JOURNAL OF ALGEBRA 197, 506᎐520Ž. 1997 ARTICLE NO. JA977117
Cosemisimple Bialgebras and Discrete Quantum Semigroups
Eduard Vaysleb
Department of Mathematics, Uni¨ersity of California, Los Angeles, California 90095-1555
Communicated by Susan Montgomery
Received December 20, 1996
We show for a cosemisimple bialgebra that a standard *-operation making it into a discrete quantum semigroup must be unique. It may not exist: we prove such an
operation on a cosemisimple OqŽŽ..SL2 ރ exists if and only if the parameter q is real. We also conclude that discrete quantum groups form a more restrictive class than cosemisimple *-Hopf algebras. ᮊ 1997 Academic Press
INTRODUCTION
Let A s Ž.A, m,1,⌬, be a bialgebra over ރ, i.e., a vector space with a multiplication m, a multiplicative identity 1, a comultiplication ⌬, and a counit . It is called cosemisimple wxSw, Sect. 14; Mon, Sect. 2.4 if as a coalgebra it is a direct sum of simple subcoalgebras. d Recall that on the full dual algebra M s A of any bialgebra A the multiplication is definedwx Sw, Sect. 1.1 by ²:²xy, a s x m y, ⌬Ž.a :, where x, y g M; a g A. DEFINITION 1. Let M be an associative algebra over ރ. We say it is a *-algebra if there is a conjugate linear involution x ª x*Ž called a *-oper- ation. on M such that 1* s 1, Ž.xy * s y*x*, Ž.x**sx for all x, y g M. The *-operation on M s Ž.M, ) is called standard Žor a standard involu- tion. if for x g M, x*x s 0 « x s 0. 506
0021-8693r97 $25.00 Copyright ᮊ 1997 by Academic Press All rights of reproduction in any form reserved. COSEMISIMPLE BIALGEBRAS 507
Followingwx ER we define a *-operation and a co-*-operation on a bialgebra.
DEFINITION 2. Let A be a bialgebraŽ. not necessarily cosemisimple . A conjugate linear involution ) : A ª A is called a *-operation if it is an algebra antihomomorphism and a coalgebra homomorphism, i.e.,
Ž.i ŽA,) .is a *-algebra; Ž.ii Ž.a* s Ž.a , ⌬ Ž.a* s ŽŽ) m )( .⌬ .Ž.a for all a g A. A conjugate linear involution q : A ª A is called a co-*-operation if it is an algebra homomorphism and a coalgebra antihomomorphism, i.e.,
Ž.i1qs1, Ž.ab qqqs ab,Žaq.qsa; Ž.ii Žaqq .s Ž.a , ⌬ Ža .s Ž ( Žqmq .(⌬ .Ž.a for all a, b g A; where : A m A ª A m A is the usual flip carrying a m b to b m a. A*-Hopf algebra is defined as a bialgebra with a *-operation which has an antipode. In the case when A is a Hopf algebra with a bijective antipode S we have the followingwx ER, Lemma 7.1 : * is a *-operation if 2 and only if qs S() is a co-*-operation, in particular, Ž.S() s id. Recall that the antipode S of a cosemisimple Hopf algebra is bijective wxLar . So, in the cosemisimple situation a *-Hopf algebra can be also defined as Hopf algebra with a co-*-operation. Given a co-*-operation on a bialgebra A we see immediately that its d Ž.conjugate linear dual defined on M s A by
²:²:x*, a s x, aq ; x g M, a g A,1Ž. turns M into a *-algebra. We say that a co-*-operation on a cosemisimple bialgebra A is standard if its dual is standard in the sense of Definition 1.
DEFINITION 3. A discrete quantum semigroup is a cosemisimple bialge- bra with a standard co-*-operation. Inwx ER a discrete quantum group is defined as a cosemisimple Hopf algebra A with a standard co-*-operation. In other words, a discrete quantum group is a discrete quantum semigroup which has an antipode. A basic nonclassical example of such is a dual of a compact quantum group Žseewx ER, Sect. 10. . Let us note that finite-dimensional discrete quantum groups are exactly finite-dimensional Kac algebrasŽ studied as ‘‘ring groups’’ by George Kac and his students in the 1960swx KacPal. . We will give other finite-dimensional examples of discrete quantum semigroups in the next section. 508 EDUARD VAYSLEB
There are two natural questions asked inwx Eff : Does every cosemisimple Hopf algebra have a discrete quantum group structure? Is such a structure unique? In this paper we prove a more general uniqueness result: if the co-*-op- eration on a cosemisimple *-bialgebra is standard then it must be unique up to a bialgebra isomorphism. As a special case it implies uniqueness of a standard co-*-operation for a discrete quantum groupᎏessentially the result inwx And . In the finite-dimensional case we believeŽ and all the known examples support it. that there is no difference between cosemisimple Hopf algebras and discrete quantum groupsŽ. or Kac algebras . However, if a cosemisimple Hopf algebra is infinite-dimensional a ‘‘good’’ co-*-operation which would make it into a discrete quantum group may not exist. We prove this on the example of OqŽŽ..SL2 ރ : this Hopf algebra is cosemisimple whenever q is not a root of unitywx MMNNU ; 1 however, it has *-operations only for q g ޒ j S ; and only one of these Ž.with q real is standard. Our result also shows that cosemisimple bialge- bras form a larger class than discrete quantum semigroups.
1. EXAMPLES
EXAMPLE 1wx Sw, Sect. 3.2 . Given a semigroup G with the unit 1 we define in a usual way the cocommutative bialgebra A s ރwxG . It has a linear basis Ä4g: g g G ; in this basis the multiplication is defined as the semigroup multiplication in G; and we set up the cocommutative coalge- bra structure by writing Ž.g s 1, ⌬ Ž.g s g m g for all g g G. Then d ރŽ. Ł ރ MsAcan be identified with the G s g g G , i.e., the algebra of all complex functions on G; its one-dimensional matrix blocks are generated by the delta-functions ␦g . Suppose there is an antipode S : A ª A. Then for every basis element ggGwe have ⌬Ž.g s g m g and the map S must satisfy Sgg Ž.sgS Ž. g sŽ.g1s1. Thus if some g in G does not have an inverse Ž i.e., G is not a group. , the bialgebra A does not have an antipode, and so A is not a Hopf algebra. The natural co-*-operation on A is given by gqs g, g g G.Itis Ž. U standard as its dual on M s ރ G is defined by ␦ggs ␦ . Indeed for Ž. Ž. xsxgg,ggGwe write x* s x , g g G and so for each g g G
²: 2 x*x,gs< In the finite-dimensional situation A is a cosemisimple bialgebra if and d only if M s A is a semisimple bialgebraŽ see, e.g.,wx Mon, Sect. 5.1. . So we can directly define a discreteŽ. or, rather, finite quantum semigroup in this case as a semisimple bialgebra with a standard *-operation as in Defini- tion 1. Ž. EXAMPLE 2. Let us take M s C [ M2 ރ ᎏobviously it is a semisimple algebra with identity. Denote the elements of M as Ž., x where g ރ Ž. Ž. Ž . Ž. and x g M2 ރ . Now set E s 1, 0 , x s 0, x , and let us have 1 s 0, 1 . Then the identity is I s E q 1. Define the counit on M by Ž.,xs and the comultiplication by ⌬Ž.E s E m E, ⌬Ž.x s E m x q x m E q x m 1. Then the axioms of bialgebra are easy to checkwx V . Note that this 5-dimensional bialgebra is nontrivialŽ i.e., neither commu- tative nor cocommutative. . As we knowwx Wi, Zh there is only one 5-dimensional Hopf algebra; it is both commutative and cocommutative and is given by the cyclic group of order 5 in the sense of Example 1.Ž It is also knownwx Wi that the first nontrivial cosemisimple Hopf algebra is the 8-dimensional example of Kac and Palyutkinwx KacPal .. t The usual matrix *-operation on M given by Ž., x * s Ž, x .is of d course standard; and we see that A s M is a discrete quantum semi- group but not a discrete quantum group since it cannot be a Hopf algebra. As an aside let us note two differences between finite-dimensional cosemisimple bialgebras with and without antipode. Ž.1 We know fromwx LR that a finite-dimensional cosemisimple Hopf algebra over ރ is always semisimple as an algebra. This happens to be false d for bialgebras: the cosemisimple bialgebra A s M from Example 2 is not a semisimple algebra. To see it consider the dual basis of A consisting of Ä4 the elements , eij , where : M ª ރ is the counit of M and thus the multiplicative unit of A, and ²: ²: E,eij s0, x, eijs x ij; i, j g Ä41,2 ŽŽ.. here, as before, x g M2 ރ . The multiplication on A is given by the formulas ²:²:²:²:²E,ab s E, aE,b; x,ab s E m x q x m E q x m 1, a m b : ²: for all a, b in A. We compute E, eeij kl s0, and ²:² :²: x,eeij kl s EmxqxmEqxm1, eijm e kls x ij1, e kl and is equal to xij if k s l and to zero otherwise. We also see that 510 EDUARD VAYSLEB Ä4 Ä 4 eeij 12 seeij 21s0 for all i, j g 1, 2 . Denoting J s span e 12, e 21 we con- 2 clude that AJ s J, J s 0. Hence J is a nilpotent left ideal of A, and A is not semisimple. Ž.2 Unlike finite-dimensional Hopf algebras Ž see, e.g.,w Sw, Corollary 5.1.6x.Ž which always have one left integral and one right integral up to a scalar multiple. , the bialgebra M from Example 2 has more than one left integral and no right integrals at all. d The axiom for the left integral ⌳ l g A s M is Ž⌳ llm id .Ž.⌬ F s ⌳ Ž.FI, for each F g M. For the right integral ⌳ r the definition is symmetric: Ž.Ž.Ž.id m ⌳⌬rrFs⌳FI, FgM. Since ⌬Ž.E s E m E and E / I we get Ž⌳ lllm id .Ž.⌬ E s ⌳ Ž.EEs⌳ Ž.EI, Ž. Ž. therefore ⌳ lrE s 0. The same argument gives ⌳ E s 0. Now take xsŽ.0, x g M and compute Ž.Ž.Ž.Ž⌳ llm id ⌬ x s ⌳ m id E m x q x m E q x m 1. s⌳lllŽ.xEq⌳ Ž.x1s⌳ Ž.ŽxEq1 .s⌳ l Ž.xI. Thus any nonzero functional on M2Ž.ރ defines a left integral. On the other hand Ž.Ž.Ž.Žid m ⌳⌬rrxsid m ⌳ E m x q x m E q x m 1. s⌳rrŽ.xEq⌳ Ž.1x, Ž. Ž.Ž . which should be equal to ⌳ rrxIs⌳ xEq1 . Then for x / 1 we must Ž. Ž. Ž. Ž. have ⌳ rr1 x s ⌳ x 1, therefore ⌳ rr1 s ⌳ x s 0; and so ⌳ rs 0on M. Let us note that the same phenomena could also be exhibited in the cocommutative case of Example 1 with a finite semigroup G Žsee, e.g.,w CP, Sect. 5.2x. . 2. UNIQUENESS A cosemisimple bialgebra A can be writtenŽ see, e.g.,wx ER. as a direct sum of simple matrix coalgebras. That means A s A, [g⌳ COSEMISIMPLE BIALGEBRAS 511 Ä 4 where ⌳ is a set of indices, and each A s span eij; i, j s 1, 2, . . . , n with n ⌬Ž.eij sÝeikme kj,Ž.eijs␦ ij. ks1 The basis elements eij of a coalgebra A satisfying the relations above are called comatrix units. Any subcoalgebra of a cosemisimple bialgebra A is completely reducible Sw, Lemma 14.0.1 ᎏand so is equal to A A , where ⌳ is a wx s [␥ g ⌳ 0 ␥ 0 subset of ⌳. For a comatrix units basis Äeij4 of a cosemisimple A let us define the Ä 4 d dual basis Eij of M s A as usual by ² ␣ : Eij,e rs s␦␦␦␣ ir js. Then for a fixed the elements ÄEij4 are matrix units of a simple matrix algebra M. The dual algebra M can be identifiedŽ seewx ER, Sect. 3. with the direct product M Ł M , where each M M Ž.ރ as an algebra. s ( n Ł Suppose M s g ⌳ M has a standard *-operation as in Definition 1. The standardness condition easily implies that for each g ⌳ the *-oper- ation maps the simple matrix algebra M onto itself. Furthermore, Kaplan- sky showedwx Kap that for a standard involution on a simple matrix algebra Ž . Ž M there exists a choice of matrix units for which Eij* s E ji this explains the use of the word ‘‘standard’’ in Definition 1. . Conversely, this Ž.t formula defines the usual *-operation on M given by x* s x , which is standard. The following result was suggested by E. Kirschberg. LEMMA. Suppose a simple matrix algebra M M Ž.ރ has two standard s n Ä4 Ž. in¨olutions:*and †. If for the matrix units Eij of M we ha¨eEij*sE ji, then there is a positi¨e matrix C g M such that † y1 x s C x*C for all x g M. Proof. The map s †() is an automorphism of a simple matrix algebra M, so by the Skolem᎐Noether theorem it has to be inner. Thus Ž. y1 xsTxTfor some invertible T g Mn. That means that for all Ž.† y1† xgM we have x* s TxTor, replacing x with x* we get x s Ty1 x*T. Since †(† s idŽ. † is an involution we see that †††111 xsŽ.xs ŽTxy*T .sTTy*xT Žy .*T Ž.2 512 EDUARD VAYSLEB Ž. y1 for all x g M. If we substitute x s T* into 2 we get T* s TT*T, hence TT* s T*T, and so T must be normal. Consider its polar decompo- 1 sition T s UC s CU with unitary U and positive C. Then Ty T* s y1 y1 y2 Ž. y22 2 UCCU*sU and 2 means x s UxUfor all x g M.SoU i 1 must be a scalar; assume it equals i g S . Suppose U itself is not a scalar. Then it has two eigenvalues "ei r2, and 1 2 U H for some nonzero vector . We take s Cy r and compute 1 2 1 2 Ž.T N s Ž.CUCy r N Cy r s Ž.U N s 0. U Ä4 † Consider Ps Pg M ᎏthe orthogonal projection onto . Then PP y1 sTPTP s0 and † cannot be standardᎏa contradiction. † 1 1 1 Thus we see U is a scalar and x s Cy Uy x*UC s Cy x*C, i.e., † is implemented by a positive C. Ž. Remark 1. The inner automorphism : M ª M given by x s y1r21r2 Ž. CxCdefines the new matrix units Fijs E ij on M. Our new standard involution † acts on them as † y1 y1r2 1r2 Fij s C Ž.C ECij *C y1 1r2 y1r2 y1r2 1r2 sC C ECji CsC ECji sFji. This demonstrates that the standard *-operation on M is unique up to an algebra automorphismᎏso we have given aŽ different fromwx Kap. proof of the Kaplansky result quoted in the beginning of this section. d Let us consider a simple matrix coalgebra A s M and look at the dual d map ⌽ s : A ª A given by ²:²:²y1r21r2: x,⌽Ž.as Ž.x,asCxC,a; xgM,agA. It follows from the general theory of bialgebraswx Sw, Sect. 3 that ⌽ is a coalgebra automorphism. Now we can prove uniquenessŽ. up to a bialgebra automorphism of a discrete quantum semigroup structure for any cosemisimple bialgebra. THEOREM 1. Let A s [ A be a cosemisimple bialgebra. Suppose it has two standard co-*-operations q : A ª A and ( : A ª A. Then there exists a bialgebra automorphism ⌽ of A such that ⌽Žaq.Ž.s ⌽Њ a for all a in A. Ž. d Ł Proof. As in 1 , on the dual algebra A s M s g ⌳ M we consider the dual *-operations given by † ²:²:²:²:x*, a s x, aq ; x , a s x, aЊ ; x g M, a g A. COSEMISIMPLE BIALGEBRAS 513 Since the two given co-*-operations are standard we have that ) and † are standard involutions on M. Therefore ) : M ª Mand † : M ª M for every g ⌳. Taking the duals again we obtain the mappings q : A ª A and ( : Aª A for every g ⌳. Ž. Assume the matrix units Eij are chosen such that Eijs E ji on each M. From the lemma we know that the standard involution † : M ª M can † y1 U be written on each M as xs C x C , where the Care positive matrices in M. Let us consider the algebra automorphisms of M y1r2 1r2 defined by : xª C xC . d By the previous discussion, the adjoint maps ⌽ s on A are coalgebra automorphisms. We define the direct sum coalgebra automor- phism ⌽ s [ ⌽ on the whole A. We will also use the dual algebra d automorphism ⌽ s s Ł of M. We need to show that ⌽ is in fact a bialgebra automorphism intertwin- ing the two involutions: q and (. Ž.i Let us start by showing that ⌽ is unital, i.e., for the multiplica- tive identity 1 g A we have ⌽Ž.1 s 1. Since 1 is a group-like element Ž that is, ⌬Ž.1 s 1 m 1 . , it generates a one-dimensional subcoalgebra. So there is a fixed index 0 g ⌳ such that no s 1. Then of course M0 ( ރ, and for an Ž. arbitrary x s x g ⌳ g M, ²: x,1 sx0 gރ. Ž. y1r21r2 Then we see that 00000x s CxCsx, and so ²:²: ²: x,⌽Ž.1s Žx .,1 sx0s x,1 for all x g M. Ž.ii To prove ⌽ commutes with the multiplication of A it suffices to show that for arbitrary ␣, , ␥ g ⌳, a g A␣ , b g A␥, x g M the follow- ing holds ²:²:x,⌽Ž.ab s x, ⌽ Ž.Ž.a ⌽ b .3Ž. Let us fix indices ␣,  g ⌳. Since A␣ , A are subcoalgebras of the bialgebra A, the restriction of the multiplication map m : A m A 2 a m b ª ab g A to the coalgebra A␣ m A is a coalgebra homomorphism. The image AA␣  of this restriction is a finite-fimensional subcoalgebra of A, which is a cosemisimple coalgebra. Bywx Sw, Lemma 14.0.1 , AA␣  is completely reducible, and so is equal to A for some finite [␥ g ⌳ 0 ␥ ⌳ 0 ; ⌳. Thus we have a coalgebra epimorphism: m : A A A .4Ž. ␣m ␥ª [ ␥g⌳0 514 EDUARD VAYSLEB Ž. Ž. Ž.Ž. Note that ⌽ A s A for each , hence ⌽ ab , ⌽ a ⌽ b are in A for all a A , b A . In particular if x M and ␥ ⌳ [␥g⌳0␥ g ␣g g ␥f 0 both sides ofŽ. 3 are equal to zero. Define the algebra monomorphism ⌫ as the dual map to the coalgebra epimorphismŽ. 4 , ²:²: ⌫Ž.x,ambsx,ab ; a g A␣, b g A, x g M0; it acts from M M to M M . 0 s [␥g⌳0␥␣m ŽWe note that our ⌫ is a finite-dimensional truncation of the ‘‘comulti- plication’’ defined inwx ER on the whole algebra M as the dual to the multiplication on A.. We can rewriteŽ. 3 as ²:Ž⌫( .Ž.x,ambs²: Žm .Ž.⌫ Ž.x,amb; agA␣,bgA,xgM0. Now in order to prove ⌽ commutes with the multiplication we need only to show that ⌫( s Ž. m (⌫ Ž.5 as maps from M M to M M . 0 s [␥g⌳0␥␣m Ž. iii Since each A is closed under the two co-*-operations q and (, Definition 2 shows that the mapŽ. 4 commutes with q and (. This implies its dual map ⌫ commutes with the dual involutions ) and †. So the map ⌫ is an injective *-homomorphism between finite-dimen- Ž. sional C*-algebras. Also it is unital, i.e., ⌫ 1 s 1 m 1, because for a g A␣ , b g A we can write ²:1,ab Ž.ab Ž.Ž.a b ²:²:1,a1,b. MM0s s s ␣M Denoting by n the size of M␣we fix some *-isomorphism M m M( MŽ.ރ. Then ⌫ can be described as follows ER, Proposition 8.3 : there nn␣  wx is a sequence of integer multiplicities m and a unitary matrix V M Ž.ރ ␥ g nn␣  Ž. such that for every x s x␥ g M0 we have ⌫Ž.x Vy1 x[m␥ V s ž/[␥ ␥g⌳0 Žthe exponents m␥ determine the ‘‘Bratteli diagram’’ of the map ⌫ wxBr. . COSEMISIMPLE BIALGEBRAS 515 Since the mapping ⌫ commutes with both ) and †, the algebraic automorphism of M given by y1 s †() : Ž.x; g ⌳ ª Ž.CxC;g⌳ must also commute with ⌫. It implies Ž.Ž.Ž. m Ž.Ž.⌫ x s ⌫ x for all x g M0. Note that on M␣m M the map m acts by x␣m y ª Žy1y1.Ž .Ž . C␣ m Cx␣␣myCmC. Therefore we have the equality 1 yy1 [m␥ Ž.C␣mCV␥␣ž/[␥xVC Ž.mC [m␥ y1y1 y1y1[m␥ sVž/[␥Ž.CxC␥␥␥ VsVCž/[␥xCV␥ , [m␥ [m␥ where the positive matrix C s [␥␥C␥ is in [ M␥ . Ž. Since this must hold for an arbitrary x s x␥ g M0 we conclude that the matrix D VCŽ.CVy1 Cy1is in the commutant of M[m␥. s ␣ m ␥[␥g⌳0 In particular D commutes with C. This implies y1 y1 y1 y1 VCŽ.␣mCV␣CCsCV Ž. C m CVC , Ž.y1 meaning the positive matrices VC␣ mCV and C commute, and D is a positive matrix. The automorphism could be viewed as the ‘‘square root’’ of . Repeating the previous argument we see thatŽ. 5 is equivalent to having VCŽ 1r21CVCr2.y1y1r2commute with all x M[m␥. But we can ␣ m ␥g[g⌳0␥ write 1r2 1r21r2y1y1r2 y1y11r2 VCŽ.␣mCVC␣s ŽVCŽ.mCVC .sD , and observe that the matrix D1r2 is in the commutant of M[m␥ [␥ g ⌳ 0 ␥ since the positive matrix D is in the commutant. Ž.iv From above, ⌽ is a bialgebra automorphism of A. Recall from Remark 1 that †( s () for all g ⌳. We want to show that Ž q.Ž. ⌽as⌽Њ a on A. For arbitrary x g M, a g A, ²:²:²:qq† x,⌽Ž.as Ž.x,as Ž.x,a ²:²:²: sŽ.x*,as x*, ⌽ Ž.a s x, ⌽Њ Ž.a . Since the pairing²: , : M = A ª ރ is nondegenerate the proof is completed. 516 EDUARD VAYSLEB Remark 2. Suppose A is a Hopf algebra with an antipode S. Recall wxSw that S is the unique map satisfying the axiom ⌺aS12Ž. a s⌺Sa Ž. 12 as Ž.a1 for all a g A. It follows that every bialgebra automorphism ⌽ of A must also be a Hopf automorphism, i.e., it must commute with S.Ž This is probably ‘‘folklore,’’ though it does not appear inwx Sw or w Mon x .. Indeed we write ⌬⌽Ž.Ž.a s Ž⌽m⌽⌬ . Ž. Ž.as⌺⌽ Ža12 .m ⌽ Ža ., and so ⌺⌽Ž.aS12Ž.⌽ Ž.a s⌺SŽ⌽ Ž. Ž.a 12⌽ Ž.a sŽ.⌽ Ž.a1s Ž.a1. Taking ⌽y1 of this equality and using the result that ⌽y1 commutes with the multiplication on A we obtain y1 y1 ⌺a12Ž.Ž.Ž.Ž.Ž.⌽ (S(⌽ a s ⌺⌽ (S(⌽ aa 12sa1, 1 and conclude that ⌽y (S(⌽ s S. Thus if a bialgebra A in Theorem 1 happens to be a Hopf algebra the bialgebra isomorphism ⌽ constructed in the proof automatically becomes Hopf isomorphism. This way the uniqueness of a standard involution for a discrete quantum group immediately follows from our result. 3. EXAMPLE OF NONEXISTENCE In this section we show that cosemisimple Hopf algebras form a wider category than discrete quantum groups. It follows that, respectively, cosemisimple bialgebras form a wider cate- gory than discrete quantum semigroups. Let A s [ A be a cosemisimple Hopf algebra with the antipode S. d On the dual algebra M s A consider the dual to S map : M ª M, defined by ²:²:Ž.x,asx,Sa Ž.; xgM,agA. The map S 2is a coalgebra automorphism of A, and, bywx Lar , S 2maps 2 each A onto itself. Then its dual is an algebraic automorphism of M mapping every simple matrix algebra M onto itself. So by the Skolem᎐Noether theorem there exists an invertible T g M such that for COSEMISIMPLE BIALGEBRAS 517 all x g M we have 2 y1 Ž.x s T xT. Inwx ER, Chap. 7Ž see alsowx V. it is proved that if A is a discrete quantum group then 2 must be implementable by positive matrices. d More precisely, in this case every simple matrix subalgebra M of M s A has a basis of matrix units with respect to which 2 y1 Ž.x s C xCfor all x g M , where C is a positive matrix in M. Ž. Ž Ž.. Let us consider the classical by now Hopf algebra Aq sOq SL2 ރ Žsee, e.g.,wx FRT .Ž.ᎏthe coordinate algebra of the quantized SL2 ރ .Asan associative algebra AqŽ.is generated by four elements a, b, c, d. The comultiplication on AqŽ.is defined by stating that its generators ee11 12 ab s ž/ee21 22 ž/cd constitute comatrix units of a simple matrix coalgebra A2 . The antipode is given by 1 SaŽ. Sb Ž. d yqby s . ž/ScŽ. Sd Ž. ž/yqc a The structure of AqŽ.depends on the value of a nonzero parameter qgރ. Inwx MMNNU Masuda et al. proved AqŽ.is cosemisimple if q is not a root of unity. In more detail, they showed that AqŽ.s Al, [lgގ where each Allis a simple matrix coalgebra of size n s l. When q is real Ž.q /"1 there is a co-*-operation which turns AqŽ. into a discrete quantum group corresponding to the compact quantum group SUqŽ.2 of WoronowiczŽ the correspondence is explained inw ER, . q Ä4 Chap. 10x . It is defined on the generators as eijs e ji; i, j g 1, 2 . The Woronowicz analog of the Peter᎐Weyl theorem for SUqŽ.2w Wor, Theorem Žl. 5.7x implies the following: one can choose comatrix units eij in each simple coalgebra Al such that Žl.Žq l. Ž.eijse ji ;i,jgÄ41,2,...,l . This means the co-*-operation q is standard; by Theorem 1 it is the only standard co-*-operation if the real value of q is fixed. 518 EDUARD VAYSLEB We can prove that q g ޒ is also necessary for existence of a standard co-*-operation on AqŽ.. THEOREM 2. The cosemisimple Hopf algebra OqŽŽ..SL2 ރ has a discrete quantum group structure only if q is a real numberŽ. q / 0, " 1. Proof. Consider the simple matrix coalgebra A22of size n s 2. In the Ž. Ä4 dual simple matrix algebra M22( M ރ with the dual basis Eij take the elements q 0 y1 1 q 0 01 00 Ks , Kys , Es , Fs . ž/0qy1 ž/0 q ž/00 ž/ 10 Ž. Ž. Note that SA22sA, and so M2s M 2. Directly using the definition of as the dual of S we compute "1 .1 1 Ž.K s K , Ž.E syqE, Ž.F syqy F; 2 "1 "1 2 2 2 2 Ž.KsK, Ž.Esyq E, Ž.F syqy F. 2 Thus in the basis Ä4Eij the inner automorphism is acting on M2 as 2 1 Ž.x sKxKy . If A has a standard co-*-operation then in some other basis the same automorphism must be implemented by a positive diagonal matrix C s Ž. diag c12, c . Multiplying by a constant, if necessary, we can assume C s Žy1. diag c, c , c ) 0. Thus there exists an invertible matrix R g M2 such that 2 1 1 1 Ž.x s RCRy xRCy Ry Ž. for all x g M2 ރ . Therefore for some nonzero constant ␣ we have 1 K s ␣RCRy . 2 Comparing determinants we get ␣ s 1, so ␣ s "1. Let su Rs , st y u¨ / 0. ž/¨ t Then write KR s "RC,or q 0 su suc 0 s" . ž/0qy1ž/ž/¨t ¨tž/0cy1 COSEMISIMPLE BIALGEBRAS 519 Then qs qu cs cy1 u 1 1 s " 1 . ž/qy ¨qty ž/c¨ cty 1 Since q / qy by assumption we conclude that either q s "c or q s "cy1.Soqmust be real. In fact one can compute directly all the possible co-*-operations on AqŽ.Žthe list is given inwx FRT , see the explicit calculation in wx V. . In the case when q is not a root of unity OqŽŽ..SL2 ރ has exactly three co-*- operations: two for a real q and one when < ACKNOWLEDGMENT I thank my adviser E. Effros for help and encouragement. REFERENCES wxAnd N. Andruskiewitsch, Compact involutions of semisimple quantum groups, Czechoslo¨ak J. Phys. 44 Ž.1994 , 963᎐972. wxBr O. Bratteli, Inductive limits of finite dimensional C*-algebras, Trans. Amer. Math. Soc. 171 Ž.1972 , 195᎐234. wxCP A. H. Clifford and G. B. Preston, ‘‘The algebraic theory of semigroups,’’ in Math. Surveys, No. 7, Vol. 1, Amer. Math. 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