then we have to prove that crossed condition (18) issatisfied For all ℎ isin 119867

1 119909 isin 119867

120572 and V isin 119881isin

119867WL119867120572 we

have(ℎ otimes 119909)

(2120572)sdot V[0]

otimes (ℎ otimes 119909)(3120582)

times V[1120582]

119878minus1

Φ120572minus1 ((ℎ otimes 119909)

(1120572120582minus1120572minus1))

= 120590 (ℎ(21)

(119886120572sdot V(0))(11)

) 119909(2120572)

sdot (119886120572sdot V(0))(0)

otimes ℎ(31)

V(11)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(119887120582minus1) 119878minus1

120595120572minus1 (119909(1120572120582minus1120572minus1))

= 120590 (ℎ(21)

V(11)

) 119909(2120572)

119886120572sdot V(0)

otimes ℎ(31)

V(21)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(119887120582minus1)

119878minus1

120595120572minus1 (119909(1120572120582minus1120572minus1))

= 120590 (ℎ(31)

V(21)

) 119909(2120572)

119886120572sdot V(0)

otimes V(11)

ℎ(21)

119878minus1

(ℎ(11)

) otimes 119909(3120582)

119878minus1

(120595120572minus1 (119909(1120572120582minus1120572minus1)) 119887120582minus1)

= 120590 (ℎ(21)

V(21)

) 119886120572119909(1120572)

sdot V(0)

otimes V(11)

119878minus1

120576119905

1(ℎ(11)

)

otimes 119909(3120582)

119878minus1

(119887120582minus1 (119909(2120582minus1)))

= 120590 (ℎ(21)

V(21)

) 119886120572119909(1120572)

sdot V(0)

otimes V(11)

119878minus1

120576119905

1(ℎ(11)

)

otimes 119878minus1

120576119905

120582minus1 (119909(21)

) 119878minus1

(119887120582minus1)

= 120590 (11015840

(21)ℎ V(21)

) 1198861205721(1120572)

119909 sdot V(0)

otimes V(11)

11015840

(11)

otimes 119878minus1

(1(2120582minus1)) 119878minus1

(119887120582minus1)

= 120590 (11015840

(21) V(21)

) 120590 (ℎ V(31)

) 1198861205721(1120572)

119909 sdot V(0)

otimes V(11)

11015840

(11)otimes 119878minus1

(119887120582minus11(2120582minus1))

= 120576 (V(21)

1(21)

) 120590 (ℎ V(31)

) 119886120572119909 sdot V(0)

otimes V(11)

1(11)

otimes 119878minus1

(119887120582minus1)

= 120590 (ℎ V(21)

) 119886120572119909 sdot V(0)

otimes V(11)

otimes 119878minus1

(119887120582minus1)

= 120590 (ℎ V(11)

) 119886120572sdot (119909 sdot V

(0))(0)

otimes (119909 sdot V(0))(11)

otimes 119878minus1

(119887120582minus1)

= 120575119881

120582((ℎ otimes 119909) ⊳

120572V)

(52)

Finally it follows from Proposition 6 the braiding onWYD(119867 otimes 119867) that the braiding on

119867WL119867 is as the

following

120591119881119882

(V otimes 119908) =120572

(119878120573minus1 (V[1120573minus1])) ⊳120573119908otimesV[0]

= 120590 (119878 (V(11)

) 119908(11)

)120572

(119887120573sdot 119908(0)) otimes119886120572sdot V(0)

(53)

for all 119881isin119867WL119867120572119882isin119867WL119867120573 V isin 119881 and 119908 isin 119882

This completes the proof

10 Abstract and Applied Analysis

Acknowledgments

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

References

[1] V G Turaev ldquoHomotopy field theory in dimension 3 andcrossed group categoriesrdquo httparxivorgabsmath0005291

[2] P J Freyd and D N Yetter ldquoBraided compact closed categorieswith applications to low-dimensional topologyrdquo Advances inMathematics vol 77 no 2 pp 156ndash182 1989

[3] A Virelizier Algebres de Hopf graduees et fibres plats sur les3-varietes [PhD thesis] Universite Louis Pasteur StrasbourgFrance 2001

[4] A Virelizier ldquoInvolutory Hopf group-coalgebras and flat bun-dles over 3-manifoldsrdquo Fundamenta Mathematicae vol 188 pp241ndash270 2005

[5] A J Kirillov ldquoOn G-equivariant modular categoriesrdquohttparxivorgabsmath0401119

[6] S Wang ldquoNew Turaev braided group categories and groupSchur-Weyl dualityrdquo Applied Categorical Structures vol 21 no2 pp 141ndash166 2013

[7] T Yang and S Wang ldquoConstructing new braided 119879-categoriesover regular multiplier Hopf algebrasrdquo Communications inAlgebra vol 39 no 9 pp 3073ndash3089 2011

[8] M Zunino ldquoYetter-Drinfelrsquod modules for crossed structuresrdquoJournal of Pure and Applied Algebra vol 193 no 1-3 pp 313ndash343 2004

[9] A Van Daele ldquoMultiplier Hopf algebrasrdquo Transactions of theAmerican Mathematical Society vol 342 no 2 pp 917ndash9321994

[10] A Van Daele and S H Wang ldquoWeak multiplier Hopf algebrasI The main theoryrdquo Journal fur die reine und angewandteMathematik 2013

[11] A Van Daele and S Wang ldquoNew braided crossed categoriesandDrinfelrsquod quantumdouble for weakHopf group coalgebrasrdquoCommunications in Algebra vol 36 no 6 pp 2341ndash2386 2008

[12] G Bohm F Nill and K Szlachanyi ldquoWeak Hopf algebras IIntegral theory and 119862

lowast-structurerdquo Journal of Algebra vol 221no 2 pp 385ndash438 1999

[13] X Zhou and S Wang ldquoThe duality theorem for weak Hopfalgebra (co) actionsrdquo Communications in Algebra vol 38 no12 pp 4613ndash4632 2010

[14] A Virelizier ldquoHopf group-coalgebrasrdquo Journal of Pure andApplied Algebra vol 171 no 1 pp 75ndash122 2002

[15] M E Sweedler Hopf Algebras Mathematics Lecture NoteSeries W A Benjamin New York NY USA 1969

[16] X Zhou and T Yang ldquoKegelrsquos theorem over weak Hopf groupcoalgebrasrdquo Journal of Mathematics vol 33 no 2 pp 228ndash2362013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

Acknowledgments

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

References

[1] V G Turaev ldquoHomotopy field theory in dimension 3 andcrossed group categoriesrdquo httparxivorgabsmath0005291

[2] P J Freyd and D N Yetter ldquoBraided compact closed categorieswith applications to low-dimensional topologyrdquo Advances inMathematics vol 77 no 2 pp 156ndash182 1989

[3] A Virelizier Algebres de Hopf graduees et fibres plats sur les3-varietes [PhD thesis] Universite Louis Pasteur StrasbourgFrance 2001

[4] A Virelizier ldquoInvolutory Hopf group-coalgebras and flat bun-dles over 3-manifoldsrdquo Fundamenta Mathematicae vol 188 pp241ndash270 2005

[5] A J Kirillov ldquoOn G-equivariant modular categoriesrdquohttparxivorgabsmath0401119

[6] S Wang ldquoNew Turaev braided group categories and groupSchur-Weyl dualityrdquo Applied Categorical Structures vol 21 no2 pp 141ndash166 2013

[7] T Yang and S Wang ldquoConstructing new braided 119879-categoriesover regular multiplier Hopf algebrasrdquo Communications inAlgebra vol 39 no 9 pp 3073ndash3089 2011

[8] M Zunino ldquoYetter-Drinfelrsquod modules for crossed structuresrdquoJournal of Pure and Applied Algebra vol 193 no 1-3 pp 313ndash343 2004

[9] A Van Daele ldquoMultiplier Hopf algebrasrdquo Transactions of theAmerican Mathematical Society vol 342 no 2 pp 917ndash9321994

[10] A Van Daele and S H Wang ldquoWeak multiplier Hopf algebrasI The main theoryrdquo Journal fur die reine und angewandteMathematik 2013

[11] A Van Daele and S Wang ldquoNew braided crossed categoriesandDrinfelrsquod quantumdouble for weakHopf group coalgebrasrdquoCommunications in Algebra vol 36 no 6 pp 2341ndash2386 2008

[12] G Bohm F Nill and K Szlachanyi ldquoWeak Hopf algebras IIntegral theory and 119862

lowast-structurerdquo Journal of Algebra vol 221no 2 pp 385ndash438 1999

[13] X Zhou and S Wang ldquoThe duality theorem for weak Hopfalgebra (co) actionsrdquo Communications in Algebra vol 38 no12 pp 4613ndash4632 2010

[14] A Virelizier ldquoHopf group-coalgebrasrdquo Journal of Pure andApplied Algebra vol 171 no 1 pp 75ndash122 2002

[15] M E Sweedler Hopf Algebras Mathematics Lecture NoteSeries W A Benjamin New York NY USA 1969

[16] X Zhou and T Yang ldquoKegelrsquos theorem over weak Hopf groupcoalgebrasrdquo Journal of Mathematics vol 33 no 2 pp 228ndash2362013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of