Hopf Modules in the Category of Yetter-Drinfeld Modules

Keywords: Weak Hopf Algebra, Hopf Module, Fundamental Theorem

We give the Fundamental Theorem for Hopf modules in the category of Yetter-Drinfeld modules L L YD , where L is a quasitriangular weak Hopf algebra with a bijective antipode. We also show that H * has a right H-Hopf module structure in the Yetter-Drinfeld category. As an application we deduce the existence and uniqueness of right integral from it. 

Introduction

Weak Hopf algebras were introduced by G. Böhm and K. Szlachányi as a generalization of usual Hopf algebras and groupoid algebras [1] [2]. A weak Hopf algebra is a vector space that has both algebra and coalgebra structures related to each other in a certain self-dual fashion and possesses an analogue of the linearized inverse map [3]- [5]. The main difference between ordinary and weak Hopf algebras comes from the fact that the comultiplication of the latter is no longer required to preserve the unit (equivalently, the counit is not requires to be a homomorphism) and results in the existence of two canonical subalgebras playing the role of "noncommutative bases". 
Paper [6] was shown what is a weak Hopf algebra in the braided category of modules over a weak Hopf algebra. In [7] we prove a Fundamental Theorem of Hopf modules for the categorical weak Hopf algebra motivation to study quasitriangular weak Hopf algebras is the so-called biproduct construction and interpreted in the terms of braided categories. More precisely, we are interested in a specific type of quaitriangular weak Hopf algebras. 
we prove the Fundamental Theorem for Hopf modules in the category of Yetter-Drinfeld modules according to the fact that the matrix R gives rise to a natural braiding for L M and L L YD . Furthermore uniqueness of integrals for a finite dimensional weak Hopf algebra in L L YD . 

Preliminaries

Throughout this paper we use Sweedler's notation for comultiplication, writing ( ) 1 2 h h h ∆ = ⊗ . Let k be a fixed field and all weak Hopf algebras are finite dimensional. 
Definition 1. A weak Hopf algebra is a vector space L with the structure of an associative unital algebra ( ) 
, , 
L m µ with multiplication : m L L L ⊗ → and unit 1 L ∈ and a coassociative coalgebra ( ) 
, , L ε ∆ with comultiplication : L L L ∆ → ⊗ and counit : L k ε → such that 1)
The comultiplication ∆ is a (not necessarily unit-preserving) homomorphism of algebras such that 
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 1 . id ∆ ⊗ ∆ = ∆ ⊗ ⊗ ∆ = ⊗ ∆ ∆ ⊗
2) The counit satisfies the following identity 
( ) ( ) ( ) ( ) ( ) 1 2 2 1 , , , . kgl kg g l kg g l k g l L ε ε ε ε ε = = ∀ ∈
3) There is a linear map : 
L S L L → called an antipode, such that, for all l L ∈ ( ) ( ) ( ) ( )( ) ( ) 1 1 , L m id S l id l ε ⊗ ∆ = ⊗ ∆ ⊗ ( ) ( ) ( ) ( ) ( ) ( ) 1 1 , L m S id l id l ε ⊗ ∆ = ⊗ ⊗ ∆ ( ) ( ) ( ) 1 2 3 . 

L L L S l S l l S l =

The linear map defined in the above equations are called target and source counital maps and denoted by t ε and s ε 
1 2 1 2 1 1 1 1 , t L l l S l ε ε ε = = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 2 1 1 1 1 . s L l l S l ε ε ε = = respectively: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 
For all l L ∈ , we have 
( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 2 1 2 1 2 1 1 , 1 1 , t s l l l l l l ε ε ⊗ = ⊗ ⊗ = ⊗ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 2 1 2 1 2 1 1 , 1 1 . s L t L l l l S l l S l ε ε ⊗ = ⊗ ⊗ = ⊗
We will briefly recall the necessary definitions and notions on the weak Hopf algebras. 

Definition 2.

A quasitriangular weak Hopf algebra is a pair ( ) , L R where L is a weak Hopf algebra and ( )( ) ( ) 
1 1 op R L L ∈ ∆
⊗ ∆ (called the R-matrix) satisfying the following conditions: 
( ) ( ) 
op l R R l ∆ = ∆
for all l L ∈ , where op ∆ denotes the conditions apposite to ∆ , 
( )( ) 13 12 , id R R R ⊗ ∆ = ( )( ) 13 23 . id R R R ∆ ⊗ = where 12 23 1, 1 R R R R = ⊗
= ⊗ , etc. as usual, and such that there exits ( )( ) ( ) 
1 1 op R L L ∈ ∆ ⊗ ∆ with ( ) ( ) 1 , 1 . op RR RR = ∆ = ∆
where we write 
1 2 1 2 1 2 R R R r r T T = ⊗ = ⊗ = ⊗
. By [3], we can obtain the following results. Proposition 2.1. For any quasitriangular weak Hopf algebra ( ) 
, L R , we have ( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) 1 1 1 1 , 1 , 1 , 1 , , , 1. 
s s L op t t L L L L L id R id R S id id R id R S id S id R id S R R S S R R id R id R ε ε ε ε ε - - - ⊗ = ∆ ⊗ = ⊗ ∆ ⊗ = ∆ ⊗ = ⊗ ∆ ⊗ =⊗ = ⊗ = ⊗ = ⊗ =

Weak Hopf Algebras in the Yetter-Drinfeld Module Category

Let L be a quasitriangular weak Hopf algebra with a bijective antipode L S . Suppose H is a weak Hopf algebra in L M . Paper [7] show that H is also a weak Hopf algebra in L L YD with a left L-coaction via 
1 0 2 1 : , H H L H h h h R R h σ - → ⊗ ⊗ = ⊗ → 
. Bing-liang and Shuan-hong introduce the definition of Weak Hopf algebra in the braided monoidal category L L YD in [6]. Moreover they have showed that if H is a finite-dimensional weak Hopf algebra in L L YD , then its dual * H is a weak Hopf algebra in L L YD . 
Definition 3. Let ( ) , L R be a quasitriangular weak Hopf algebra. An object L L H ∈ YD is called a weak bialgebra in this category if it is both an algebra and a coalgebra satisfying the following conditions: 
1) ∆ and ε are not necessarily unit-preserving, such that 
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 2 1 1 1 2 2 2 1 1 2 2 1 2 2 1 1 2 1 2 1 1 2 2 , , 1 1 11 1 1 1 1 1 . xy x R y R x y xyz xy y z x R y R y z R R ε ε ε ε ε ∆ = → ⊗ → = = → → ′ ′ ′ ′ ∆ = ⊗ ⊗ = ⊗ → → ⊗ 2) H is a left L-module algebra and left L-module coalgebra if H is a left L-module via l x l x ⊗ →  such that ( )( ) ( ) 1 2 , 1 1, , , , t l xy l x l y l l x y H l L ε → = → → → = → ∈ ∈ ( ) ( ) ( ) ( ) ( ) 1 1 2 2 1 2 , . s l x l x l x l x x l x ε ε ∆ → = → ⊗ → → = → 3) H is a left L-comodule algebra and left L-comodule coalgebra if H is a left L-comodule via ( ) 1 0 2 1 H x x x x R R x σ - = ⊗ = ⊗ →  such that ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 0 0 2 2 1 1 2 1 2 1 2 1 1 2 2 1 1 1 2 1 2 1 2 1 2 , 1 , , . H s 
t xy x y x y R r R x r y R x R R R x R R x R x R r R x r x R x R R x R σ ε ε ε ε - - = ⊗ = ⊗ → → ⊗ → = ⊗ → ⊗ → ⊗ → = ⊗ → ⊗ → → = → 4)
Furthermore, H is called a weak Hopf algebra in l L YD if there exists an antipode : S H H → (here S is left L-linear and left L-colinear i.e., S is a morphism in the category of L L YD ) satisfying 
( ) ( ) ( ) 2 1 1 2 1 2 2 1 1 2 1 2 1 2 3 1 1 , ( ) ( )( ) ( ) ( ) ( )( ) ( ) 
, . 
x 
S x R R x S x x R x R S x x S x S x x H ε ε = → → = → → = ∀ ∈
Similar to the definition of weak Hopf algebra, we denote 
( ) ( ) 1 2 , t x x S x ε = ( ) ( ) 1 2 . s x S x x ε = If 1 x = one can obtain ( ) ( ) 1 1 1 t s ε ε = = .
According to the definitions of , t s ε ε one obtains explicit expressions for these coproducts 
( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 2 1 1 , 1 1 . t t s x x x x ε ε ε ε ∆ = ⊗ ∆ = ⊗
Paper [7] give the following results: Proposition 3.1. Suppose H is a weak Hopf algebra in L L YD . For all x H ∈ we have the identities 
( ) ( ) ( ) ( ) 1 2 1 2 1 2 1 2 1 1 , 1 1 . s t x x x S x x S x ε ε ⊗ = ⊗ ⊗ = ⊗
Since a weak Hopf algebra H in the weak Yetter-Drinfeld categories L L YD is both algebra and coalgebra, one can consider modules and comodules over H. As in the theory of Hopf algebras, an H-Hopf module is an H-module which is also an H-comodule such that these two structures are compatible (the action "commutes" with coaction): 
Definition 4. Let H be a weak Hopf algebra in L L YD . A right H-Hopf module M in L L YD is an object L L M ∈ YD such that it is both a right H-module : M M H M ϕ ⊗ → and a right H-comodule via ( ) 0 1 : , M M M M H m m m ρ ρ → ⊗ = ⊗ and the following equations hold for , , m M h H l L ∈ ∈ ∈ : 1) ( ) ( ) ( ) 2 1 0 1 1 2 , M mh m R h R m h ρ = → ⊗ → 2) ( ) ( )( ) 1 2 , l mh l m l h → =→ → 3) ( ) ( ) ( ) ( ) 2 1 1 2 2 1 1 0 1 0 1 , R R m R m R r R m r m ⊗ → ⊗ → = ⊗ → ⊗ → 4) ( ) ( )( ) 2 2 1 1 , M mh R r R m r h σ = ⊗ → → 5) ( ) ( )( ) 1 0 2 1 . M l m l m l m ρ → = → → We remark that t M H ⊗ is a right H-module by ( ) ( ) ( ) 2 1 1 2 m h x m R x R h x ⊗ = → ⊗ → and a right H-comodule ( ) ( ) 2 1 0 1 1 2 M H m h m R h R m h ρ ⊗ ⊗ = ⊗ → ⊗ → . The condition (1) means that the H-comodule structure : M M M H ρ → ⊗ is H-linear, or equivalently the H-module structure map : M M H M ϕ ⊗ → is H- colinear. Also, (4) (resp. (2)) ⇔ M ϕ is L-colinear (resp. L-linear); (3)(resp. (5)) ⇔ M ρ is L-colinear (resp. L-linear). Example 3.2. H itself is a right H-Hopf module (in L L YD ) in the natural way. If V is an object in L L YD , then so is t V H ⊗ by ( ) ( ) ( ) 1 2 l v h l v l h → ⊗ = → ⊗ → and ( ) ( ) ( ) 2 2 1 1 V H v h R r R v r h σ ⊗ ⊗ = ⊗ → ⊗ → . It
is also both a right H-module and a right H-comodule by ( ) 
v h x v hx ⊗ = ⊗ and
( ) 
1 2 V H v h v h h ρ ⊗ ⊗ = ⊗ ⊗ . One easily checks that t V H ⊗
is an right H-Hopf module. when H is a weak Hopf algebra in L M and M a right H-Hopf module in L M , we prove the Fundamental Theorem 3.3 [7]. Furthermore we will show 
( ) { } 1 2 | 1 1 coH M M m M m m ρ = ∈ = ⊗ is a L-subcomodule of M. Applying ( ) ( ) ( ) ( ) 2 1 2 1 1 2 1 1 1 1 1 t R R R R ε ⊗ → = ⊗ → = ⊗ → we obtain ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 1 (1) 
(2) 1 2 1 2 1 2 1 2 1 2 1 2 2 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 . t R R R R R ε ⊗ → ⊗ → = ⊗ → ⊗ = ⊗ → ⊗ = ⊗ → ⊗ = ⊗ → ⊗
For coH n M ∈ we do a calculation: 
( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) (
) G m P m m = . 
1 0 0 2 1 1 1 2 0 1 2 2 1 1 1 2 2 2 2 1 1 1 1 2 2 2 1 1 1 1 2 2 2 1 1 1 2 2 1 1 2 1 0 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 . n n n R R n R n R r R n r R T r R n T r R r R n r r R r R n r R R n n n - - ⊗ ⊗ = ⊗ → ⊗ → = ⊗ → ⊗ → = ⊗ → → ⊗ → = ⊗ → → ⊗ → = ⊗ → → ⊗ = ⊗ → ⊗ = ⊗ ⊗ This implies that ( ) 1 0 coH n n n L M σ - = ⊗ ∈ ⊗ . So coH L L M ∈ YD . It is clearly to prove F is a left L-colinear by the following equation ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 1 2 1 1 . F n h nh n R R h id F n h σ σ σ ⊗ = = ⊗ → = ⊗ ⊗

Fundamental Theorem for H * in L L YD

In [4] * H has the contragredient left L-module structure by 
( )( ) ( ) ( ) * , , , . L l f h f S l h l L f H h H → = → ∈ ∈ ∈
Since H is a finite-dimensional left L-comodule, * H has the transposed right L-comodule structure and so it becomes a left L-comodule via ( ) 
* * * * 1 0 2 1 : , . H H H L H f f f R R f σ σ - → ⊗ = ⊗ = ⊗ → i.e. ( ) 
( ) ( ) ( ) ( ) 0 1 0 1 1 1 1 2 , . L f h f f h S h f R h S R h H - - - - = = → ∈ Now assume that H is finite-dimensional. We will show that * H becomes a right H-Hopf module in L L YD . First * H is a right H-module by ( )( ) ( ) * , , . fh x f hx f H h H = ∈ ∈ Second, *
H is a right H-comodule using the identification 
( ) * : , H H H Hom H H θ ⊗ ≅ , ( )( ) ( )( ) 2 1 H f h x f R x R h θ ⊗ = → → as follows: ( ) ( )( ) ( ) ( ) * * * * 1 2 : , , . H H H Hom H H H H f x f S x ρ ρ → ≅ ⊗ = That is ( ) * 0 1 H f f f ρ = ⊗ means ( ) ( ) ( ) ( )( ) 1 0 2 1 1 2 0 1 1 0 1 , . f x S x f f x f f R x R f x H - = → = → → ∈ Proposition 4.1. * H is a right H-comodule by H H θ ⊗ . Proof. Now for * , f H x H ∈ ∈ , we have ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 0 0 1 0 1 2 2 1 1 0 0 1 0 1 2 2 1 0 1 2 1 1 2 2 1 1 3 2 1 2 2 1 1 0 1 1 1 2 1 2 2 2 1 1 0 1 1 1 2 0 1 1 1 2 , , , , , , 
. 
H H H H H f f f x f r R x r f R f f R x S R x R f f x R S x R S x f x S x f R x R f R f f R r x R f r f f f f x θ θ ⊗ ⊗ ⊗ ⊗ = → → → ⊗ → = → → ⊗ → = → ⊗ → = ∆ = → → ⊗ → = → → ⊗ → = ⊗ ⊗ It implies that ( ) ( ) ( ) ( ) 0 0 1 0 1 1 1 2 0 1 f f f f f f ⊗ ⊗ = ⊗ ⊗ . Accord to ( ) * L L H f ρ ∈ YD we have ( ) ( ) ( ) ( ) 0 1 0 1 1 2 1 1 f f f f → ⊗ → = ⊗ . Applying the equality ( ) ( ) ( ) 1 2 1 2 t R x R R x R ε ε ε → = → we obtain ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 
2 1 0 1 2 1 0 1 0 1 1 2 0 1 1 2 0 1 1 1 1 1 , . t L f x f R x R f f R x R f f S x f f x f f f x ε ε ε ε ε ε = → → = → → = → → = → → = Hence ( ) ( ) * H id f f ε ρ ⊗ = . Thus * H becomes a right H-comodule. Y. M. Yin 1 1 1 1 . H s f R h R f h x f R h r T x r R f T h f R h T x R f T h f h T x S h T x T h f h R x R S x R h f h R x r S x r R S f h R x S R x f h f h x θ ε → ⊗ → = → → → → = → → → → = → → → = → → → = → → → →     = ⋅ → → = ⋅ ⊗ ⋅    
Next we want to check ( ) ( )( ) 
1 2 l fh l f l h → = → → for 1 1 1 1 . L L L s L L L L L l f l h x l f l h x f S l l h x f S l l h S l x f l h S l x f S h S l x f h S l x fh S l x l fh x ε → → = → → = → → = → → = → → = → → = → → → = → = → Applying the equality ( ) ( ) ( ) 2 2 1 1 0 1 2 1 2 1 1 0 1 1 2 2 2 1 1 1 0 1 2 1 2 1 1 1 2 1 2 1 2 2 1 1 1 1 2 3 1 2 2 1 1 1 2 2 1 2 1 1 2 2 1 2 1 1 2 2 1 1 0 1 1 1 . H t L L H R r R f r f x R f S R r x r R f R f r S R x R r f R f S R x R S S R x R f S R x R S R S x R f S R x R S x R f S R S x S S x R R f x S x R R f R f x θ ε θ ⊗ → ⊗ → = → → = → → →   = → → →     = → → = → → = → → = → = ⊗ → ⊗ → It implies that ( ) ( ) 2 1 1 2 2 1 1 * 0 1 0 1 , R R f R f R r R f r f f H ⊗ → ⊗ → = ⊗ → ⊗ → ∈ . Using the equality ( )( ) 1 2 l fh l f l h → = → → we compute ( )( ) ( ) ( ) ( ) ( ) 1 1 0 0 2 2 1 1 2 1 1 0 . f h f h R r R f r h R R fh fh fh fh σ -- - ⊗ = ⊗ → → = ⊗ → = ⊗ = Finally we show that ( ) * 1 0 2 1 H l f l f l f ρ → = → ⊗ → . Since for 1 1 1 1 H L L L L L L L L L L t L L L l f l f x l f R x R l f f S l R x R l f f R S l x l R f R S l x l R f f S l x l S S l x f S l x l S S l x f S l x l S l S x f S l x l S x f S l x S S x f S l x S x f θ ε → ⊗ → = → → → = → → = → → = → → → → = → → → = → → → = → → = → → = → → = → → → = ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )

Applications

As a consequence the space of coinvariants of the finite dimensional Hopf algebra is free of rank one. This is the case for the weak Hopf algebra in the category of the Yetter-Drinfeld modules. ( ) ( ) ( ) ( ) ( ) 
( ) ( ) H tx x φ ε = a) ( ) ( ) H tx x φ ε = , b) ( ) ( ) ( ) L l x l x φ χ φ → = , c) ( ) ( ) ( ) 1 1 2 L L x g R x S R φ φ - = → d) ( ) L l t l t χ → = , for all l L ∈ . 4)
( ) ( ) ( ) 1 2 L l h l l h θ χ θ → = → , ( ) ( ) ( ) ( ) ( ) 1 0 2 1 . L h h h g R R h σ θ θ θ θ - = ⊗ = ⊗ → Proof. 1) Since *CoH H is a right H-Hopf module in L L M ,
1 0 1 0 2 1 L L h h h g h h g R R h σ θ φ φ φ θ - - = ⊗ = ⊗ = ⊗ →



  op 




  group-like element L g in L. This implies that 


Acknowledgements

The author would like to thank the referee for many suggestions and comments, which have improved the overall presentations.


Funding

Research supported by the Project of Shandong Province Higher Educational Science and Technology Program (J12LI07) and the Project of National Natural Science Foundation of China (51078225).


Proof. Now we prove that * H is a right H-Hopf module. First we will show that ( ) ( ) ( )
Yanmin M Yin
Hopf Modules in the Category of Yetter-Drinfeld Modules