Rudiment of weak Doi-Hopf π-modules *

144 Jia et al / J Zhejiang Univ SI 2005 6(Suppl I:144-154 Journal of Zhejiang University SIEE ISS 1009-3095 http://wwwzjueducn/jzus E-mail: jzus@zjueducn Rudiment of weak Doi-Hopf π-modules * JI Ling ( 贾玲 LI Fang ( 李方 (Department of athematics Zhejiang University Hangzhou 310027 hina E-mail: jialing471@126com; fangli@zjueducn Received ar 21 2005; revision accepted June 24 2005 bstract: The notion of weak Doi-Hopf π-datum and weak Doi-Hopf π-module are given as generalizations of an ordinary weak Doi-Hopf datum and weak Doi-Hopf module introduced in (Böhm 2000 also as a generalization of a Doi-Hopf π-module introduced in (Wang 2004 Then we also show that the functor forgetting action or coaction has an adjoint Furthermore we explain how the notion of weak Doi-Hopf π-datum is related to weak smash product This paper presents our preliminary results on weak Doi-Hopf group modules ey words: Weak semi-hopf π-coalgebra Weak Doi-Hopf π-modules doi:101631/jzus2005s0144 Document code: L number: O1533 ITRODUTIO Turaev recently introduced the notion of group coalgebra as a generalization of an ordinary coalgebra for the study of Homotopy quantum field theory aturally as a generalization of Hopf algebra the notion of Hopf group coalgebra was defined and used to construct Hennings-like and uperberg-like invariants of principal π-bundles over link complements and over 3-manifolds It has been shown that some theories of Hopf algebras can be extended to the setting of Hopf group-coalgebras which one can find in (Virelizier 2002 One of the attractive aspects of weak Doi-Hopf module (Böhm 2000; Böhm et al 1999 is that many notions of modules studied appear as its special cases otivated by the ideas of weak Hopf algebras and Doi-Hopf group modules we want to generalize the results of weak Doi-Hopf modules to weak Doi-Hopf group modules orresponding author * Project supported by the Program for ew entury Excellent Talents in University (o 04-0522 the ational Science Foundation of Zhejiang Province of hina (o 102028 and the ultivation Fund of the ey Scientific and Technical Innovation Project inistry of Education of hina (o 704004 The organization of the paper is as follows: in Section 1 we recall some basic definitions and properties; in Section 2 we introduce the notion of weak Doi-Hopf π-datum and give some examples; in Section 3 we first define the notion of weak Doi-Hopf π-modules and then state the functor forgetting action or coaction has an adjoint; in Section 4 we explain how the notion of weak Doi-Hopf π-datum is related to weak smash product onventions We work over a ground field k We denote by I the unit of the group π We use the standard algebra and coalgebra notation ie is a coproduct ε is a counit m is a product and 1 is a unit The identity map V from any k-space to itself is denoted by id V We write a for any element in and [a] for an element in =/ker f where f is a k-linear map For a right -comodule we write ρ(m=m [0] m [1] For a left -comodule we write ρ(m=m [1] m [0] For a weak bialgebra B we have the notations b l =ε( 1 B b 1 B 2 and b r =ε(b1 B 2 1 B for any b B We let (H denote the category which has the finite dimensional left weak Doi-Hopf modules over the left weak Doi-Hopf datum (H as objects and morphisms being both left -colinear and right -linear as arrows

Jia et al / J Zhejiang Univ SI 2005 6(Suppl I:144-154 145 Definition 1 π-coalgebra over k is a family of ={ π of k-spaces endowed with a family k-linear maps ={ : π and a k-linear map ε: I k such that for any π (1 ( id ( id γ γ γ γ = (1 (2 ( id ε I = ( ε id I = id (2 Here we extend the Sweedler notation for comultiplication we write (c=c 1 c 2 for any π and c Remark 1 ( I II ε is a coalgebra in the usual sense of the word We say a π--coalgebra is finite dimensional if is finite dimensional for any π Let ={ ε π be a π-coalgebra and be an algebra with multiplication m and unit element 1 For any f Hom k ( and g Hom k ( we define their convolution product by f *g=m (f g Hom k ( In particular for =k the π-graded algebra onv( k= * is called dual to and is denoted by * Definition 2 weak semi-hopf π-h-coalgebra is a family of algebras {H m l π and also a π-coalgebra {H ε π satisfying the following conditions for any π (1 ( hg = ( h ( g ε(1 = 1 (3 (2 I 2 (1 εγ = 1εγ 11 1γ 1 2 1γ 2γ = 1 1 1 1 = (3 ε( xyz I I I = ε( xy I I1 ε( yi2zi = ε( xy ε( y z (5 1 γ 1γ 2 γ 2γ 11 1 21γ 1 1 γ 2γ (4 I I2 I1 I We call a weak semi-hopf π-h-coalgebra finite type if H is finite dimensional for any π Remark 2 Eqs(4 and (5 imply that the unit preserving property of and the multiplication preserving property of ε are not required Eq(4 can be regarded as a generalization of ( (1 1(1 (1= (1 (1( (1 1= 2 (1 for a weak bialgebra Obviously a semi-hopf group coalgebra is a weak semi-hopf group coalgebra and when the group π is trivial it is just an ordinary weak bialgebra Definition 3 Let ={ ε π be a π-coalgebra right π--comodule over is a family of k-spaces ={ π endowed with a family of k-linear maps {ρ : π such that the following holds: (1 ( ρ id ρ = ( id ρ γ γ λ γ for any γ π (6 (2 ( id ε ρ I = id for any π (7 THE WE DOI-HOPF GROUP DTU Definition 4 Let ={ ε π be a π-coalgebra and a k-vector space right π--comodulelike object is a couple ( { ρ where π ρ : is a k-linear map and written by ρ ( m = m((0 m((1 for any π such that the following holds: (1 ( ρ id ρ ( id = ρ for any π (8 (2 ( id ε ρ = id (9 I Similarly a left π--comodulelike object is a couple ( { ρ where ρ : is π a k-linear map and written by ρ ( m m = ((1 m ((0 for any π Such that the following holds: (1 ( id ρ ( id ρ = ρ for any π (10 (2 ( ε id ρ = id (11 I Let and be two left π--comodulelike objects k-linear map f: is called a left π-n comodulelike morphism if ρ f = ( id f ρ for any π Definition 5 Let H={H m 1 ε be a weak semi-hopf π-coalgebra and let be an algebra is

146 Jia et al / J Zhejiang Univ SI 2005 6(Suppl I:144-154 called a right π-h-comodule algebra if is a right π-h-comodulelike object ( { ρ π Such that the following holds: (1 ρ ( ab = ρ ( a ρ ( b for any π and ab (12 (2 (1 (1 ( ρ(1 1 = ( ρ (1 1 (1 (1 = ( ρ (1 (13 id Similarly a left π-h-comodule algebra is a left π-h-comodulelike object ( { ρ π such that the following holds: (1 ρ ( ab = ρ ( a ρ ( b for any π and a b (14 (2 ( (1 1 (1 ρ(1 = (1 ρ (1 ( (1 1 = ( ρ (1 (15 id Remark 3 Eq(15 implies that the unit preserving property of ρ is not required and generalizes (1 (1( ρ(1 1=(id ρ(1 for an ordinary right comodule algebra over a weak bialgebra introduced in (Böhm 2000 endowed with ρ I is an ordinary H I -comodule algebra introduced in (Böhm 2000 Definition 6 Let H={H m 1 ε be a weak semi-hopf π-h-coalgebra and ={ ε π be a π-coalgebra couple ( { ϕ π is called a left π-h-module coalgebra where ϕ : H is a k-linear map for any π if the following holds: (1 ( φ is a left H -module for any π; ( h c = h c h c (2 1 1 2 2 for any π c h H ; (16 (3 ε( h c = ε(1 I1 c ε( h1 I2 for any c I h H I (17 Similarly a couple ( { ϕ π is called a right π-h-module coalgebra where ϕ : H is a k-linear map for any π if the following holds: (1 ( φ is a right H -module for any π; (2 ( c h = c1 h1 c2 h2 for any π c h H ; (18 (3 ε( c h = ε( c 1 I2 ε(1 I1h for any c I h H I (19 Remark 4 Eq(19 implies that the multiplication preserving property of ε is not required and I endowed with ϕ I is an ordinary H I -module coalgebra introduced in (Böhm 2000 Remark 5 In contrast to the case when H is a Hopf group coalgebra the unit preserving property of {ρ and the counit preserving property of {φ are not required Example 1 Let H={H m 1 ε be a weak H semi-hopf π-h-coalgebra Then ( H{ m π is a right π-h-module coalgebra Definition 7 Let H={H m 1 ε be a weak semi-hopf π-coalgebra triple (H is called a right weak Doi-Hopf group datum or a right weak Doi-Hopf π-datum if is a left π-h-comodule algebra and is a right π-h-module coalgebra Similarly a triple (H is called a left weak Doi-Hopf group datum or a left weak Doi-Hopf π-datum if is a right π-h-comodule algebra and is a left π-h-module coalgebra We call a weak Doi-Hopf group datum (H finite dimensional if H are all of finite dimension Remark 6 If the group π is trivial then they are just the notions of weak Doi-Hopf datum introduced in (Böhm 2000 Example 2 Let H={H m 1 ε be a weak semi-hopf π-coalgebra such that H λ =H λ for a fixed element λ π and any π Let =H λ together with the { ρ : Hλ Hλ H given by ρ ( h h h = 1λ 2 for any h H λ π Let =H together with the {m π it is not hard to verify that the triple (H is a left weak Doi-Hopf group datum Similar to 134 in (Virelizier 2002 we have: Lemma 1 Let H={H m 1 ε be a finite type weak semi-hopf π-h-coalgebra Then the π-graded algebra H = H dual to H inherits a weak bialgebra structure by setting ( f = m ( f

Jia et al / J Zhejiang Univ SI 2005 6(Suppl I:144-154 147 ε ( f = f(1 for any π f H Theorem 1 For a finite dimensional right weak Doi-Hopf π-datum (H the triple (H * * * is a left weak Doi-Hopf datum which we call the dual of (H with the right coaction on every summand ρ : H given by ρ ( f = f[ 0] f[ 1] = x f X for any π f where ( x X is a dual basis in H and H and (h f(c=f(c h for any c h H ; the left -module structure is given by (f g(a= f{a ((1 g{a ((0 for any f g * a Similarly for a finite dimensional left weak Doi-Hopf π-datum (H the triple (H * * * is a right weak Doi-Hopf datum which we call the dual of (H with the left coaction on every summand ρ : H given by ρ (f=f (1 f (0 =X f x for any π f where ( x X is a dual basis in H and H and (f h(c=f(h c for any c h H the right -module structure is given by (g f(a=f{a ((1 g{a ((0 for any f H g * a Proof Firstly we claim that ρ (f=x f X for any π and f exactly defines a right coaction In fact ( id ( ρ f = x f X1 X2 (20 ( ρ id ρ ( f x ( x l f X X l H = (21 For any h g H c from Eqs(20 and (21 we get ( x f( c X1( h X2( g = f( c hg = f(( c h g = ( x ( x f( c X ( h X ( g l l So ( id ρ = ( ρ id ρ H It is very easy to verify ( id ε ρ id = Secondly we claim that * is a right H * -comodule algebra For any π f g ρ ( fg = x fg X (22 ρ( f ρ ( g = ( x f( xn g XXn (23 From Eqs(22 and (23 we have = = ( (( x f( xn g( c( XXn( h ( x f( c1 ( xn g( c2 Xn( h2 X( h1 = f( c h g( c h = ( fg( c h 1 1 2 2 x fg ( c X ( h So Eq(22=Eq(23 ow we show H H ( id ρε ( = ( id ( ε ( ρε ( ε H H (24 For any c I h g H I we have H H ( id ( ε ( ρ( ε ε ( c h g H H H = ( xi ε ( c( ε1 XI ( h( ε2 ( g H = ε ( c xi ε ( hg 1 XI ( h2 H l = ε ( c h2 ε ( hg 1 = ε ( c hg = ε ( c hg = ( id ρε ( ( c h g H ext we claim that * is a left H * -module In fact for any f H g H a a ( f ( g a ( a = f( a((1 g( a((0 ((1 a ( a((0 ((0 = f( a((1 1 g( a((1 2 a ( a((0 = ( fg a ( a ( ε a ( a = ε( a a ( a = a ( a ((1 I ((0 Finally we claim that * is a left H * -module coalgebra For any f H a a b ( ( f a ( a b = ( f a ( ab = f(( ab ((1 a (( ab ((0 = f ( a f ( b a ( a a ( b 1 ((1 2 ((1 1 ((0 2 ((0 = ( f a f a ( a b 1 1 2 2

148 Jia et al / J Zhejiang Univ SI 2005 6(Suppl I:144-154 For any f H a π ε ( f a = ε (1 a ε ( f1 r HI1 HI 2 = f(1 1 ε(1 ((1 1 2 I a (1 ((0 = f(1 ((1 1 ε (1 ((1 2 I a (1 ((0 = f(1 a (1 = ε ( f a ((1 ((0 THE WE DOI-HOPF ODULES Definition 8 k-space is called a right weak Doi-Hopf π-module over the right weak Doi-Hopf π-datum (H if it is a right -module and at the same time a left π--comodulelike object such that for any m a π ρ ( m a = m((0 a((0 m((1 a((1 (25 Similarly k-space is called a left weak Doi-Hopf π-module over the left weak Doi-Hopf π-datum (H if it is a left -module and at the same time a right π--comodulelike object such that for any m a π ρ ( a m = a([0] m([0] a([1] m([1] (26 Definition 9 Let and be two right weak Doi-Hopf π-modules over the right weak Doi-Hopf π-datum (H a k-linear map f: is called a right weak Doi-Hopf π-module morphism if the following holds: (1 f is a right -module map; (2 f is a π--comodulelike map ie ρ f = ( id f ρ for any π (27 Similarly a k-linear map f: is called a left weak Doi-Hopf π-module morphism if the following holds: (1 f is a left -module map; (2 f is a π--comodulelike map ie ρ f = ( f id ρ for any π (28 (π H denotes the category which has finite dimensional right weak Doi-Hopf π-modules over the right weak Doi-Hopf π-datum as objects and right weak Doi-Hopf π-module morphisms as arrows Similarly the category (π H which has the finite dimensional left weak Doi-Hopf π-modules over the left weak Doi-Hopf π-datum as objects and left weak Doi-Hopf π-module morphisms as arrows Let π denote the category which has left π--comodulelike object as objects and left comodulelike maps as morphisms Example 3 Let H={H m 1 ε be a weak semi-hopf π-coalgebra such that H λ =H λ for a fixed element λ π and π Let (HH λ H be the same as H Example 2 and let = H λ ρ = λ and we define the left action of H λ on by the left multiplication of H it is not hard to verify that is a left weak Doi-Hopf π-module Theorem 2 Let (H be a finite dimensional right weak Doi-Hopf π-datum and (H * * * its dual Then the category (π H and ( H are equivalent Proof Firstly we define a functor G: (π H ( H For any (π H we put G(= * with the right coaction of * on * given by k ρ ( f = f f = a f a [ 0] [ 1] k where {a k a k is a dual basis in and * and the left * -module structure is given by ( g u ( m = gm ( ((1 u ( m for any π g u m ((0 Obviously * is a right * -comodule ow we only show that * is a left * -module For any π g f u m ( g ( f u ( m = g( m((1 ( f u ( m((0 = gm ( ((1 f( m((0 ((1 u( m((0 ((0 = gm ( ((1 1 f( m((1 2 u( m((0 = ( gf u ( m nd we claim that the compatibility condition holds ie ρ( f u = f(0 u[ 0] f(1 u[ 1] In fact for any f u m a ( f u f u ( m a (0 [0] (1 [1]

Jia et al / J Zhejiang Univ SI 2005 6(Suppl I:144-154 149 = f(0 ( m((1 f(1 ( a((1 u[0] ( m((0 u[1] ( a((0 = f( m((1 a((1 u ( m((0 a((0 = f(( m a ((1 u (( m a ((0 = ρ( f u ( m a For any f: (π H we define G(f=f t where f t means the transposition of linear map It is t easy to prove f ( H ext we define a functor F: ( H (π H Let F(= * for any ( H and we define the right action of on * by ( u a( m = m[1] ( a u ( m[0] for any a u * * where ρ(m=m [0] m [1] and a comodulelike structure { ρ : by ρ ( u = xk u Xk for any u * * m π a where {x X is a dual basis in and Obviously * is a right -module here we only show * is a right π--comodulelike object For any g f m (( id ρ ( u ( f g m = f( x 1 g( x 2 u ( X m = f( x g( x u ( X ( X m l l = {(( id ρ ρ ( u ( f g m (( ε id ρ ( u ( m = ε ( x u ( X m = ( u ( m Ij nd we claim that the compatibility condition holds ie ρ( u a = u a((0 u ((1 a((1 ((0 Ij I ((0 ( u a u a ( m a ((0 ((1 ((1 = (( u X a ( m f( x a k ((0 k ((1 = u (( a((1 f m[0] m[1] ( a((0 = ( u a( f m = ρ( u a( m a for any u m a f For any f : ( H let F(f=f t One can easily verify that (π H and ( H are equivalent via the functors F and G Theorem 3 Let (H be a right weak Doi-Hopf π-datum Then the forgetful functor F: (π H has a right adjoint functor Proof Before defining a functor G: (π H we first set ϖ : ((1 ((0 ϖ ( c m = c 1 m 1 Then we claim ( ϖ In fact 2 ( ϖ ( c m for any π c m (29 2 = ϖ = c 1 1 m 1 1 = c 1 m 1 = ( c m ((1 ((1 ((0 ((0 ((1 ((0 ϖ So we can define G(= G( where G( =( /ker ϖ s a k-space the right action of on G( given by [ c m] a= [ c a((1 m a((0 ] for any a π m c and the left π--comodulelike structure is defined by { ρ : G( G( 1 2 ρ ([ c m] = c [ c m] for any m π c (30 Firstly we claim that the above action is well-defined In fact for any m c ab ( ϖ ( c m c m a = {( c 1 ((1 a((1 ( m 1 ((0 a((0 c a((1 m a((0 = { c a m a c a m a = 0 ((1 ((0 ((1 ((0 ([ c m] a b = ( c a b c a ( m a b = c ( ab ((1 m ( ab ((0 = [ c m] ab ((1 ((1 ((1 ((0 ((0 So G( is a right -module

150 Jia et al / J Zhejiang Univ SI 2005 6(Suppl I:144-154 Secondly we claim that the above comodulelike structure is well-defined In fact for any c m G ( ϖ ρ ϖ ( id ( ( c m c m = { c1 1((1 1 c2 1 ((1 2 1 ((1 m 1 ((0 1 ((0 c1 c2 1((1 m 1 ((0 = { c1 1((1 c2 1((0 ((1 1 ((1 m 1 1 c c 1 m 1 = { c1 1((1 1 c2 1((1 2 m 1((0 c1 c2 1((1 m 1 ((0 = { c1 c2 1((1 m 1((0 c1 c2 1((1 m 1 ((0 = 0 ((0 ((0 ((0 1 2 ((1 ((0 For any c γ m G( γ G( γ γ ρ ( id ρ ([ c m] = c1 c2γ1 γ [ c2γ2 m] = c c [ c m] 1γ 1 1γ 2γ 2 G( γ G( ρ γ = ( id ([ c m] m c ϖ ( G( f ( ϖ ( c m c m = { c 1 1 f( m 1 1 c 1 ((1 f( m 1 ((0 = { c 1 f( m 1 = 0 ((1 ((1 ((0 ((0 ( ( 1 ( ( 1 ((0 c 1 f( m 1 ((0 nd we also claim that G(f is a right -module map and a left π--comodulelike map In fact for any m c a G( f ([ c m] a = [ c a((1 f( m a((0 ] = [ c a((1 f ( m a((0 ] = G( f ([ c m] a For any m c ρ G( ( G( f ([ c m] = c [ c f( m] 1 2 G( = ( id G( f ρ ([ c m] So G( is a left π--comodulelike object Thirdly we claim that the compatibility condition holds For any m c a [ c m] a [ c m] a ((1 ((1 ((0 G( ((0 = c1 a((1 [ c2 m] a((0 = c1 a((1 [ c2 a((0 ((1 m a((0 ((0 ] = ( c a((1 [( c a 2 ((1 1 m a((0 ] = ([ c m] a ρ Therefore G( (π H Finally for any f : G(f = G(f where G(f :G( G( G( f ([ c m] = [ c f( m] for any m c (31 We claim G(f is well-defined In fact for any We still need to prove that F and G are adjoint functors We define the unit natural Homomorphism ϑ id G F and the counit natural π : ( H Homomorphism τ : F G id by the following formulas: ϑ : G( ϑ ( m = m m ((1 ((0 (32 τ : G ( τ ( [ c n ] = ( ε id (33 G( I The existence of τ comes from the fact that H H ( H m 1 ε is a usual weak bialgebra and I I I I ( I I I ε is a usual coalgebra We still have to show for any (π H G( τ ϑ G( = id G( and F( F( idf( τ ϑ =

Jia et al / J Zhejiang Univ SI 2005 6(Suppl I:144-154 151 In fact for any ( G( G( τ ϑ ( ( c m G( = G( τ γ = ( c1 γ ( c2 γ m = ε( c1i ( ( c2 m = ( c m For any n τ F( ϑ ( n = ε( n n = n F( ((1 I ((0 Thus we complete the proof Theorem 4 Let (H be a right weak Doi-Hopf π-datum Then the forgetful functor F: (π H π has a left adjoint functor Proof Its proof is dual to Theorem 3 here we only give the construction of the right adjoint functor of F Before defining the functor we first define a k-linear map for any π δ: ε ((1 I ((1 I ((0 ((0 (34 δ( m a = ( m a m a We claim δ 2 =δ In fact for any m a 2 δ ( m a = {( ε m((1 I a((1 I ( ε m((0 ((1 I a((0 ((1 I m((0 ((0 a((0 ((0 = {( ε m((1 I1 I a((1 I1 I( ε m((1 I2 I a((1 I2I m((0 a((0 = ε( m((1 I a((1 I m((0 a((0 = δ ( m a So we can define the adjoint functor G: π (π H by G(=( /ker δ G(f=f id for an object and a morphism f in π The action of on G( is given by [m a] b=[m ab] The left π--comodulelike structure is given by ρ ([ m a] = m a [ m a ] ((1 ((1 ((0 ((0 (35 We claim that the above action is well-defined In fact for any m a b δ(( m a b δ( m a b = ε( m((1 I ( ab ((1 I m((0 ( ab ((0 { ε( m((1 I a((1 I ε( m((0 ((1 I ( a((0 b ((1 I m((0 ((0 ( a((0 b ((0 = ε( m((1 I a((1 I b((1 I m((0 a((0 b((0 { ε( m((1 I a((1 I ε( m((0 ((1 I a((0 ((1 I b((1 I m((0 ((0 a((0 ((0 b((0 = ε( m((1 I a((1 I b((1 I m((0 a((0 b((0 { ε( m((1 I2 I a((1 I2 I ε( m((1 I1I a((1 I1 Ib((1 I m((0 a((0 b((0 = 0 We claim that the above comodulelike structure is well-defined In fact for any a m π ρ ( m a δ( m a = m a m a { ε( m a m a m a = m a m a a((1 2 m((0 a((0 = 0 ( id ρ([ m a] = {( m((1 a((1 1 ( m((1 a((1 2 m((0 a((0 = { m((1 1 a((1 1 m((1 2 a((1 2 m((0 a((0 = { m(( 1 a((1 m((0 ((1 a m a = ( id ρ ρ ([ m a] ((1 ((1 ((0 ((0 ((1 I ((1 I ((0 ((1 ((0 ((1 ((0 ((0 ((0 ((0 ((1 ((1 ((0 ((0 { ε( m((1 1 I a((1 1 I m((1 2 ((0 ((1 ((0 ((0 ((0 ((0 We still claim that the compatibility condition holds In fact for any π m a b [ m a] b [ m a] b ((1 ((1 ((1 ((0 ((0 = m a b [ m a ] b ((1 ((1 ((0 ((0 ((0 = {( ε m a b m ((0 ((1 I ((0 ((1 I ((0 ((1 I ((1

152 Jia et al / J Zhejiang Univ SI 2005 6(Suppl I:144-154 a b [ m a b ] ((1 ((1 ((0 ((0 ((0 ((0 ((0 ((0 = m a b [ m a b ] ((1 = ρ([ m a] b ((1 ((1 ((0 ((0 ((0 The unit and counit natural Homomorphisms ϑ : id π F G and τ : G F id are ( π H given by ϑ( m = ε( m((1 I 1 ((1 I m((0 1 ((0 (36 τ ([ n a] = n a (37 We have to show for any (π H π F( τ ϑ F( = idf( and τg( G( ϑ = idg( In fact for any [m a] G( τg( G( ϑ([ m a] = {( ε m((1 I a((1 I 1 ((1 I ( ε m((0 ((1 I a((0 ((1 I [ m((0 ((0 a((0 ((0 1 ((0 ] = {( ε m((1 I2 a((1 I2 ( ε m((1 I1 a((1 I1 [ m((0 a ] ((0 = [ m a] For any n F( τ ϑf( ( n = ε( n((1 I 1 ((1 I n((0 1((0 = n Thus we complete the proof THE WE SSH PRODUT Lemma 2 Let (H be a right weak Doi-Hopf π-datum then : given by χ ((0 ((1 (38 χ ( a f = 1 a 1 f 2 for any a f is a projection ie χ = χ Lemma 3 Let (H be a right weak Doi-Hopf π-datum then 1 a ((0 1 l l ((1 I = a a ((0 ((1 I for any a (39 h fg = ( h1 f( h2 g for any h f g (40 H Proof Eq(40 is obvious we only prove Eq(39 a a l ((0 ((1 I = 1 ((0 a((0 ε(1i1 1 ((1 I a((1 I 1I2 = 1 a ε(1 1 ε(1 a 1 = {1 ((0 ((0 a((0 ε(1 I1 1 ((1 I1 ε(1 ((0 ( (1 I a((1 I 1 I2 = 1 ((0 ((0 a ε(1i1 1 ((1 I 1I2 l = 1 1 ((0 ((0 I1 ((1 I1 ((1 I2 ((1 I I2 ((0 ((1 I Given a right weak Doi-Hopf π-datum (H we define # = (( / ker χ as a k-space and its multiplication by [ a# f][ b# g] = [ a b# f( a g] ((0 ((1 for any a b f g π (41 Theorem 5 Let (H be a right weak Doi-Hopf π-datum then # is an associative algebra with the unit [1 #ε] Proof First we claim that the above multiplication is well-defined In fact for any a b f g π χ (( a# f χ ( a# f( b# g = {( a((0 b# f( a((1 g((1 ((0 a ((0 b#(1 ((1 f((1 ((0 a ((1 g = {( a b# f( a g (1 a b #(1 ((1 f(1 ((0 ((1 a((1 g = {( a((0 b# f( a((1 g (1 ((0 a((0 b#(1 f(1 a g ((0 ((1 ((0 ((0 ((0 ((1 1 ((1 2 ((1 = {( a((0 b# f( a((1 g (1 ((0 a((0 b#1 ((1 f(( a((1 g = 0 ( a# f( b# g χ ( b# g = {( a b# f( a g ((0 ((1

Jia et al / J Zhejiang Univ SI 2005 6(Suppl I:144-154 153 ( a((0 1 ((0 b# f( a((1 1 ((1 g = {( a((0 b# f( a((1 g ( a((0 b# f( a((1 g = 0 Therefore it is well-defined ext we claim that it is associative In fact for any a b c f g u γ ([ a# f][ b# g][ c# u] = {( a((0 b ((0 c# f( a((1 g (( a b u ((0 ((1 γ = { a((0 ((0 b((0 c# f( a((1 g ( a((0 ((1 γ b((1 γ u = { a((0 b((0 c# f( a((1 γ 1 g ( a((1 γ 2 γ b((1 γ u = ( a((0 b((0 c# f( a((1 γ g( b((1 γ u = [ a# f]( b((0 c# g( b((1 γ u = [ a# f]([ b# g][ c# u] Obviously [ a# f][1 # ε] = [1 # ε][ a# f] = [ a# f] Lemma 4 Let (H be the same as mentioned above then we have: [ a# ε][ b# ε] = [ ab# ε] for any a b (42 [1 # f][1 # g] = [1 # fg] for any f g (43 Proof = [ a# ε][ b# ε] = [ a b# a ε] = [1 #1 ε] = [ ab # ε] ((0 ((1 l ((0 ab ((1 [1 # f][1 # g] [1 ((0 # f(1 ((1 g] = [1 ((0 ((0 # f(1 ((1 g(1 ((0 ((1 I ε] = [1 ((0 # f(1 ((1 1 g(1 ((1 2I ε] = [1 # f(1 g] = [1 # fg] ((0 ((1 Theorem 6 Let (H be a right weak Doi-Hopf π-datum such that is finite dimensional Then the category of (π H is isomorphic to the category of # Proof We define the functor F: (π H # by F(= F(f=f for an object and a morphism f in (π H We define m [ a# c] = c( m((1 m((0 a for any m a c We claim that the above map is well-defined and define an action of # on In fact for any a c m ( a# c χ( a# c = { c( m((1 m((0 a c( m((1 1 ((1 m((0 1 ((0 a = ( c( m((1 m((0 a c( m((1 m((0 a = 0 For any m a c c ((1 I ((0 ( m ( a# c ( b# c = c( m((1 c(( m((0 a ((1 ( m((0 a ((0 b = { c( m((1 c( m((0 ((1 a((1 m((0 ((0 a((0 b = c( m((1 1 c( m((1 2 a((1 m((0 a((0 b = m [ a((0 b# c( a((1 c] = m [( a# c( b# c] m (1 # ε = ε( m m 1 = m So is a right # -module We also claim f is a right # -module map In fact for any a c m f( m [ a# c] = f( m((0 a c( m((1 = f( m((0 ac( m((1 = f( m ((0 ac( f( m ((1 = f( m [ a# c ]

154 Jia et al / J Zhejiang Univ SI 2005 6(Suppl I:144-154 Because is finite dimensional there exists a dual basis {x X in and It is easy to verify ( x X = x x l X X l (44 We define the functor G: # (π H by G(= G(f=f for an object and f a morphism in # The right action of on G( is given by m a=m [a#ε] for any a m The left comodulelike structure is given by ( ε id ρi ( m = ε( xi m [1 ((0 #1 ((1 I XI ] = m [1 #1 ε] = m ((0 ((1 I Obviously f is a right -linear and we only need to show that f is a right π--comodulelike map In fact for any m π ρ ( f( m = x f( m [1 ((0 #1 ((1 X] = x f( m [1 #1 X ] = ( id f ( m ((0 ((1 ρ ρ ( m = x m [1 ((0 #1 ((1 X] for any m (45 ow we claim that it exactly defines a comodulelike structure For any π and m ( id ρ ρ = { x xl m ((1 ((0 #1 ((1 X (1 ((0 #1 ((1 Xl = { x xl m (1((0 ((0 1 ((0 #(1 ((1 X (1 ((0 ((1 1 ((1 Xl = { x xl m (1((0 ((0 #(1 ((1 X (1 ((0 ((1 Xl = { x xl m (1 ((0 #(1 ((1 1 X (1 ((1 2 Xl = x xl m (1 ((0 #1 ((1 ( X Xl = x 1 x 2 m (1((0 #1 ((1 X = ( id ρ ( m OWLEDGEET We are very grateful to the referees for their valuable suggestions on deeper research into weak Doi-Hopf group modules They pointed out that we can consider the new prospective of a symmetric monoidal category which we call the Turaev s category We are doing this work and have achieved some aims References Böhm G 2000 Doi-Hopf modules over weak Hopf algebras omm lgebra 28(10:4687-4698 Böhm G ill F Szlachányi 1999 Weak Hopf algebras I: integral theory and * -structure Journal of lgebra 221:385-438 Virelizier 2002 Hopf group-coalgebras Journal of Pure and pplied lgebra 171(1:75-122 Wang SH 2004 Group twisted smash product and Doi-Hopf modules for T-coalgebras omm lgebra 32:3417-3436