Weak Hopf Algebras and Some New Solutions of the Quantum Yang-Baxter Equation*

JOURNAL OF ALGEBRA 208, 72᎐100Ž. 1998 ARTICLE NO. JA987491

Weak Hopf Algebras and Some New Solutions of the Quantum Yang᎐Baxter Equation*

Fang Li

Department of Mathematics, Hangzhou Uni¨ersity, Hangzhou, Zhejiang, 310028, People’s Republic of China

Communicated by Efim Zelmano¨

Received April 25, 1997

In this paper, the concepts of a weak Hopf algebra and a quasi-braided almost bialgebra are introduced. It is shown that the quantum quasi-doubles of some weak Hopf algebras are quasi-braided almost bialgebras. This fact implies that some new solutions of the quantum Yang᎐Baxter equation can be constructed from some weak Hopf algebras, in particular, when the weak Hopf algebra is a finite Clifford monoid algebra. ᮊ 1998 Academic Press

INTRODUCTION

In this paper, the notations and concepts about Hopf algebras and semigroups can be found respectively inwx 1, 8, 4, 9 . The modules in this paper always are unital. In Section 1, first we introduce the concept of a weak Hopf algebra and discuss some properties of weak Hopf algebrasŽ. Propositions 1.2 and 1.4 . As a generalization of a Hopf algebra, some characterizations of weak Hopf algebras have been studied by the authorwx 7 . I think that the algebraic meaning of a weak Hopf algebra is to relate its structure as a bialgebra with its monoid of group-like elements. Then, we define a quasi-module-algebra and quasi-module-coalgebra and give some conditions such that some weak Hopf algebras H and their duality Ž H op.* are quasi-module-coalgebras over each otherŽ Proposition 1.6, Corollary 1.8, and Theorem 1.10. .

* Project 19501007 supported by National Natural Science Foundation of China.

0021-8693r98 $25.00 Copyright ᮊ 1998 by Academic Press All rights of reproduction in any form reserved. WEAK HOPF ALGEBRAS 73

In Section 2, first we introduce the concepts of a quasi-matched pair of bialgebras and an almost bialgebra and then construct the quasi-bicrossed product on a quasi-matched pair of bialgebras, in particular, of some weak Hopf algebras with invertible weak antipode. The main result is that for some weak Hopf algebras, we can construct the quasi-bicrossed products Ž.called quantum quasi-double which are quasi-braided with the quasi-R- matrices as the solutions of the quantum Yang᎐Baxter equation. As an example, we turn out that every finite Clifford monoid algebra is such a weak Hopf algebra. It means that we can get a solution of the QYBE from every finite Clifford monoid algebra. As is well known, a Clifford semigroup is a generalization of a group. Hence, the main meaning of our work is to generalize the method of constructing the solution of the QYBE from a group algebra to a semigroup algebra. However, for an arbitrary Clifford monoid, the solution of the QYBE constructed may not be invertible generally. This point is different from the case of groups.

1. WEAK HOPF ALGEBRA

Let H be a linear space over a field K. H is called an algebra if H has a multiplication m: H m H ª H and a unit u: K ª H such that mŽ.Ž.Ž.Id m m s mmm Id associativity and Id s muŽ.Ž.Žm Id s m Id m u unitary property . , where Id is the identity map of H. H is called a coalgebra if H has a comultiplication ⌬: H ª H m H and a counit ␧: H ª K such that Ž.Ž.Ž⌬ m Id ⌬ s Id m ⌬⌬ coassociativity of ⌬.Ž.Ž.Ž.and Id s ␧ m Id ⌬ s Id m ␧ ⌬ counitary property . H is called a bialgebra if H is an algebra and also a coalgebra, and ⌬ and ␧ are both algebra maps. In what follows, H always denotes a bialgebra. The set of group-like elements GHŽ.s Ä g g H: ␧Ž.g s 1 and ⌬ Ž.g s g m g4 is a monoid with the multiplication m. Set R s Hom K Ž.H, H . For f, g g R, define f ) g s mf Ž.m g ⌬.Itis easy to see that R is a K-algebra with a multiplication ) and u␧ is the identity of the algebra R. The bialgebra H is called a Hopf algebra if there exists S g R such that S)Id s Id)S s u␧. S is called an antipode. For a Hopf algebra H, the monoid GHŽ.is a group Ž seewx 1. . The bialgebra H is called a leftŽ. right, resp. Hopf algebraw3x if there exists S g R such that S)Id s u␧ Ž.resp. Id)S s u␧ . S is called a left antipode Ž.Ž.resp. right antipode . Generally, GH may not be a group. But when the left Hopf algebra H is pointed as a coalgebra, GHŽ.is a groupw3x . 74 FANG LI

Now we define the bialgebra H as a weak Hopf algebraw7x if there exists T g R such that Id)T )Id s Id and T )Id)T s T. T is called a weak antipode. Some properties of a weak Hopf algebra are shown inwx 7 ; in particular, we have known that if the weak Hopf algebra H is pointed as a coalgebra, GHŽ .is a regular monoid. Obviously, T Ž.1 s 1 since 1T Ž.11s 1. An example of a weak Hopf algebra is the regular monoid algebra KS where K is a field, S is a regular monoid. We can show that for a field K and a monoid S, the monoid algebra KS is a weak Hopf algebra if and only if S is a regular monoid. First, if KS is a weak Hopf algebra, then since KS is pointed we know that S s GKSŽ.is regular by the result ofwx 7 . Conversely, if S is regular, for s g S let sЈ be a fixed element in VsŽ.s Ä4t g S: tst s t, sts s s . Define T: KS ª KS such that TsŽ.s sЈ, X TŽ.Ýksiis Ýks ii. Then T g Hom KŽ.KS, KS and for x g KS we have ŽId)T )Id .Ž.x s Id Ž.x s x, ŽT )Id)Tx .Ž.s Tx Ž.. It follows that KS is a weak Hopf algebra with weak antipode T. It is obvious that every leftŽ. resp. right Hopf algebra with a left Ž resp. right. antipode S is a weak Hopf algebra and S is a weak antipode. In this section, we discuss some properties which are useful to construct some new solutions of the quantum Yang᎐Baxter equation. Every Hopf algebra possesses a unique antipode. However, usually, a weak Hopf algebra may have one or more weak antipodes; a leftŽ resp. right. Hopf algebra is similar. A bialgebra H is a weak Hopf algebra iff w7x there is TЈ g Hom K Ž.H, H such that Id)TЈ)Id s Id. Let H and N be both K-weak Hopf algebras with the weak antipodes

THNand T , respectively, ␴ a K-bialgebra morphism from H to N such that ␴ THNs T ␴ . Then ␴ is called a K-weak Hopf algebra morphism. Let ␶ : H m H ª H m H be the twist mapping defined by ␶ Ž.h12m h s op op h21m h , for all h 12, h g H and let m : H m H ª H, ⌬ : H ª H m H op op be the maps defined by m s m␶ and ⌬ s ␶⌬ Žthe opposite multiplication and the opposite comultiplication.Ž . If H s H, m, u, ⌬, ␧ .is a bialge- op op cop op opcop bra, then H s Ž H, m , u, ⌬, ␧ ., H s Ž H, m, u, ⌬ , ␧ ., H s Ž H, mop, u, ⌬ op, ␧ . are bialgebraswx 6 . Moreover we have

LEMMA 1.1. Let H s Ž.H, m, u, ⌬, ␧ be a bialgebra, P g Hom K Ž.H, H , and P be in¨ertible. Then

Ž.i for e¨ery a, b g H, Pab Ž .s Pb Ž.s P Ž. b P Ž. a iff for e¨ery a, b g 1 1 1 H, Paby Ž.s PbPay Ž.y Ž.; 1 1 1 Ž.ii ⌬ P s ␶ ŽP m P .⌬ iff ⌬ Py s ␶ Ž Py m Py .⌬;

Ž.iii assume P Ž ab .s PbPa Ž.Ž. fore¨ery a, b g H, then ŽId H ) PH .Ž. op y1 : CŽ.Ž H the center of H . iff ŽId HH) PH.Ž .: CH Ž ., ŽP )Id .ŽH . : WEAK HOPF ALGEBRAS 75

y1 op op y1 CŽ. H iff Ž P ) Id HHH.ŽH .: C Ž H . where ŽId ) P .Žs m Id m Py1 .⌬op and so on;

Ž.iv assume ⌬ P s ␶ ŽP m P .⌬, then ŽId H ) PH .Ž.: C Ž H . iff y1 op op y1 Ž P ) id HHH.ŽH .: CH Ž ., ŽP )Id .ŽH .: C Ž H . iff ŽId ) PH.Ž . : CHŽ.; Ž.v assume P Ž ab .s PbPa Ž . Ž . fore¨ery a, b g H and ⌬ P s ␶ ŽP m P.⌬, then the following Ž1 .᎐ Ž4 .are equi¨alent:1 Ž . Ž IdH ) PH .Ž .: CH Ž .; op y 1 Ž.Ž2 P )Id HH .Ž.H : CH Ž.Ž.Ž;3 Id) PH.Ž .: CH Ž .;4 Ž . y1 op Ž P ) Id H .ŽH .: CH Ž ..

1 1 1 Proof. Ž.i Ž« .. ab s ab « PPy Ž. ab s PPy Ž. a PPy Ž. b « 1 1 1 1 1 1 PPy Ž. ab s PP Žy Ž. bPy Ž.. a « Paby Ž.s PbPay Ž.y Ž.. Ž¥ .. This is similar. 1 1 Ž.Ž.Ž.ii « . ⌬ P s ␶ P m P ⌬ « ⌬ s ␶ Ž.P m P ⌬ Py « Ž Py m 1 1 1 1 1 Py .⌬ s ␶ ⌬ Py « ␶ Ž Py m Py .⌬ s ⌬ Py . Ž.¥ . This is similar. Ž.Ž.Ž.Ž.iii If Id) PH: CH, i.e., for every h, x g H,Id Ž.) P Ž.hxs x ŽId) Ph .Ž.« Ý Ž h .ŽhЈ Ph Ž.Љ x s Ý h . xhЈ Ph Ž.Љ ,then y1 y1 y1 y1 y1 op y1 PŽÝŽ h.ŽPxhŽ.Љ PhŽ..Ј s P ŽÝ h. hЉ PhŽ.Ј PxŽ..« PxŽ.ŽId) P . op 1 1 op 1 1 Ž.h s ŽId) PhPxy .Ž . y Ž.« ŽId) PHy .Ž .: CH Ž . since PHy Ž.s H. The converse result is similarly proved. 1 op Similarly Ž.Ž.Ž.ŽP )Id H : CH m Py ) Id.ŽH .: CH Ž .. y1 op Ž.iv Assume Ž Id HH) PH .Ž.: CH Ž .. For h g H, ŽP ) Id .Žh . y1 y1 y1 y1 y1 sÝŽh.ŽPhŽ.Љ hЈsÝ h.ŽPhŽ.ŽЉ PPŽ.. hЈ sÝ Py1 Ž h.. ŽPhŽ..ŽЈPPŽ.. h Љ y1 y1 op s Ž.ŽId HH) PPŽ.. hg CH Ž .. Then ŽP ) Id .ŽH .: CH Ž .. The con- op y1 verse result is similar. Similarly Ž.Ž.Ž.ŽP )Id HHH : CH iff Id ) PH.Ž . : CHŽ.. Ž.v This follows from Ž iii . and Ž iv . .

PROPOSITION 1.2. Let H s Ž.H, m, u, ⌬, ␧ denote a weak Hopf algebra with weak antipode T. Then

opcop op op Ž.i H s Ž H, m , u, ⌬ , ␧, T. is a weak Hopf algebra. If more- oër, for eëry a, b g H, TabŽ.s T Ž.Ž. b T a and ⌬T s ␶ ŽT m T .⌬, then T: opcop H ª H is a weak Hopf algebra morphism; Ž.ii If T is inërtible and T Ž ab .s TbTa Ž. Ž. foreëry a, b g H, then op op 1 cop op 1 H s Ž H, m , u, ⌬, ␧, Ty . and H s Ž H, m, u, ⌬ , ␧, Ty . are weak 1 Hopf algebras with weak antipodes Ty . If moreoër, ⌬T s ␶ Ž.T m T ⌬, then opT cop H ( H is a weak Hopf algebra isomorphism. 76 FANG LI

opcop Proof. Ž.i B s H is a bialgebra. We have now Id H s Id HH) T ) HHId , T s T ) HHHId ) T. Then for every x g H, op op op op Ž.Ž.Ž.Ž.Id BB) T ) BId Bx s m Ž.m Id Bm T ⌬ m Id B⌬ x op op s Ý m Ž.m Ž.xٞ m TxŽ.Љ m xЈ Ž.x s Ý xЈTxŽ.Љ xٞ Ž.x

s Ž.Ž.Id HH) T ) HId H x

s Id HBŽ.x s Id Ž.x .

Hence Id BB) T ) BId Bs Id B . Similarly, T ) BId BB) T s T. opcop op op It means that H s Ž H, m , ␮, ⌬ , ␧, T . is a weak Hopf algebra. T g Hom KKŽ.H, H s Hom Ž.H, B . T is a bialgebra morphism from H to B since TabŽ.s TbTa Ž.Ž.for every a, b g H and ⌬T s ␶ ŽT m T .⌬. But THBs T s T, then T и T HBs T и T. So, T is a weak Hopf algebra morphism. op cop Ž.ii H and H are both bialgebras. For every x g H,

y1 op op y1 Ž.Ž.Id HH) opT ) HopId HŽ.x s m Ž.m Id Hm T ⌬ m Id H⌬ Ž.x 1 s Ý xٞTy Ž.xЉ xЈ Ž.x

1 . s Ty ž/Ý TxŽ.Ј xЉ Tx Žٞ Ž.x

y1 s T Ž.Ž.T )HHId) Tx 1 s Ty Ž.TxŽ.s x

s Id H Ž.x .

y1 y1 y1 y1 Hence Id HH) opT ) HHopId s Id. Similarly, T ) HHHopId ) opT s T . Therefore H op is a weak Hopf algebra with weak antipode Ty1. Similarly, H cop is too.

If moreover, ⌬T s ␶ Ž.T m T ⌬, then for every x, y g H, Tx Ž.иop y s op TmŽ Ž xm y ..Ž.Ž.Ž.Ž.Ž.s Tyxs TxTys TxиcopTy, op ⌬TxŽ.s ␶ ŽT m T .Ž.⌬ x s ŽT m T .⌬ Ž.x , op i.e., ⌬T s Ž.T m T ⌬ . It follows that T is a bialgebra isomorphism from H op to H cop since T y1 is invertible. And THHop s T cop s T .SoT и T HHop s T cop и T. Hence T is a weak Hopf algebra isomorphism. WEAK HOPF ALGEBRAS 77

o LEMMA 1.3. For a bialgebra H s Ž.H, m, u, ⌬, ␧ , let H s Ä g g H*: Ker g contains a cofinite ideal of H4. If X, Y g Hom K Ž.H, H such that oo X*, Y * g Hom KHHŽ H , H .Ž, then X ) Y .* s X*) o Y *. o o Proof. Bywx 8 , H s Ž H , ⌬*, ␧*, m*, u*. is a bialgebra. Obviously, oo X*, Y * g Hom KkŽ.H*, H* . Hence X*, Y * g Hom ŽH , H . implies o o o oo X*Ž H . : H , Y *Ž H . : H . For every f g H , x g H, Ž.Ž.Ž.X )H Y * fx s fXŽŽ)H Yx .Ž ..s f ŽÝŽ x. Xx ŽЈ .Yx ŽЉ .., and

Ž.Ž.Ž.X*)H o Y * fx s ⌬*Ž.Ž.Ž.Ž.Ž.Ž.X* m Y * m* fxs Ý Ž.X* fЈ m Y * fЉ⌬x Ž.f

s¦;ÝÝX*Ž.fЈ m Y * ŽfЉ ., xЈ m xЉ Ž.fx Ž.

s ÝÝX*Ž.Ž.ŽfЈ xЈ Y * fЉ .Ž.xЉ s fЈŽ.Ž.Xx Ž.Ј fЉ Yx ŽЉ . Ž.Ž.x , fx Ž.Ž., f

s Ý ²:²:fЈ, XxŽ.Ј fЉ , Yx ŽЉ . Ž.Ž.x , f

s¦;ÝÝfЈ m fЉ , XxŽ.Ј m Yx ŽЉ . Ž.fx Ž.

s¦;¦;m*Ž.f , ÝÝXx Ž.Ј m Yx ŽЉ .s f , Xx Ž.Ž.Ј YxЉ Ž.xx Ž.

s Ž.Ž.Ž.X )H Y * fx.

Hence Ž.X )HHY * s X*) o Y *. PROPOSITION 1.4. Let H s Ž.H, m, u, ⌬, ␧, T be a weak Hopf algebra and TŽ. ab s T Ž.Ž. b T a for eëry a, b g H. Then o o Ž.i H s Ž H , ⌬*, ␧*, m*, u*, T*. is a weak Hopf algebra with weak antipode T*; Ž.ii If moreoër, ⌬T s ␶ ŽT m T .⌬, then T* Žfg .s T* ŽgT .* Žf . for o eëry f, g g H and ⌬ HHHoooT* s ␶ Ž.T* m T* ⌬ where ⌬ s m*.

Proof. Ž.i Of course T* g Hom k ŽH*, H* . . Now, we need to verify o o that T*Ž H . : H . In fact, for every a, b g H, f g H*, ²Ž.:a © T* f , b s ²T* Žf ., ba :s ²f, Tba Ž .:s ²f, TaTb Ž . Ž .:s ²f £ Ta Ž ., Tb Ž .: s ²ŽT* f £ Ta Ž..:, b . Hence H © T* Ž.f s T* Žf £ TH Ž..: T* Žf £ H ..If o f g H , then dimŽ.f £ H - qϱ bywx 8, Lemma 9.1.1 , thus dimŽŽT* f £ H ..- qϱ. It follows that dim ŽH © T* Žf ..- qϱ. It means that T* Žf . o o o o o g H . Hence T*Ž H . : H . Obviously,Ž.Ž IdHH * H . s H , i.e.,Ž. Id * s o o Id H on H . 78 FANG LI

Since Id HH) T ) HId Hs Id H , T ) HId HH) T s T, by Lemma 1.3 we have Id HHo ) oT*) Ho Id Ho s Id Ho , T*) Ho Id HHo ) oT* s T*. Therefore, T* is a weak antipode of H o such that H o is a weak Hopf algebra. Ž.ii For every x g H, ²T * Žfg ., x :s ²fg, Tx Ž.:s ²f m g, ÝŽ x.ŽTxŽЉ .m Tx ŽЈ .:s Ý x.Ž ²f, Tx ŽЉ .:²g, Tx ŽЈ .:s Ý x. ²T* Žf ., xЉ .²T* Žg ., xЈ:²Ž.Ž.:s T* gT* f , x . Hence T* Ž.fg s T* Ž.Ž.gT* f . For every a, b g H, ²Ž.:²Ž.:⌬ H o T * f , a m b s m*T * f , a m b s²T* Žf ., ab :s ²f, Tab Ž .:s ²f, TbTa Ž . Ž .:s ²ÝŽ f . fЈ m fЉ,Tb Ž .m Ta Ž .: s ÝŽ f .Ž²fЈ, Tb Ž .:² fЉ, Ta Ž .:s Ý f .Ž ²T* ŽfЈ ., bT :²* ŽfЉ ., a :s ²Ý f .T* ŽfЉ . mT*Ž.fЈ , a m b :. Hence ⌬ H oT* Ž.f s ÝŽ f .T* ŽfЉ .m T* Ž.fЈ s ␶ ŽT* m T*.Ž.⌬ HHHooof . It follows that ⌬ T* s ␶ ŽT* m T* .⌬ .

DEFINITION 1.5.Ž. i Let H be a bialgebra, A an algebra. If A is a left

H-module and habŽ.s ÝŽ h. ŽhЈah .Ž.Љ b for every h g H, a, b g A, then we call the algebra A a left quasi-module-algebra over H. The right quasi-module-algebra can be defined similarly. Ž.ii Let H be a bialgebra, C a coalgebra. If C is a left H-module and ⌬Ž.hc s ⌬ Ž.Ž.h ⌬ c for every h g H, c g C, then we call the coalgebra C a left quasi-module-coalgebra over H. The right quasi-module-coalgebra can be defined similarly.

PROPOSITION 1.6. Let H s Ž.H, m, u, ⌬, ␧, T be a weak Hopf algebra with TŽ. ab s T Ž.Ž. b T a for a, b g H. For a, x g H, set a и x s ÝŽ a. aЈxTŽ. aЉ . If Ž T )Id H .Ž.H : CH Ž., then the map Ž a, x .ª a и x endows H with the structure of a left quasi-module-algebra o¨er the bialgebra H. We denote byad H the left H-module defined this way and call this action the left a adjoint representation of H. Similarly for a, x g H, set x s ÝŽ a.TaŽ.Ј xaЉ. a If Ž.Ž.Ž.Id)TH: CH, then the map Ž. a, x ª x endows H with the structure of a right quasi-module-algebra o¨er the bialgebra H. We denote by Had the right H-module defined this way and call this action the right adjoint representation of H.

Proof. 1 и x s ÝŽ1.1ЈxTŽ.1Љ s x since T Ž.1 s 1 and b и Ža и x .s ÝŽ a.Ž b.ŽbЈaЈxTŽ.Ž. aЉ TbЉ s Ý ba. Ž.ba ЈxT Ž. ba Љ s Ž.ba и x for a, b, x g H. Thus, H is a left H-module. For a, x, y g H, ÝŽ a.Ž.Ž.aЈиxaЉиy s Ž4. . ÝŽ a.ŽaЈxTŽ. aЉ aٞ yT Ž a .s Ý a.ŽaЈxT Ž)Id .ŽaЉ .yT Ž aٞ .s Ý a. aЈxy Ž T )Id Ž.Ž.aЉ Taٞ s ÝŽ a.ŽaЈxy Ž T )Id)Ta .Ž.Љ s Ý a. aЈxyT Ž. aЉ s a и Ž.xy . Hence H is a left quasi-module-algebra over H. Similarly, the right adjoint representation can be proved.

LEMMA 1.7. Let H s Ž.H, m, u, ⌬, ␧, T be a weak Hopf algebra with inërtible weak antipode T such that ⌬T s ␶ Ž.T m T ⌬ and a K-algebra A be a leftŽ. resp. right H-quasi-module algebra. Put on the dual ëctor space A* WEAK HOPF ALGEBRAS 79 the leftŽ. resp. right H-module structure giën by

1 x Ty1 Ž x. ²:²xf , a s f , Txay Ž.:²Ž.resp. f , a:²s f , a : op o for all a g A, x g H, and f g A*. Then the coalgebraŽ A . s o op Ž A , Žm ..*, u* is a left Ž resp. right . quasi-module-coalgebra o¨er H. Proof. First, we check that Ž Aop.o is a left H-module. For every op o y1 h g H, f12, f g Ž A . , ²Žhf 12q f ., a :s ²f 12q f , ThaŽ.:s y1 y1 ² f12, ThaŽ.:q ²f , ThaŽ.:²:²s hf 1212, a q hf , a :²s hf q hf , a :. Hence hfŽ.12q f s hf 1q hf 2. op o For every h 12, h g H , f g Ž A . , ²Žh 1q hf 2 ., a : s y1 y1 y1 y1 ² f, ThŽ.:²12q ha s f, ThaŽ. 1q ThaŽ.: 2s ²f, ThaŽ.: 1q y1 ² f, ThaŽ.:²21s hf, a :²q hf 2, a :²s hf 12q hf, a :. Hence Žh 1q y 1 hf21212.²Ž.:²s hfq hf. hh f, a s f , ThhaŽ.:² 12s f , y1 y1 ThThaŽ.2Ž.:²Ž 1s hhf 12 .:, a . Hence Žhh 12 . fs hhf 12 Ž .. It follows that Ž Aop.o is a left H-module which is unital trivially. op o Now for h g H, f g Ž A . , a, b g A, op ²:²:⌬Ž.hf , a m b s Žm .Ž.* hf , a m b s ²:hf , ba 1 1 1 s² f , Thbay Ž.Ž.: s Ý²:f , Ž.Ž.Thy ŽЉ .bTy Ž hЈ .a Ž.h

1 1 1 Žsince ⌬Thy Ž.s ÝThy Ž.Љ m Thy Ž.Ј and A is a left H-quasi-module- algebra.

op 1 1 s Ý²:f , Ž.m Ž.Ty Ž.hЈ a m Ty Ž.hЉ b Ž.h

op 1 1 s Ý² Ž.Ž.Ž.m * f , Thy Ј a m Thy Ž.Љ b: Ž.h

1 1 s Ý² ⌬Ž.f , Thy ŽЈ .a m Thy ŽЉ .b: Ž.h

1 1 s ÝÝ² fЈ m fЉ , Thy Ž.Ј a m Thy ŽЉ .b: Ž.Ž.hf

1 1 s ÝÝ² fЈ, Thy Ž.Ј af:² Љ , Thy ŽЉ .b: Ž.Ž.hf

s ÝÝ²:²:hЈfЈ, ahЉ fЉ , b Ž.Ž.hf

s¦;ÝÝhЈfЈ m hЉ fЉ , a m b . Ž.Ž.hf

Therefore ⌬Ž.hf s ÝÝŽh.Žf . hЈfЈ m hЉ fЉ s ⌬ Ž.Ž.h ⌬ f . 80 FANG LI

Hence Ž Aop.o is a left H-quasi-module-coalgebra. Similarly, Ž Aop.o is a right H-quasi-module-coalgebra when A is a right H-quasi-module- algebra.

COROLLARY 1.8. Let H s Ž.H, m, u, ⌬, ␧, T be a weak Hopf algebra with T in¨ertible, Ž.Ž.Ž.Ž.Ž.Ž.T )Id H H : CH, Tabs T b T a for a, b g H op o and ⌬T s ␶ Ž.T m T ⌬. For a, x g H, f g ŽH . , set x и a s ÝŽ x. xЈaTŽ. xЉ y1 x x and²:² xf, a s f, TxŽ.и a :w resp. a s ÝŽ x.TxŽ.Ј axЉ and ² f , a: s Ty1 Ž x. op o o op ² f, a :x. Then the weak Hopf algebraŽ H . s Ž H , ⌬*, ␧*, Žm .*, u*, ŽTy1 ..* is a leftwx resp. right H-quasi-module-coalgebra by this action on Ž H op.o.

x Proof. By Proposition 1.6, the map Ž.x, a ª x и a wresp. Ž.x, a ª a x endows H with the structure of a leftwx resp. right H-quasi-module- algebra. Then by Lemma 1.7, the coalgebra Ž H op.o is a leftwx resp. right H-quasi-module-coalgebra. These actions in Corollary 1.8 will be called the left and right coadjoint representations of H.

LEMMA 1.9. For a bialgebra H s Ž.H, m, u, ⌬, ␧ , in the dual bialgebra o op op op op op o Hwehaë thatŽ. m* s Žm .Ž.*, ⌬* s Ž⌬ .Ž*, then moreoër, H . o cop cop o o op s Ž H . , Ž H . s Ž H . . o Proof. For every f, g g H , x, y g H, set m*Ž.f s ÝŽ f . fЈ m fЉ. op Then Ž.m* Ž.Žfxm y .s ÝŽ f .Ž Žf Љ m f Ј .Žx m y .s Ý f . f Љ Ž.xfЈ Ž.y s op op ÝŽ f . fЈŽ.yfЉ Ž.x ; Žm .* Žfx .Žm y . s fmŽ.Ž.Ž. x m y s fyxs fm y m x s op Ž.Ž.Žm*fym x s ÝŽ f .ŽfЈ m fЉ .Ž.Ž.Ž.Ž.y m x s Ý f . fЈ yfЉ x s m* Ž.Žfxm op op op op y.Ž.. Hence m* s Žm .Ž.*. And ⌬* Žf m gx .Ž.s Žgf .Ž.Ž x ; ⌬ .Ž* f m op gx.Ž .s Žf m g .Ž⌬ .Žx .s ÝŽ x.Ž Žf m gx .ŽЉ m xЈ .s Ý x. fx ŽЉ .gx ŽЈ . s op ÝŽ x.Ž.Ž.Ž.Ž.Ž.Ž.Ž.g m fxЈ m xЉ s ⌬* g m fxs gf x s ⌬* Ž.Ž.f m gx. op op Hence Ž⌬ .Ž.* s ⌬*. op o op o o op o Ž H . s Ž H, m , u, ⌬, ␧ . s Ž H , ⌬*, ␧*, Žm ..Ž*, u* s H , ⌬*, ␧*, op o cop Ž.m*,u*.Žs H . . cop o op o o op Ž H . s Ž H , m , u, ⌬ , ␧ . s Ž H , Ž⌬ ..*, ␧ *, m*, u* s oop oop ŽŽ.H , ⌬*,␧*, m*, u* .Ž.s H . The special case of Lemmas 1.3, 1.7, 1.9, Corollary 1.8, and Proposition 1.4 is for a finite dimensional bialgebra H Ž.or algebra A in Lemma 1.7 in o o which H s H*orŽ A s A*. and so on. THEOREM 1.10. Let H s Ž.H, m, u, ⌬, ␧, T be a finite dimensional co- commutatië weak Hopf algebra with T inërtible, ⌬T s ␶ Ž.T m T ⌬, and f TabŽ.s T Ž.Ž. b T a for eëry a, b g H. For a g H, f g H*, set a s y1 op op y1 ÝŽ a. fTŽ Ž.. aٞ aЈ aЉ. Then Ž H .Ž* s H*, ⌬*, ␧*, Žm .Ž*, u*, T ..* is a weak Hopf algebra and H is a rightŽ H op.*-quasi-module-coalgebra by this action on H. WEAK HOPF ALGEBRAS 81

Proof. By Proposition 1.2, H op and H cop are weak Hopf algebras with weak antipode Ty1. By Proposition 1.4, Ž H op.Ž* and H cop.* are weak Hopf algebras with invertible weak antipode ŽTy1 .*. cop 1 1 For every x g H, f, g g Ž H .Ž*, Ty .* Žfg .Ž x .s Žfg .ŽŽ Ty .Ž x .. s op y1 op y1 y1 Ž⌬ .Ž* f m gT .Ž Ž.. x s Žf m g .Ž⌬ ŽTxŽ ...s ÝŽ x. Žf m gT .Ž Ž. xЈ m y1 y1 y1 y1 y1 TxŽ..Љ s ÝŽ x.ŽfT Ž Ž..Ž xЈ gTŽ.. xЉ s Ý x.wŽT .Ž* fx .xŽ.Ј wŽT .Ž* g .x 1 1 1 1 op Ž.xЉ s wŽTy .Ž* f .m ŽTy .Ž* g .x⌬Ž.x s wŽTy .Ž* g .m ŽTy .Ž* f .x⌬ Ž.x s op 1 1 1 1 Ž⌬ .ŽŽ* Ty .Ž* g .m ŽTy .Ž* fx ..Ž.s wŽTy .Ž* gT .Ž y .Ž* fx .xŽ..Hence 1 1 1 ŽTy .Ž* fg .s ŽTy .Ž* gT .Ž y .Ž* f .. cop o 1 1 For f g Ž H . , x, y g H, Ž.Žm * Ty .Ž* fx .Žm y .s ŽTy .* Žfxy .Ž . s 1 1 1 1 1 fTŽ y Ž.. xy s fT Žy Ž. yTy Ž.. x ; ␶ ŽŽTy .Ž* m Ty ..Ž.Ž* m * fx .Žm y .s 1 1 1 1 1 Žm .* ŽfT .ŽŽ y .Ž y . m Txy Ž..s fT Žy Ž. yTy Ž.. x s Žm .Ž* Ty .* Žfx .Žm y .. 1 1 1 cop Hence Ž.Žm * Ty .ŽŽ* s ␶ Ty .Ž* m Ty ..Ž.* m *in ŽH .*. cop For f, g g Ž H .*, x g H, y1 g Ž.Id H cop* )H cop* Ž.Ž.Ž.T * fx op 1 s g Ž.⌬ *Ž. Id* m ŽTy .* m* Ž.Ž.fx op 1 s Ý gxŽ.Љ⌬*Ž. Id* m ŽTy .* m* Ž.Ž.fxЈ Ž.x

1 op s Ý gxŽ.Љ Ž.Id* m ŽTy .* m* Ž.Ž.f ⌬ xЈ Ž.x

1 . s Ý gxŽ.ٞ Ž.Id* m ŽTy .* m* Ž.ŽfxЉ m xЈ Ž.x 1 1 ; s ÝÝgxŽ.ٞ m* Ž.fxŽ.Ž.Љ m Ty Ž.xЈ s gx Ž.ٞ fxЉ Ty Ž.xЈ Ž.xx Ž.

y1 Ž.Id H cop* )H cop* Ž.Ž.Ž.T * fg x op 1 s Ý Ž.⌬ *Ž. Id* m ŽTy .* m* Ž.Ž.Ž.fxЉ gxЈ Ž.x

1 s Ý Ž.Id* m Ž.Ty * m* Ž.Žfxٞ m xЉ .Ž.gxЈ Ž.x 1 . s Ý m*Ž.fxŽ.ٞ m Ty ŽxЉ .gx ŽЈ Ž.x 1 s Ý fxŽ.ٞTy Ž.Ž.xЉ gxЈ Ž.x 1 . s Ý gxŽ.ٞ fxŽ.ЉTy Ž.xЈ Žsince H is cocommutation Ž.x

y1 s g Ž.Id H cop* )H cop* Ž.Ž.Ž.T * fx. y1 y1 Hence gŽŽId H copcop* )H * T ..Ž.* f s ŽId H copcop* )H * ŽT ..Ž.* fg, it means y1 cop cop thatŽŽ Id H cop* )H cop* T ..Ž* H *.Ž: cH *.. 82 FANG LI

Thus, by Lemma 1.1Ž v . and Corollary 1.8, ŽŽŽ H cop..*op. * is a right Ž H cop.*-quasi-module-coalgebra. But by Lemma 1.9, ŽŽŽ H cop..**op. ␸ s cop cop ŽŽŽH . .** .s ŽH* . *, and H ( ŽH** .s H** as vector spaces by ␸ Žx . s x** such that x**Ž.f s fx Ž.for every f g H*. H** s ŽH**, m**, u**, ⌬**, ␧**, T**. . For every x, y g H, f g H*, wm**Ž.Ž.␸ m ␸ x m yfxŽ.s wm**Ž.x** m y** xŽ.f s Žx** m y** .m* Ž.f s ÝŽ f . x** ŽfЈ .y** ŽfЉ .s ÝŽ f .ŽfЈŽ.xfЉ Ž.y s ²Ý f . fЈ m fЉ , x m y :s ²m* Ž.f , x m y :s fxy Ž .s Ž.Ž.xy ** f s ␸ Ž.Ž.xy f s wŽ.Ž.␸mxm yfxŽ.. Hence m** Ž␸ m ␸ .s ␸m, i.e., ␸ is a ring morphism, and, for every k g K, ␸Ž.kx s Ž.kx **, k␸ Ž.x s kx**. But for f g H*, Ž.Ž.kx ** f s fkx Ž.s kf Ž. x s kx**. Hence ␸ Ž.kx s k␸ Ž.x , i.e., ␸ is a K-morphism. Thus, ␸ is a K-algebra morphism. For x g H, f, g g H*, wŽŽ.⌬**␸ xfxŽ.m g s Ž⌬**x** .Ž.f m g s x**⌬*Žf m g .s x** Ž.fg s Ž.Ž.fg x s w⌬*Ž.f m gxxŽ.s Žf m g .Ž.⌬ x s ÝŽ x.ŽfxŽ.Ž.Ј gxЉ s Ý x.Ž ŽxЈ** m xЉ** .Žf m g .s wÝ x. ␸Ž.xЈ m ␸ ŽxЉ .xŽ.f m g s wŽ.Ž.␸ m ␸ ⌬ xfxŽ.m g . Hence ⌬**␸ s Ž␸ m ␸ .⌬. Since ␧ g H*, ␧** g Ž.H* ** and for x g H, x** g Ž.H* *, we have ␧** Ž.x** s x** Ž.␧ for H*. Then Ž.Ž.Ž.Ž.Ž.␧**␸ x s ␧** x** s x** ␧ s ␧ x since ␧ g H*. Thus, ␧**␸ s ␧. Therefore, ␸ is a coalgebra morphism. For x g H, f g H*, ŽŽ␸Tx .Ž ..Ž f .s ␸ ŽTx Ž ..Ž f .s Tx Ž .** Žf . s fTŽ Ž x .., ŽŽT**␸ .Žxf ..Ž .s ŽT** Ž␸ Žxf ...Ž .s ␸ ŽxT .Ž* Žf ..s x** ŽT* Žf .. s T*Ž.Ž.fxs fT ŽŽ.. x . Hence ␸T s T**␸. Therefore, ␸ is a weak Hopf ␸y1 cop op algebra morphism. Thus H **s H** ( H as weak Hopf algebras. It cop follows that H is a right Ž H .²:*-quasi-module-coalgebra by a и f, g s y1 y1 cop ÝŽ f .²ŽŽa, fЉ gT ..* Ž.:fЈ for f, g g H *, a g H. 1 1 1 1 But ŽTy .Ž*T* s T и Ty .Ž* s 1, i.e., Ty .Ž.* s T*y . Now, ²:a и f , g s Ý²:a, fЉ gT*Ž.fЈ Ž.f

op op s Ý² a, ⌬ *Ž.⌬ * m 1 Ž.fЉ m g m T* Ž.fЈ : Ž.f

op op s Ý²Ž.Ž.Ž.⌬ m 1 ⌬ a , fЉ m g m T* fЈ : Ž.f s Ý ²:aٞ m aЉ m aЈ, fЉ m g m T*Ž.fЈ Ž.Ž.fa s Ý ²:²:aٞ, fЉ aЉ , ga²:Ј, T*Ž.fЈ Ž.Ž.fa s Ý ²:TaŽ.Ј , fЈ ²:²:aٞ, fЉ aЉ , g Ž.Ž.fa s Ý ²:TaŽ.Ј m aٞ, fЈ m fЉ ²:aЉ , g Ž.Ž.fa WEAK HOPF ALGEBRAS 83 s Ý²:TaŽ.Ј aٞ, fa²:Љ , g Ž.a

. s Ý²:fTaŽ.Ž.Ј aٞ aЉ , g Ž.a

.So, a и f s ÝŽ a. fTaŽŽЈ .aٞ .aЉ T op cop By Proposition 1.2, H ( H as weak Hopf algebras. Then copT* op op 1 Ž.Ž.H * ( H * as weak Hopf algebras. For f g H *, Ž.Ž.T* y f g cop op f 1 H *. We get a right action of Ž H .*on H given by a s a и Ž.T* y Ž.f y1 y1 y1 y1 .s a и ŽT .Ž* f .s a и fT s ÝŽ a.ŽfTŽŽ T aЈ .aٞ .aЉ s Ý a. fT Ž Ž.. aٞ aЈ aЉ op For f, g g H *, fg 1 1 a s ÝÝŽ.fgŽ. Ty Ž.aٞ aЈ aЉ s ⌬* Žf m gT . Ž.y Ž.aٞ aЈ aЉ Ž.aa Ž.

1 s Ý Ž.f m g ⌬Ž.Ty Ž.aٞ aЈ aЉ Ž.a

1 Ž5. 1 Ž4. s Ý Ž.Ž.f m gTŽ.y a aЈ m Ty Ž.a aЉ aٞ Ž.a

1 Ž5. 1 Ž4. s Ý fTŽ.Ž.y Ž.a aЈ gTy Ž.a aЉ aٞ Ž.a

1 g s Ý fTŽ.y Ž.aٞ aЈ aЉ Ž.a g 1 s ž/Ý fTŽ.y Ž. aٞ aЈ aЉ Ž.a g f s Ž.a . op Hence with this action, H is a right Ž H .Ž*-module. Since Tm s mTm T .␶ , for a, b g H, f g H*, Žm*T* .Žfa .Žm b .s T* Žfma .Ž Žm b .. s fTmaŽ Žm b ..; ␶ ŽT* m T* .m* Žfa .Žm b .s m* ŽfTb .Ž Ž .m Ta Ž ..s fmT Ž Ž m T .␶ Ža m b ..s fTma Ž Žm b ..s m*T* Žfa .Žm b .. Hence m*T* s ␶ ŽT* 1 1 1 m T*.m*. It follows that m*T*y s ␶ ŽT*y m T*y .m*. f y1 y1 ⌬Ž.a s ⌬Ž.a и ŽT* .Ž.f s ⌬ Ž.a и⌬H cop* Ž.T* Ž.f Ž.since H is a right H cop*-quasi-module-coalgebra ,

1 1 1 s ⌬Ž.a и m*Ž.T*y Ž.f s ⌬ Ž.a и ␶ ŽT*y m T*y .m* Ž.f 1 1 s ⌬Ž.a и ŽT*y m T*y .Ž␶ m* .Ž.f 1 1 op s ⌬Ž.a и ŽT*y m T*y .Žm .Ž.* f

⌬ op*Ž.f s ⌬Ž.a H . 84 FANG LI

op f Hence H is a right Ž H .*-quasi-module-coalgebra given by a s y1 .ÝŽ a. fTŽŽ.. aٞ aЈ aЉ

2. QUASI-BICROSSED PRODUCT AND SOLUTIONS OF THE QUANTUM YANG᎐BAXTER EQUATION

DEFINITION 2.1. A pair Ž.X, A of bialgebras is called quasi-matched if there exist linear maps ␣: A m X ª X and ␤: A m X ª A turning X into a left A-quasi-module-coalgebra and turning A into a right X-quasi- x module-coalgebra such that if one sets ␣Ž.a m x s a и x, ␤ Ž.a m x s a , the following conditions are satisfied:

xЉ a и Ž.xy s Ý ŽaЈиxЈ .Ž.aЉ и y Ž.Ž.ax

a и 1 s ␧ Ž.a 1 Ž 1 is the identity of X . x bЈи xЈ Ž.ab s Ý abЉ xЉ Ž.Ž.bx

x 1 s ␧ Ž.x 1 Ž 1 is the identity of A . xЈ xЉ ÝÝaЈ m aЉиxЉ s aЉ m aЈиxЈ Ž.Ž.ax Ž.Ž. ax for all a, b g A and x, y g X. DEFINITION 2.2. Given a K-linear space H, suppose that there are K-linear maps, m: H m H ª H, u: K ª H, ⌬: H ª H m H, ␧: H ª K such that Ž.H, m, u is a K-algebra, Ž.H, ⌬, ␧ is a K-coalgebra. If ⌬ is a K-algebra morphism, i.e., ⌬Žxy .s ⌬ Ž.Ž.x ⌬ y for every x, y g H, then one calls H an almost bialgebra.

THEOREM 2.3. LetŽ. X, A be a quasi-matched pair of bialgebras. There exists an almost bialgebra structure on the ëctor space X m A with unit equal to 1 m 1 such that its product is giën byŽ.Ž. x m aym b s ÝŽa.Ž y. xa Ž.ЈиyЈ yЉ m aЉ b, its coproduct by ⌬Ž.x m a s ÝŽ x.Ž a. ŽxЈ m aЈ .Ž.m xЉ m aЉ , and its counit by ␧Ž.Ž.Ž.x m a s ␧XAx ␧ a for all x, y g X and a, b g A. Equipped with this almost bialgebra structure, X m A is called the quasi-bicrossed product of X and A and denoted XϱA. Furthermore, the injectië maps ixXAŽ.s x m 1 and i Ž. a s 1 m a from X and from A into XϱA are bialgebra morphisms. Also x m a s Ž.Ž.x m 11m a for a g A and x g X.

Proof. For x g X, a g A, Ž.Ž.⌬ m 1 ⌬ x m a s ÝŽ a.Ž x. ŽxЈ m aЈ .Žm xЉ m aЉ .Žm xٞ m aٞ .Žs 1 m ⌬⌬ .Žx m a ., thus Ž⌬ m 1 .⌬ s Ž1 m ⌬⌬ .Ž. ␧ m 1 .и ⌬Ž.Ž.ŽŽx m a s ␧ m 1 ÝŽ x.Ž a.ŽxЈ m aЈ .Ž..m xЉ m aЉ s Ý x.Ž a. w␧Ž.xЈ m aЈ xŽ xЉ WEAK HOPF ALGEBRAS 85 maЉ .s ÝŽ x.Ž a.Ž␧ ŽxЈ .␧ ŽaЈ .ŽxЉ m aЉ .s Ý x.Ž a. ␧ ŽxЈ .xЉ m ␧ ŽaЈ .aЉ s x m a similarly Ž.Ž.1 m ␧ ⌬ x m a , thus Ž.Ž.␧ m 1 ⌬ s 1 m ␧ ⌬. Hence ŽXϱA, ⌬, ␧ . is a coalgebra.

Ž.Ž.Ž.Ž.x m aym bzm c

yЉ s ž/Ý xaŽ.ЈиyЈ m aЉ bz Ž.m c Ž.Ž.ay

yЉ yЉ zЉ s Ý xaŽ.Ž.ЈиyЈ aЉ b ЈиzЈ m Ž.aЉ b Љ c Ž.Ž.Ž.ayz yЉ y ٞ zЉ s Ý xaŽ.Ž.ЈиyЈ aЉ bЈиzЈ m Žaٞ bЉ .c Ž.Ž.Ž.ayz yЉ y ٞ bЉиzЉ z ٞ s Ý xaŽ.Ž.ЈиyЈ aЉ bЈиzЈ m Ž.aٞ bٞ c Ž.Ž.Ž.ayz yЉ y ٞŽ bЉиzЉ . z ٞ ;s Ý xaŽ.Ž.ЈиyЈ aЉ bЈиzЈ m aٞ bٞ c Ž.Ž.Ž.ayz

Ž.Ž.Ž.x m ayŽ.m bzm c zЉ s Ý Ž.Ž.x m aybЈиzЈ m bЉ c Ž.Ž.bz

Ž yŽbЈи zЈ..Љ zЉ s Ý xaŽ.ЈиŽ.ybŽ.ЈиzЈЈm aЉ bЉ c Ž.Ž.Ž.abz

yЉ Ž bЈи zЈ.Љ zЉ s Ý xaŽ.ЈиŽ.yЈŽ.bЈиzЈЈ m aЉ bЉ c Ž.Ž.Ž.Ž.ab yz yЉ y ٞŽ bЉиzЉ . z ٞ s Ý xaŽ.ЈиyЈ Ž.aЉ и Ž.bЈиzЈ m aٞ bٞ c Ž.Ž.Ž.Ž.ab yz yЉ y ٞŽbЉиzЉ . z ٞ s Ý xaŽ.Ž.ЈиyЈ aЉ bЈиzЈ m aٞ bٞ c Ž.Ž.Ž.Ž.ab yz

s Ž.Ž.Ž.Ž.x m aym bzm c ; xЉ Ž.Ž.1 m 1 x m a s ÝÝ11 Ž.и xЈ m 1 a s xЈ m ␧ Ž.xЉ a Ž.xx Ž.

s Ý xЈ␧ Ž.xЉ m a s x m a, Ž.x

1 Ž.Ž.x m a 1 m 1 s ÝÝxa Ž.Ји1 m aЉ s x␧ Ž.aЈ m aЉ s x m a. Ž.aa Ž. 86 FANG LI

Hence 1 m 1 is the identity of X m A.So XϱA is a K-algebra. ⌬Ž.Ž.x m a ⌬ y m b s Ý Ž.Ž.Ž.Ž.xЈ m aЈ m xЉ m aЉ yЈ m bЈ m yЉ m bЉ Ž.Ž.Ž.Ž.xyab

s Ý Ž.Ž.Ž.Ž.xЈ m aЈ yЈ m bЈ m xЉ m aЉ yЉ m bЉ Ž.Ž.Ž.Ž.xyab

yЉ Ž4. y Ž4. , s Ý xЈŽ.aЈиyЈ m aЉ bЈ m xЉ Žaٞиyٞ .m a bЉ Ž.Ž.Ž.Ž.xyab

⌬Ž.Ž.Ž.x m aym b yЉ yЉ s ⌬ ÝÝxaŽ.ЈиyЈ m aЉ b s ⌬Ž.xa Ž.ЈиyЈ m aЉ b Ž.Ž.ay Ž.Ž. ay

yЉ yЉ s Ý Ž.xaŽ.Ž.ЈиyЈЈm aЉ b Ј m Ž.xa Ž.Ž.ЈиyЈЉm aЉ b Љ Ž.Ž.ay

.y ٞ Ž4. y Ž4 s Ý xЈŽ.aЈиyЈ m aٞ bЈ m xЉ ŽaЉиyЉ .m a bЉ Ž.Ž.Ž.axy

yЉ Ž4. y Ž4. . s Ý xЈŽ.aЈиyЈ m aЉ bЈ m xЉ Žaٞиyٞ .m a bЉ Ž.Ž.Ž.axy

Hence ⌬Žx m a .⌬ Žy m b .s ⌬ ŽŽx m ay .Žm b ... It means that XϱA is an almost bialgebra.

yЉ ixyX Ž .s xy m 1 s Ý x Ž1 и yЈ .m 1 1 s Žx m 1 .Žy m 1 .s ixiyxx Ž. Ž., Ž.y

ikxxxŽ.s kx m 1 s kx Žm 1 .s ki Ž. x ,

⌬ixx Ž.s ⌬ Žx m 1 .s Ý ŽxЈ m 1 .m ŽxЉ m 1 . Ž.x

s Ý ixxxŽ.Ј m ix ŽЉ .s Ži xxm i .Ž.⌬ x , Ž.x

␧ ixx Ž.s ␧ Žx m 1 .s ␧ Ž.Ž.x ␧ 1 s ␧ Ž.x .

It follows that i xAis a bialgebra morphism. Similarly, i is a bialgebra morphism. At last,

1Љ Ž.Ž.x m 11m a s Ý x Ž.1Ји1Ј m 1Љ a s x m a for x g X, a g A. Ž.1 WEAK HOPF ALGEBRAS 87

Note that for two arbitrary bialgebras X and A with the quasi-bicrossed product XϱA, in general, XϱA is noncommutative as an algebra; and if either X or A is noncocommutative, XϱA is noncocommutative as a coalgebra. Hence we can construct many noncommutative and noncocom- mutive almost bialgebras in the form of the quasi-bicrossed product XϱA.

LEMMA 2.4. Let H s Ž.H, m, u, ⌬, ␧, T be a weak Hopf algebra with weak antipode T. Then ␧ и T s ␧. If T is in¨ertible and TŽ. ab s TbTa Ž.Ž. 1 for a, b g H, then ␧ и Ty s ␧.

Proof. Id)T )Id s Id « h s ÝŽ h. hЈThŽ.Љ hٞ for h g H. Then, ␧ Ž.h s ,ÝŽ h.Ž␧Ž.ŽŽ..Ž.hЈ ␧ ThЉ ␧ hٞ . But, h s Ý h.Ž␧ Ž.hЈ hЉ s Ý h. ␧ Ž.hЈ hЉ ␧ Žhٞ .. So ThŽ.s ÝŽ h.Ž␧ ŽhЈ .ŽThЉ .Ž␧ hٞ .« ␧ ŽŽ..Th s Ý h. ␧ ŽhЈ .ŽŽ␧ ThЉ ..Ž␧ hٞ .s ␧Ž.h for h g H. Hence ␧ и T s ␧. By Proposition 1.2, H op is a weak Hopf algebra with weak antipode Ty1. y1 y1 Then Id HH) opT ) H opId Hs Id H« h s ÝŽ h. hٞThŽ.Љ hЈ « ␧ Ž.h s y1 y1 ÝŽ h.Ž␧Ž.Žhٞ ␧ ThŽ..Ž.Љ ␧ hЈ . But h s Ý h. ␧ Ž.hٞ hЉ ␧ Ž.hЈ . So, ThŽ.s y1 y1 y1 ÝŽ h.Ž␧Ž.hٞ ThŽ.Ž.Љ ␧ hЈ « ␧ ŽThŽ..s Ý h. ␧ Žhٞ .Ž␧ ThŽ..Ž.Љ ␧ hЈ s ␧ Ž.h 1 for h g H. Hence ␧ и Ty s ␧.

THEOREM 2.5. Let H s Ž.H, m, u, ⌬, ␧, T be a finite dimensional co- commutatië weak Hopf algebra with T inërtible, Ž.Ž.Ž.T )Id H H : CH, ⌬T s ␶ Ž.T m T ⌬, and T Ž.Ž.Ž. ab s T b T a for a, b g H. Then for X s op op 1 Ž H .Ž* s H*, ⌬*,␧*, Žm .Ž*, u*, Ty ..*, the pair Ž X, H . of bialgebras is quasi-matched in the sense of Definition 2.1 with the linear maps ␣ and ␤ y1 f giën by ␣Ž.a m f s a и f s ÝŽ a. fT ŽŽ. aЉ ?aЈ .and ␤ Ža m f .s a s y1 y1 ÝŽ a. fTŽ Ž.. aٞ aЈ aЉ for a g H and f g X where f Ž TŽ. aЉ ?aЈ .means the 1 map x ª fTŽ y Ž. aЉ xaЈ .for x g H.

Proof. By Corollary 1.8, for a g H, f g X, ␣Ž.a m f s a и f s y1 op ÝŽ a. fTŽ Ž. aЉ ?aЈ .Ž, H .* is a left H-quasi-module-coalgebra; by Theo- f y1 op -*. rem 1.10, for ␤Ž.a m f s a s ÝŽ a. fTŽ Ž.. aٞ aЈ aЉ, H is a right ŽH quasi-module-coalgebra. For x g H and g g X,

f Љ f Љ ¦;ÝÝŽ.aЈиfЈ Ž.aЉ и g , x s Ž.Ž.aЈиfЈ xЈ Ž.aЉ и gx Ž.Љ Ž.Ž.af Ž.Ž. af

1 1 Ž5. Ž4. s Ý fЈŽ.Ty Ž.aЉ xЈaЈ Ž.fЉ Ž.Ty Ža .aٞ a и gx Ž.Љ Ž.Ž.Ž.af x 88 FANG LI

1 1 Ž5. Ž4. s Ý fЈŽ.Ty Ž.aЉ xЈaЈ fЉ Ž.Ty Ža .aٞ Ž.a и gx Ž.Љ Ž.Ž.Ž.af x

1 1 Ž6. 1 Ž5. Ž4. s Ý fЈŽ.Ty Ž.aЉ xЈaЈ fЉ Ž.Ž.Ty Ž.a aٞ gTy Ž.a xЉ a Ž.Ž.Ž.af x

1 1 Ž6. 1 Ž5. Ž4. s ÝÝ¦;fЈ m fЉ , Ty Ž.aЉ xЈaЈ m Ty Ž.a aٞ gTŽ.y Ž.a xЉ a Ž.Ž.ax Ž. f

op 1 1 Ž6. 1 Ž5.Ž4. s Ý ² Žm .Ž.Ž.* f , Tay Љ xЈaЈ m Taay Ž.ٞ:gTŽ.y Ž. a xЉ a Ž.Ž.ax

1 Ž6. 1 1 Ž5. Ž4. s Ý fTŽ.Ž.y Ž.a aٞTy Ž.aЉ xЈaЈ gTy Ž.a xЉ a Ž.Ž.ax

1 Ž4. 1 1 Ž5. Ž6. s Ý fTŽ.Ž.y Ž.a aٞTy Ž.aЉ xЈaЈ gTy Ž.a xЉ a Ž.Ž.ax Ž.since H is cocommutative

1 1 Ž4. ; s Ý fTŽ.Ž.y Ž.aЉ xЈaЈ gTy Ž.aٞ xЉ a Ž.Ž.ax and

1 1 ²:a и Ž.fg , x s ÝÝ Ž.fgŽ. Ty Ž. aЉ xaЈ s ²:⌬* Žf m g ., Tay Ž.Љ xaЈ Ž.aa Ž.

1 s Ý²:f m g, ⌬Ž.Ty Ž.aЉ xaЈ Ž.a

1 1 . s Ý ²:f m g, Ž.Ty Ž.aٞЈ m Ty Ž.Žaٞ xЈ m xЉ .ŽaЈ m aЉ Ž.Ž.ax

1 Ž4. 1 s Ý fTŽ.y Ž.a xЈaЈ gTŽ.y Ž.aٞ xЉ aЉ Ž.Ž.ax

1 1 Ž4. s Ý fTŽ.Ž.y Ž.aЉ xЈaЈ gTy Ž.aٞ xЉ a Ž.Ž.ax

f Љ s¦;Ý Ž.aЈиfЈ Ž.aЉ и g , x . Ž.Ž.af

f Љ Hence a и Ž.fg s ÝŽ a.Ž f . ŽaЈиfЈ .ŽaЉиg.. Žf␧ .Ž.x s Ž⌬* Žf m ␧ ..Ž.x s Žf m ␧ .Ž.⌬ x s ÝŽ x . fx ŽЈ .Ž␧ xЉ .s ÝŽ x.ŽfxŽЈ␧ ŽxЉ ..s f ŽÝ x. xЈ␧ ŽxЉ ..s fx Ž .. It follows that f␧ s f. Similarly, op ␧ f s f. It means that ␧ is the identity of Ž H .*. WEAK HOPF ALGEBRAS 89

1 1 a и ␧ s ÝÝ␧ Ž.Ž.Ty Ž.aЉ ?aЈ s ␧ Ty Ž.Ž.aЉ ␧ aЈ ␧ s Ý␧ Ž.Ž.aЉ ␧ aЈ ␧ Ž.aa Ž. Ž. a Ž.by Lemma 2.4

s Ý ␧ Ž.aЈ␧ Ž.aЉ ␧ s ␧ Ž.a ␧ . Ž.a

bЈи f Ј f Љ 1 f Љ ÝÝa bЉ s Ž.Ž.bЈиfЈ Ž.Ty aٞ aЈ aЉ bЉ Ž.Ž.bf Ž.Ž.Ž. abf

1 1 1 Ž5. Ž4. s Ý fЈŽ.Ty Ž.bЉ Ty Ž.aٞ aЈbЈ aЉ fЉ Ž.Ty Žb .bٞ b Ž.Ž.Ž.ab f

1 1 1 Ž5. Ž4. s Ý fЈŽ.Ty Ž.bЉ Ty Ž.aٞ aЈbЈ fЉ Ž.Ty Žb .bٞ aЉ b Ž.Ž.Ž.ab f

1 Ž5. 1 1 Ž4. s Ý fTŽ.y Žb .bٞTy Ž.bЉ Ty Ž.aٞ aЈbЈ aЉ b Ž.Ž.Ž.ab f

1 Ž4. 1 1 Ž5. s Ý fTŽ.y Žb .bٞTy Ž.bЉ Ty Ž.aٞ aЈbЈ aЉ b Ž.Ž.ab Ž.since H is cocommutative

1 1 s Ý fTŽ.y Ž.bЉ Ty Ž.aٞ aЈbЈ aЉ bٞ Ž.Ž.ab

1 f . s Ý fTŽ.y Ž. aٞ bٞ aЈbЈ aЉ bЉ s Ž.ab Ž.Ž.ab

f 1HHHkHs f Ž11 .s fuŽ. Ž.11s Ž.u* Ž.Ž.f 11 kHs u* Ž.f 1 H since u*Ž.f g K.

f Ј 1 1 Ž5. Ž4. ÝÝaЈ m aЉиfЉ s fЈŽ.Ty Ž.aٞ aЈ aЉ m fЉ Ž.Ty Ža .?a Ž.Ž.af Ž.Ž. af

1 1 Ž5. Ž4. . s Ý aЉ m fЈŽ.Ty Ž.aٞ aЈ fЉ ŽTy Ža ?a Ž.Ž.af

y1 Ž5. Ž4. y1 s ÝŽ a. aЉ m fTŽ.Ž.a ?a T Ž.aٞ aЈ

1 Ž5. 1 Ž4. s Ý aЉ m fTŽ.y Ž.aЈ ?a Ty Ža .aٞ Ž.a Ž.since H is cocommutative

1 1 ; s ÝÝaЉ m fTŽ.y Ž.aЈ ?aٞ s aٞ m fT Ž.y ŽaЉ .aЈ Ž.aa Ž. 90 FANG LI

f Љ 1 Ž5. Ž4. 1 ÝÝaЉ m aЈиfЈ s fЉ Ž.Ty Ž.a aٞ a m fЈŽ.Ty Ž.aЉ ?aЈ Ž.Ž.af Ž.Ž. af

Ž4. 1 1 Ž5. s Ý a m fЈŽ.Ty Ž.aЉ ?aЈ fЉ Ž.Ty Ža .aٞ Ž.Ž.af

Ž4. 1 Ž5. 1 s Ý a m fTŽ.y Ž.a aٞTy Ž.aЉ ?aЈ Ž.a

Ž5. 1 Ž4. 1 s Ý a m fTŽ.y Ž.a aٞTy Ž.aЉ ?aЈ Ž.a Ž.since H is cocommutative

1 f Ј . s ÝÝaٞ m fTŽ.y Ž.aЉ ?aЈ s aЈ m aЉиfЉ Ž.aaf Ž.Ž.

By Definition 2.1, we have shown that the pair ŽŽ H op..*, H of bialgebras is quasi-matched. In Theorem 2.5, when H is noncommutative, the quasi-bicrossed product Ž H op.*ϱH is a noncommutative and noncocommutative almost bialgebra since Ž H op.* is noncocommutative in this case.

It is well knownwx 2 that for an algebra A with identity 1 and R s Ýi, jis m tj g A m A, the equation RRR12 13 23s RRR 23 13 12 is called the quantum Yang᎐Baxter equation, briefly QYBE, where R12 s Ýsijm t m 1, R13 s Ýsijm 1 m t , R23 s Ý1 m sijm t . An element R g A m A satisfying the QYBE is called a quasi-R-matrix.

DEFINITION 2.6. Let H s Ž.H, m, u, ⌬, ␧ be an almost bialgebra. We call it almost quasi-cocommutati¨e if there exists an element R of the op algebra H m H such that for all x g H we have ⌬ Ž.xRs R⌬ Ž.x .An element R satisfying this condition is called a uni¨ersal quasi-R-matrix. DEFINITION 2.7. An almost quasi-cocommutative almost bialgebra H s Ž.H, m, u, ⌬, ␧ with a universal quasi-R-matrix R is called quasi-braided if R satisfies the two relations

Ž.Ž.⌬ m Id H R s RR13 23 and

Ž.Ž.Id H m ⌬ R s RR13 12 .

PROPOSITION 2.8. Let H s Ž.H, m, u, ⌬, ␧ be a quasi-braided almost bialgebra with a uni¨ersal quasi-R-matrix R. Then the uni¨ersal quasi-R-matrix R satisfies the QYBE, i.e., R is a quasi-R-matrix. WEAK HOPF ALGEBRAS 91

op Proof. We have ⌬ Ž.xRs R⌬ Ž.x for every x g H, Ž⌬ m Id H .Ž.R s RR13 23 andŽ.Ž. Id H m ⌬ R s RR13 12. Let R s Ýsijm t . Then RRR12 13 23 s R12Ž.Ž.Ž.Ž.Ž.⌬ m Id HHijiR s R m 1 ⌬ m Id Ýs m t s Ý , jijR⌬ Ž.s m t s op op Ýi, ji⌬ Ž.sRm t js Ž⌬ m Id H.ŽRR .12 s ŽŽ␶ m 1 .Ž⌬ m Id H .ŽRR .. 12 s ŽŽ␶m1 .RR13 23 . R 12 s ŽŽ␶m1 .ŽÝsijm1mt .ŽÝ1 m s ijmtR .. 12s RRR 23 13 12.

DEFINITION 2.9. Let H s Ž.H, m, u, ⌬, ␧, T be a finite dimensional cocommutative weak Hopf algebra with T invertible, Ž.Ž.T )Id H H : CHŽ., ⌬T s ␶ ŽT m T .⌬, and Tab Ž.s TbTa Ž.Ž.for a, b g H. The quasi- bicrossed product Ž H op.Ž*ϱH of H op.Ž* and H by Theorems 2.3 and 2.5 . is called the quantum quasi-double of H, denoted as DHŽ., i.e., DH Ž.s Ž H op.*ϱH.

LEMMA 2.10. Let H be as in Definition 2.9. Then the multiplication in op DHŽ.s ŽH .*ϱHisgi¨en by 1 Ž.Ž.fϱagϱb s Ý fgŽ. Ty Ž.aٞ ?aЈϱaЉ b Ž.a ␣ op 1 for f, g g Ž.H *, a, b g H where g ŽŽ.. Ty aٞ ?aЈ means the map: x ª .. gTŽ y1 Ž. aٞ xaЈ 1 1 . Proof. For k g K, k␣Ž.x s kg Ž Ty Ž. aٞ xaЈ .s gT Žy Ž. aٞ kxaЈ .s ␣ Ž.kx y1 y1 For x12, x g H, ␣Ž.Žx 1q x 2s gTŽ.Ž aٞ x 1q xa 2 ..Ј s gT Ž Ž. aٞ xa 1Ј q y1 y1 y1 . TaŽ.ٞ xa21212Ј .s gT Ž Ž. aٞ xaЈ .q gT Ž Ž..Ž.Ž. aٞ xaЈ s ␣ x q ␣ x op Hence ␣ g Ž H .*. g Љ Ž.Ž.fϱagϱb s Ý fa ŽЈиgЈϱ .aЉ b Ž.Ž.ag

1 1 Ž5. Ž4. s Ý fgЈŽ.Ty Ž.aЉ ?aЈϱgЉ Ž.Ty Ža .aٞ a b Ž.Ž.ag

1 1 Ž5. Ž4. s Ý fgЈŽ.Ty Ž.aЉ ?aЈ gЉ Ž.Ty Ža .aٞϱa b Ž.Ž.ag

1 Ž5. 1 Ž4. s Ý fgŽ. Ty Ž.a aٞTy Ž.aЉ ?aЈϱa b Ž.a

1 Ž4. 1 Ž5. s Ý fgŽ. Ty Ž.a aٞTy Ž.aЉ ?aЈϱa b Ž.a Ž.since H is cocommutative 1 s Ý fgŽ. Ty Ž.aЉ ?aЈϱaٞ b Ž.a

1 .s Ý fgŽ. Ty Ž.aٞ ?aЈϱaЉ b Ž.a 92 FANG LI

Now we have the following main result:

THEOREM 2.11. Let H s Ž.H, m, u, ⌬, ␧, T be a finite dimensional co- commutatië weak Hopf algebra with T inërtible, Ž.Ž.Ž.T )Id H H : CH, ⌬T s ␶ Ž.T m T ⌬, and T Ž.Ž.Ž. ab s T b T a for a, b g H. Then the quantum quasi-double DŽ. H of H is quasi-braided equipped with the quasi-R-matrix Ž.Žϱ iϱ .Ž.Ž. Ä4 R s Ýig Ii1 e m e 1 g DH m D H where e iig I is a basis of the Ä i4 Ž op. K-ëctor space H together with its dual basis eig I in H *. Hence R is a solution of the QYBE. Proof. We must prove

Ž.Ž1 ⌬ m Id H .Ž.R s RR13 23 and Ž Id H m ⌬ .Ž.R s RR13 12 , op op Ž.2 for all f g ŽH .*, a g H, we have ⌬ Ž.fϱaRs R⌬ Ž.fϱa .

i Ž.Ž.Ž.Ž.Ž.⌬ m Id HHiR s Ý ⌬ m Id 1ϱe m e ϱ1 igI X Y i s ÝÝŽ.Ž.Ž.1ϱeiim 1ϱe m e ϱ1, igI Ž.ei

RR Ž.Ž.Ž.1ϱe 1ϱ1 eiϱ11 Ž.Ž.Ž.ϱ1 1ϱe eiϱ1 13 23 s ž/ž/ÝÝiim m m m igIigI i j s Ý Ž.1ϱeijm Ž.1ϱe m Že e ϱ1. . i, jgI

For every a, b, c g H, t, u, ¨ g H*, let ␪ s a m t m b m u m c m ¨. Then

XYi ²:Ž.Ž.Ž.Ž.Ž.Ž.Ž.Ž.⌬ m Id H R , ␪ s ÝÝ1 ate1 1 buei e c¨ 1 igI Ž.ei

X Y i s ÝÝ␧ Ž.Ž.Ž.Ža ␧ bteiiue .e Ž.Ž.c ¨ 1 igI Ž.ei

op because 1 s ␧ inŽ H. *. Since c Ý eiŽ. c e , we get Ý cЈ cЉ ÝÝeceiŽ.XYe and s ig Ii Žc. m s ig I Že i. iim Ý tcŽ.Ž.Ј ucЉ ÝÝte ŽiŽ. ceX .Ž ueY . ÝÝeiŽ.Ž c t eX .Ž u eY . . So, Žc. s ig I Že ii. iis ig I Že . ii

i XY ²:Ž.Ž.Ž.Ž.Ž.Ž.Ž.Ž.⌬ m Id HiiR , ␪ s ␧ a ␧ b ¨ 1 ÝÝecteue igI Ž.ei

s ␧ Ž.Ž.Ž.a ␧ b ¨ 1 Ý tc ŽЈ .ŽucЉ .; Ž.c WEAK HOPF ALGEBRAS 93

ij ²:RR13 23 , ␪ s Ý 1Ž.Ž.Ž.Žateij1 bue .Ž ee .Ž.Ž. c¨ 1 i, jgI i j s ␧ Ž.Ž.Ž.a ␧ b ¨ 1 Ý te Žij .Ž ue .Ž ee .Ž.c i, jgI i j s ␧ Ž.Ž.Ž.a ␧ b ¨ 1 ÝÝte Žij .Ž ue . e ŽcЈ .e ŽcЉ . i, jgI Ž.c i j s ␧ Ž.Ž.Ž.a ␧ b ¨ 1 ÝÝteŽ. ŽcЈ .eueijŽ. ŽcЉ .e i, jgI Ž.c s ␧ Ž.Ž.Ž.a ␧ b ¨ 1 Ý tc ŽЈ .ŽucЉ .. Ž.c

Hence ²Ž⌬ m Id H .ŽR ., ␪ :s ²RR13 23, ␪ :. It follows that Ž⌬ m Id H .ŽR . s RR13 23.

²:Ž.Ž.Id H m ⌬ R , ␪

Ž.Ž.Ž1ϱe eiЈϱ1 eiЉϱ1, .a t b u c s¦;ÝÝ i m m m m m m m ¨ igI Ž.e i i i s ÝÝ1Ž.Žatei . eЈ Žbu . Ž.1 e Љ Ž.c ¨ Ž.1 igI Ž.e i

i i s ␧ Ž.au Ž.1 ¨ Ž.1 ÝÝte Ži . eЈ Žbe .Љ Ž.c . igI Ž.e i Ž.i ⌬Ž. Ј Љ It is easy to see that t s Ýig Iite e, then t s ÝŽt. t m t s ii i Ž.Ј Љ ²Ž⌬ .Ž .␪ : ÝÝi g I Ž e . teiH e m e . It follows that Id m R , s ␧Ž.Ž.Ž.au1 ¨ 1 ÝŽt. tЈ Ž.btЉ Ž.c ; but ji ²:RR13 12 , ␪ s Ý 1Ž.Žateeij . e Ž bu . Ž.1 ec Ž.¨ Ž.1 i, jgI j i s ␧ Ž.au Ž.1 ¨ Ž.1 Ý teee Žij . Žbe . Ž.c i, jgI op j i s ␧ Ž.au Ž.1 ¨ Ž.1 Ý Žm .Ž.* teŽ.jim ee Žbe . Ž.c i, jgI j i s ␧ Ž.au Ž.1 ¨ Ž.1 ÝÝtЈ Žetji .Љ Žee . Žbe . Ž.c i, jgI Ž.t j i s ␧ Ž.au Ž.1 ¨ Ž.1 ÝÝtЈŽ.e Žbe .ji tЉ Ž.e Ž.ce i, jgI Ž.t s ␧ Ž.au Ž.1 ¨ Ž.1 Ý tЈ Žbt .Љ Ž.c Ž.t

s²:Ž.Ž.Id H m ⌬ R , ␪ .

HenceŽ.Ž. Id H m ⌬ R s RR13 12. We have proved Ž. 1 . 94 FANG LI

op For f g Ž H .*, a g H,

⌬opŽ.fϱaR

i s ÝÝ Ž.Ž.Ž.Ž.fЉϱ aЉ m fЈϱaЈ 1ϱei m e ϱ1 igI Ž.Ž.fa

i s ÝÝŽ.Ž.Ž.Ž.fЉϱ aЉ 1ϱei m fЈϱaЈ e ϱ1 igI Ž.Ž.fa

y1 Ž6.Ž4.Ž5. i y1 s ÝÝ fЉ ␧ Ž.TaŽ.?a ϱaei m fЈeTŽ. Ž. aٞ ?aЈϱaЉ igI Ž.Ž.fa

y1 Ž6.Ž4.Ž5. s ÝÝ fЉ ␧ Ž.TaŽ.Ž.␧ a ␧ϱaei igI Ž.Ž.fa

i 1 m fЈeTŽ.y Ž. aٞ ?aЈϱaЉ

Ž6.Ž4.Ž5. i y1 s ÝÝ fЉ ␧ Ž.Ž.a ␧ a ␧ϱaei m fЈeTŽ. Ž. aٞ ?aЈϱaЉ igI Ž.Ž.fa

1 Ž6. Ž6. Ž.since ␧ Ž.Ty Ž.a s ␧ Ž.a by Lemma 2.4

Ž4. i y1 s ÝÝŽ.fЉϱ aei m fЈeTŽ.Ž. aٞ ?aЈϱaЉ igI Ž.Ž.fa

Ž6. Ž4. Ž5. Ž4. ž/since Ý ␧ Ž.Ž.a ␧ a a s a and fЉ ␧ s fЉ . Ž.a

op For every b, c g H, u, ¨ g Ž H .*, let ␰ s b m u m c m ¨. Then

² ⌬opŽ.fϱaR, ␰:

Ž4. i y1 . s ÝÝfЉ Ž.buaŽ. ei fЈeTŽ. Ž aٞ .?aЈ Ž.Žc ¨ aЉ igI Ž.Ž.fa

Ž4. i y1 . s ÝÝfЉ Ž.buaŽ.efi Ј ŽcЈ .e Ž.T Žaٞ .cЉ aЈ ¨ ŽaЉ igI Ž.Ž.Ž.cfa

Ž4. i y1 s ÝÝfbcŽ.Ј uaŽ.e Ž.T Ž.aٞ cЉ aЈ ei ¨ Ž.aЉ igI Ž.Ž.ac

Ž4. 1 s Ý fbcŽ.Ј uaŽ.Ty Ž.aٞ cЉ aЈ ¨ Ž.aЉ Ž.Ž.ac

Ž4. 1 . s Ý fbcŽ.Ј ucŽ.Љ a Ty Ž.aٞ aЈ ¨ Ž.aЉ Žby Lemma 1.1 Ž.Ž.ac WEAK HOPF ALGEBRAS 95

Ž4. 1 s Ý fbcŽ.Ј ucŽ.Љ a Ty Ž.aٞ aЉ ¨ Ž.aЈ Ž.Ž.ac Ž.since H is cocommutative s Ý fbcŽ.Ž.Ž.Ј ucЉ aЉ ¨ aЈ Ž.Ž.ac

s Ý fbcŽ.Ž.Ž.Ј ucЉ aЈ ¨ aЉ . Ž.Ž.ac

R⌬Ž.fϱa Ž1ϱe .Ž.eiϱ1 ŽfЈϱaЈ .ŽfЉϱ aЉ . s ž/ÝÝi m ž/m igI Ž.Ž.fa i s ÝÝŽ.Ž.Ž.Ž.1ϱefi ЈϱaЈ m e ϱ1 fЉϱ aЉ igI Ž.Ž.fa

y1 Z X Y i y1 s ÝÝŽ.␧ fЈŽ.T Ž.eiii?e ϱe aЈ m Ž.e fЉ Ž.T Ž.1?1ϱ1aЉ igI Ž.Ž.Ž.faei

y1 Z X Y i s ÝÝŽ.fЈŽ.T Ž.eiii?e ϱe aЈ m Ž.e fЉϱ aЉ . igI Ž.Ž.Ž.faei Then

y1 ZX Y i ²:R⌬Ž..fϱa , ␰ s ÝÝfЈŽ.T Ž. eii be u Ž. e i aЈ Ž.efЉ Ž.Ž.c ¨ aЉ igI Ž.Ž.Ž.faei

y1 Z X Y i s ÝÝfЈŽ.T Ž.eiibe ue Ž iaЈ .Ž.Ž.Ž.e cЈ fЉ cЉ ¨ aЉ igI Ž.Ž.Ž.Ž.faeci

y1 Z X Y i s ÝÝfcŽ.ЉT Ž.eiibe ue Ž iaЈ .Ž.Ž.e cЈ ¨ aЉ . igI Ž.Ž.Ž.aei c . But c Ý eceiŽ. . Then Ý cЈ cЉ cٞ ÝÝeceiŽ.XYZe e s ig Ii Žc. m m s ig I Žei. iim m i Hence moreover, Ž4. 1 R⌬Ž.fϱa , ␰ s Ý fcŽ. Ty Ž. cٞ bcЈ uc ŽЉ aЈ .Ž.¨ aЉ:² Ž.Ž.ac Ž4. 1 s Ý fbcŽ.Ty Ž.cٞ cЈ uc ŽЉ aЈ .Ž.¨ aЉ Ž.Ž.ac Ž4. 1 s Ý fbcŽ.Ty Ž.cٞ cЉ uc ŽЈaЈ .Ž.¨ aЉ Ž.Ž.ac

s Ý fbcŽ.Ž.Ž.Љ ucЈaЈ ¨ aЉ Ž.Ž.ac

s Ý fbcŽ.Ž.Ž.Ј ucЉ aЈ ¨ aЉ Ž.Ž.ac op s² ⌬ Ž.fϱaR, ␰:. op Therefore, R⌬Ž.fϱa s ⌬ Ž.fϱaR, i.e., Ž. 2 is true. 96 FANG LI

Now we give an example of the weak Hopf algebras satisfying the conditions of Theorem 2.11 so as to construct a solution of the QYBE. The following facts on semigroups are well knownŽ seewx 4, 9. . We know that a semilattice Y is a commutative idempotent semigroup, 2 i.e., ␣␤ s ␤␣ and ␣ s ␣ for every ␣, ␤ g Y. The natural ordering of Y is given by ␣ F ␤ m ␣ s ␣␤ s ␤␣. When <homomorphism and assume that ␸ Id , ␣, ␤␣ª ␤ ␣, ␣ s G␣ ␸␸ ␸ ␣ ␤ ␥ ␣, ␤␤, ␥␣s , ␥␣if G G . On the set S s D g Y G␣ define a multiplication by a) b s Ž.Ž.a␸␣, ␣␤b␸ ␤, ␣␤if a g G ␣, b g G ␤ and denote S s wxY; G␣ , ␸␣ , ␤ which is called a strong semilattice Y of groups G␣ . It is easy to see that the multiplication ) coincides with the given one on each Ga and that S is a semigroup with the multiplication ). Define a Clifford semigroup wx9 to be a strong semilattice of groups. A semigroup S is a Clifford semigroup if and only if S is regular and all its idempotents lie in its center CSŽ.Žseewx 4. . It is easy to see that a Clifford semigroup S s wxY; G, ␸␣, ␤ is finite with identity if and only if <

Ž.iii ŽT )Id Hiii .Ž.KS : CKS Ž ..Infact,for x s Ý ka g KS, y1 y1 Ž.Ž.T )Id Hiiiiiix s Ý ka a,but aag ESŽ.: CS Ž.: CKS Ž .,so Ž.Ž.Ž.T )Id H x g CKS; y1 y1 Ž.iv ⌬Tx Ž.s Ýiiika m a i s ␶ Ž.Ž.T m T ⌬ x , then ⌬T s ␶ ŽT m T .⌬;

Ž.vLety s Ý jjlb jwhere l jg K, b jg S.ThenTxy Ž .s y1 y1 y1 y1 y1 Ýi, jijklŽ. ab ijs Ý i, jijjklb a is ŽÝ jjjlb .ŽÝ iiika .Ž.Ž.s TyTx since 1 1 1 Ž.ab y s bay y for a, b g S Žseewx 4. . WEAK HOPF ALGEBRAS 97

According toŽ. i ᎐ Žv, . KS is a weak Hopf algebra satisfying Theorem 2.11. By Theorems 2.3, 2.5, and Definition 2.9, the quantum quasi-double DKSŽ.of KS exists. We denote DS Ž.Ž.s DKS and call DS Ž.the quantum quasi-double of the semigroup S. By Theorem 2.11, DSŽ.is quasi- braided. U S s Ä4s12, s ,...,sniis a basis of KS. Let s : KS ª K such that

U 1, i s j ssŽ. ijs ½ 0, i / j

U U U op and sxijjijjjjiŽ.s Ý ksŽ. a for x s Ý ka. Then s g wŽ.KS x* Ž.i s 1,...,n UU U op and Ä s12, s ,...,sn 4 is the dual basis in wŽ.KS x*. By Theorem 2.11, n Ž.Žϱ Uϱ .Ž. R s Ýis1 1 siim s 1 is the quasi-R-matrix of DS; then it is a solution of the QYBE. Thus, we get the result on the quantum quasi- double of a Clifford monoid algebra:

THEOREM 2.12. Let S s Ä4s1,...,sn be a finite Clifford monoid. Then o¨er a field K, the quantum quasi-double DŽ. S of S exists and is quasi-braided n Ž.Žϱ Uϱ .Ž.Ž. equipped with the quasi-R-matrix R s Ýis1 1 siim s 1 g DS m DS. It follows that R is a solution of the QYBE.

It is known that for an algebra A with identity 1 and R g A m A, R is called an R-matrix wx6Ž resp. uni¨ersal R-matrix wx6if. R is a quasi-R-matrix Ž.resp. universal quasi-R-matrix and is invertible in A m A. In Theorem 2.11, we get a quasi-braided quantum quasi-double DHŽ.with the quasi- R-matrix R. We cannot show whether this R is an R-matrix, i.e., in general, as a solution of the QYBE, this R may not be invertible in DHŽ.m DH Ž.. But in the special case of Theorem 2.12, we have the following weaker result.

DEFINITION 2.13wx 5 .Ž. i Let A be an algebra and A␣ be the subalge- ␣ ⍀ ⍀ bra of A for g .If is a semilattice, A s Ý␣ g ⍀ A␣␣␤, and AA: A␣␤␣, then A is called the semilattice sum of A for ␣ g ⍀.

Ž.ii If A is the semilattice sum of the subalgebra A␣ Ž␣ g ⍀ .and A␣ l Ý␤ / ␣␤A s Ä40 for each ␣ g ⍀, then A is called the supplementary semilattice sum of the subalgebra A␣ Ž.␣ g ⍀ . Obviously, if A is theŽ. supplementary semilattice sum of the subalgebra A␣ Ž.␣ g ⍀ and B is an algebra, then A m B is the Ž supplementary . semilattice sum of the subalgebra A␣ m B Ž.␣ g ⍀ . It is easy to prove that if S s wxY; G␣ , ␸␣ , ␤ is a Clifford semigroup, then the semigroup algebra KS is a supplementary semilattice sum of the group op op algebras KG␣ Ž.␣ g Y . Moreover, W s KS m wŽ.KS xw* m KS m Ž.KS x* is a supplementary semilattice sum of the subalgebras Wa s KS m op op wŽ.KS xw* m KG␣ m Ž.KS x* Ž.␣ g Y . 98 FANG LI

LEMMA 2.14. Let S s wxY; G␣ , ␸␣ , ␤ be a finite Clifford monoid. Then for op e¨ery ␣ g Y, wŽ.KS x*ϱKG␣ is a subalgebra of DŽ. S .

op Proof. It is enough to prove that wŽ.KS x*ϱKGa is closed under the op multiplication of DSŽ.. In fact, for f, g g wŽ.KS x*, x, y g KG␣ , by Lemma y1 Ž.Ž.fϱxgϱy s ÝŽ x. fg Ž TŽ. xٞ ?xЈϱ .xЉ y. Let x s Ýiiika. Then a ig ,2.10 G␣ for all i and ⌬Ž.x s Ýiiikam a i. Hence xЉ y g KG a.

PROPOSITION 2.15. Let S s wxY; G␣ , ␸␣ , ␤ s Ä4s1,...,sn be a finite Clif- n Ž.Žϱ Uϱ .Ž. ford monoid. Then for the quasi-R-matrix R s Ýis1 1 siim s 1 of D S , Ž. Ž. <<Ž.< there exists R g DS m D S such that RR WW␣ s RR ␣ s 1ϱ1␣ m 1ϱ1 Wa where 1␣ is the identity of the group G␣␣Ž.␣ g Y and 1ϱ1 m 1ϱ1 is the op identity of the subalgebraŽw KS. x*ϱKG␣ .Ž.Ž.Ž.m DS ofDS m DS.

op Proof. Obviously, 1ϱ1␣ m 1ϱ1 is the identity of wŽ.KS x*ϱKG␣ m DSŽ.. n Ž.ŽŽ..ϱ U и ϱ y1 Let R s Ýis1 1 siim s T 1 where T: KS ª KS such that s ª s for every s g S. Consider every element ␰ s b m u m c m ¨ g W␣ . Then c s Ýiiikcg KG␣ where ci g G␣ . ⌬Ž.c s Ýiiikcm c i.

n ²:RR, ␰ Ž.1ϱss ssUU Ž.и T ϱ1,b u c s¦;Ý ijm Ž. i j m m m ¨ i, js1 n UU s Ý ␧ Ž.Žbussij .Ž. s iŽ. s jи Tc Ž.¨ Ž.1 i, js1 n UU s ␧ Ž.Ž.b ¨ 1 Ý usž/Ž.ijŽ. sи Tc Ž.Ž ss ij . i, js1 n ␧ Ž.Ž.b 1 uscUU ŽЈ .sTcŽ. ŽЉ . Žss . s ¨ ÝÝž/ij ij i, j 1 ž/Ž.c s

␧ Ž.Ž.b 1 usU ŽcЈ .ssU Ž.Tc ŽЉ .s s ¨ ÝÝž/iiž/ Ý j j Ž.c ž/ij

s ␧ Ž.Ž.b ¨ 1 ÝÝucŽ.ЈTc ŽЉ .s ␧ Ž.Ž.b ¨ 1 kuiiŽ. cT Ž c i . Ž.c i

y1 s ␧ Ž.Ž.b ¨ 1 Ý kuiiiŽ. cc . i

X Let 1␣ be the identity of G␣ and ciithe inverse of c in G␣ . Then X X X X X X X y1 y1 cciis c iic s 1␣ « cciiic s c i, c iiiccs c i« c is c i. So, cc ii s 1␣ and y1 cciis 1forevery␣ i.Hence²:Ž.Ž.Ž.RR, ␰ s ␧ b ¨ 1 Ýiiku1␣ s WEAK HOPF ALGEBRAS 99

Ž.Ž.Ž.Ž . ² : < Ž ␧ b ¨ 1 ␧ cu1␣␣s 1ϱ1 m 1ϱ1, ␰ . It follows that RR W␣ s 1ϱ1 ␣m .< 1ϱ1,W␣

t UU ²:RR, ␰ s Ý Ž.1ϱssijm Ž..s iи Ts jϱ1,b m u m c m ¨ ¦;i, j 1 s n UU s Ý ␧ Ž.Ž.Žb ¨ 1 ussij .ŽŽ. s iи Ts . j Ž. c i, js1 n UU s Ý ␧ Ž.Ž.b ¨ 1 usŽ.Ž. Žijи Ts . Ž.Ž c ss ij . i, js1 n ␧ Ž.Ž.b 1 us ŽUUи Tc .Ž.Ј sc ŽЉ .ss s ¨ ÝÝž/ijij i, js1 Ž.c

nn UU s ␧ Ž.Ž.b ¨ 1 ÝÝusTciijjŽ. ŽЈ .ssc Ý ŽЉ .s Ž. ž/i 1 j 1 c ž/ ž/s s

s ␧ Ž.Ž.b ¨ 1 ÝÝuTŽ. Ž cЈ .cЉ s ␧ Ž.Ž.b ¨ 1 kuTiiiŽ. Ž c . c Ž.c i

y1 s ␧ Ž.Ž.b ¨ 1 ÝÝkuiiiŽ. c c s ␧ Ž.Ž.b ¨ 1 ku ia Ž1 . ii

s ␧ Ž.Ž.Ž.Žb ¨ 1 ␧ cu1␣ .s ²:1ϱ1␣ m 1ϱ1, ␰ . < Ž.< Thus, RR WW␣ s 1ϱ1␣ m 1ϱ1.␣ We say that in Proposition 2.15, R is partially in¨ertible and R is the partial in¨erse of R.

ACKNOWLEDGMENTS

I take this opportunity to express my thanks to Professor Xu YonghuaŽ. Fudan University for his kind assistance in the initial stage of my Hopf algebra study.

REFERENCES

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