On Hopf Algebras with Positive Bases

Journal of Algebra 237, 421᎐445 Ž2001. doi:10.1006rjabr.2000.8459, available online at http:rrwww.idealibrary.com on

On Hopf Algebras with Positive Bases Jiang-Hua Lu1 Department of Mathematics, Uni¨ ersity of Arizona

and Min Yan 2 and Yongchang Zhu 3 Department of Mathematics, Hong Kong Uni¨ ersity of Science and Technology, Clear Water Bay, Hong Kong Communicated by Walter Feit Received May 14, 1996

We show that if a finite dimensional Hopf algebra H over ⺓ has a basis with respect to which all the structure constants are nonnegative, then H is isomorphic to the bi-cross-product Hopf algebra constructed by Takeuchi and Majid from a finite group G and a unique factorization G s Gq Gy of G into two subgroups. We also show that Hopf algebras in the category of finite sets with correspondences as morphisms are classified in a similar way. Our results can be used to explain some results on Hopf algebras from the set-theoretical point of view. 䊚 2001 Academic Press

1. INTRODUCTION In this paper, we prove two classification theorems for finite dimensional Hopf algebras. The first theorem ŽTheorem 1. classifies finite dimensional Hopf algebras over ⺓ which admit positi¨ e bases. A basis of a Hopf algebra over ⺓ is said to be positive if the structure constants of all 1 Research was partially supported by an NSF Postdoctoral Fellowship and NSF Grant DMS 9803624. 2 Research was supported by Hong Kong Research Grant Council Earmark Grant HKUST 6071r98P. 3 Research was supported by Hong Kong Research Grant Council Earmark Grant HKUST 629r95P.

421 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.

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the structure maps of the Hopf algebra with respect to this basis are nonnegative real numbers Žsee Section 2 for more precise definitions.. This class of Hopf algebras is closed under taking tensor products, duals, and Drinfel’d doubles. Our theorem says that if a Hopf algebra has a positive basis, it must be isomorphic to the bi-cross-product Hopf algebra constructed by Takeuchi wTx and Majid wMaj1x from a finite group G and a unique factorization G s Gq Gy of G into two subgroups. The second theorem ŽTheorem 3. classifies Hopf algebras in the correspondence category. In the symmetric monoidal category of finite sets, we may formally introduce the concept of Hopf algebras by imposing all the axioms in terms of commutative diagrams. Such a definition is just that of finite groups. The correspondence category under our consideration still has finite sets as objects. But a morphism from X to Y is a subset of X = Y instead of a map from X to Y. This is a symmetric monoidal category in which the concept of Hopf algebras can be similarly defined. Our theorem says that a Hopf algebra in the correspondence category is still given by a finite group G with a unique factorization G s Gq Gy of two subgroups. There have been many other classifications of finite dimensional Hopf algebras under different assumptions. An incomplete list includes wAS1, AS2, BDG, EG, Masx. One of the motivations to consider Hopf algebras with positive bases is that there are many natural Hopf algebras with nearly positive bases. We call a basis nearly positive if all the structure constants with respect to this basis, except for those of the antipode map, are nonnegative real numbers. For example, the modified quantum group Uq Ž g . in wLux has a nearly positive basis realized in terms of certain varieties. Other examples are the direct sums [⬁1 RŽ Sn . and [⬁1 RŽ GL nŽ Fq .. of complex representation rings wZx, where positive bases are given by irreducible representations. The positivity of structural coefficients is due to the very way the product and the coproduct are defined. All the examples above are infinite dimensional Hopf algebras. In the finite dimensional case, we know of no examples of nearly positive bases that are not positive. It turns out the Hopf algebras in our classification have natural numbers as structure coefficients. It might be interesting to classify infinite dimensional Hopf algebras with natural numbers as structure coefficients. In subsequent papers wLYZ1x and wLYZ2x, we classify positive quasi-triangular structures on Hopf algebras with positive bases and discuss their relations to set-theoretical Yang᎐Baxter equations. The paper is organized as follows. We first present the basic concepts and notation, and state the first classification theorem. In the next two sections, we give detailed proof of the theorem. In the final section, we

423

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discuss the correspondence category and outline the proof of the second classification theorem.

2. HOPF ALGEBRAS WITH POSITIVE BASES Given a group G, the group algebra ⺓G has the subset G as a positive basis in the sense that all the structure constants are nonnegative. Moreover, the dual basis is also a positive basis of the dual group algebra Ž⺓G .*. In general, we say a basis B of a Hopf algebra H is positi¨ e if, with respect to this basis, Ž1. the coordinates of the unit 1 are nonnegative; Ž2. the coordinates of the counit ⑀ Žwith respect to the dual basis B*. are nonnegative, i.e., ⑀ Ž b . G 0 for all b g B; Ž3. for any b1 , b 2 g B, the coordinates of b1 b 2 are nonnegative; Ž4. for any b g B, the coordinates of ⌬Ž b . Žwith respect to the tensor product basis B m B . are nonnegative; Ž5. for any b g B, the coordinates of the antipode S Ž b . are nonnegative. It is easy to see that the duals and the tensor products of Hopf algebras with positive bases still have positive bases. For the antipode S of a finite dimensional Hopf algebra, it is well known wRx that S N s id for some positive integer N. This implies that if B is a positive basis of H, then the structure constants of Sy1 s S Ny 1 with respect to B are still nonnegative. Therefore, B* m B is a positive basis of the Drinfel’d double DŽ H . of H. The prototypical example of Hopf algebras with positive bases is the bi-cross-product Hopf algebra H Ž G; Gq, Gy . constructed by Takeuchi wTx and Majid wMaj1x from a finite group G and a unique factorization G s Gq Gy of G into two subgroups. By G s Gq Gy, we mean that G " are subgroups of G and that every g g G can be written as g s gq gy for unique gqg Gq and gyg Gy. By taking the inverse, any g g G can also be written as g s gy gq for unique gqg Gq and gyg Gy. Therefore, to any g g G is associated four uniquely determined elements

␣q Ž g . s gq ,

␤q Ž g . s gq ,

␣y Ž g . s gy ,

␤y Ž g . s gy ,

given by g s gq gys gy gq .

Ž 1.

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It is then easy to see that Gy= Gqª Gq ,

Ž 2.

Gy= Gqª Gy ,

Ž gy , gq . ¬ Ž gy gq . q s gq , Ž gy , gq . ¬ Ž gy gq . y s gy ,

Gq= Gyª Gq ,

Ž gq , gy . ¬ Ž gq gy . q s gq ,

Ž 4.

Gq= Gyª Gy ,

Ž gq , gy . ¬ Ž gq gy . y s gy ,

Ž 5.

Ž 3.

are all group actions. Because of this, we also write gqs gy gq ,

gys gygq ,

gys gq gy .

gqs gqgy ,

The third and the fourth actions satisfy the following matching relations gq

gy

Ž gy hy . s gq gy Ž gq . hy ,

g

g

Ž q gy . gq gy . Ž hq gq . y s hq

Ž 6.

It is well known that, given two actions between two groups G " satisfying the matching relations, we may reconstruct G. Similarly, G can be reconstructed from the first and the second actions, which satisfy similar matching relations. We will not use the first and the second actions in the proof of the classification theorem. The Hopf algebra H Ž G; Gq, Gy . is, by definition, the vector space over ⺓ spanned by the set G s  g 4 : g g G4 together with the following Hopf algebra structure,

 g 4  h4 s

½

 gy h4 s  ghy 4

in case gqs hq ,

0

in case gq/ hq ,

⌬ g 4 s

Ý

 h4 m  k 4 s

hkqsg , kysh y

1s

Ý

 h4 m  k 4 ,

hqksg , kysh y

Ý  gq 4 ,

gqgG q

⑀ g4 s

½

1 0

in case gqs e, in case gq/ e,

S g 4 s  gy1 4 . We have given the definition of the Hopf algebra H Ž G; Gq, Gy . that best suits the purpose of this paper. It is easy to see that the algebra structure on H Ž G; Gq, Gy . is the cross-product algebra Ž⺓Gq .* i ⺓Gy defined by the right action of Gy on Gq given in Ž5.. Similarly, the algebra structure on the dual Hopf algebra H Ž G; Gq, Gy .* Žcorresponding to the coproduct

HOPF ALGEBRAS WITH POSITIVE BASES

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in H Ž G; Gq, Gy .. is the cross-product algebra ⺓Gqh Ž⺓Gy .* defined by the left action of Gq on Gy given in Ž4.. The basis G for H Ž G; Gq, Gy . is clearly positive. The following theorem says that, up to rescaling, these are all the finite dimensional Hopf algebras with positive bases. THEOREM 1. Gi¨ en any finite dimensional Hopf algebra H o¨ er ⺓ with a positi¨ e basis B, we can always rescale B by some positi¨ e numbers, so that Ž H, B . is isomorphic to Ž H Ž G; Gq, Gy ., G . for a unique group G and a unique factorization G s Gq Gy. The proof of Theorem 1 will be given in Sections 3 and 4. A consequence of Theorem 1 is that the Drinfel’d double DŽ H Ž G; Gq, Gy .. of ˜ G˜q, G˜y . for some group G˜ and H Ž G; Gq, Gy . is also of the form H Ž G; ˜ s G˜q G˜y. By tracing our proof of the theorem, we find factorization G

˜ s G = G, G ˜qs  Ž gq , gy . : gqg Gq , gyg Gy 4 ( Gq= Gy , G ˜ys  Ž g , g . : g g G 4 ( G. G The fact DŽ H Ž G; Gq, Gy .. ( H Ž G = G; Gq= Gy, G . was previously discovered in wBGMx, and a more explicit formula for the isomorphism can be found there.

3. CLASSIFICATION UP TO RESCALING In this section, we prove the Classification Theorem 1 up to rescaling. In the next section, we work out the necessary rescaling. Let H be a finite dimensional Hopf algebra over ⺓ with a positive basis B. We denote by B* s  b*: b g B4 the dual basis of H *. The basis B* is still positive. We also have the bases B m B and B m B* of H m H and H m H *, respectively. We say an element x g H contains a term b g B if the b-coordinate of x is not zero. We can say similar things in H *, H m H, H m H *, etc. Given subsets X and Y of a basis B, the notation X *, X m Y, X m Y *, etc., carry the obvious meanings as subsets of B*, B m B, B m B*, etc. The meanings of phrases like X-term, X *-term, X m Y-term, X m Y *term, etc., are self-evident. We will often make use of the duality between H and H *, having positive bases B and B*, respectively. This allows us to turn a result about product into a result about coproduct, and vice versa. The guiding principle for doing this is the following.

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LEMMA 1. The coefficient of bU1 m bU2 in ⌬Ž b*. is the same as the coefficient of b in b1 b 2 . In particular, ⌬Ž b*. contains bU1 m bU2 if and only if b1 b 2 contains b. Similarly, we may interchange the roles of H and H *. Proof. Let b1 b 2 s Ý ␮ c c, and ⌬Ž b*. s Ý c1 , c 2 g B ␭ c1 , c 2 cU1 m cU2 . Then

␮ b s ² b*, b1 b 2 : s² ⌬ Ž b* . , b1 m b 2: s ␭ b 1 , b 2 .

The unit element 1 g H has the expression 1 s Ý b g B ⑀ Ž b*. b. Set b⬘ s ⑀ Ž b*. b if ⑀ Ž b*. / 0 and b⬘ s b if ⑀ Ž b*. s 0. Then B⬘ s  b⬘: b g B4 is also a positive basis for H and 1 s d1 q ⭈⭈⭈ qd k , where d1 , . . . , d k are distinct elements of B⬘. We will replace B by B⬘ and still denote it by B. LEMMA 2. d i d j s ␦ i j d i for i, j s 1, . . . , k, and ⺓ d1 q ⭈⭈⭈ q⺓ d k is a commutati¨ e Hopf subalgebra of H. Proof. From d i s d i ⭈ 1 s d i d1 q ⭈⭈⭈ qd i d k and positivity, d i d j is a nonnegative multiple of d i . Similarly, from d j s 1 ⭈ d j , we see that d i d j is a nonnegative multiple of d j . Therefore d i d j s 0 when i / j. Substituting this into d i s d i d1 q ⭈⭈⭈ qd i d k , we get d i s d i d i . From ⌬Ž d1 . q ⭈⭈⭈ q⌬Ž d k . s ⌬Ž1. s 1 m 1 s Ý i, j d i m d j and positivity, we see that ⺓ d1 q ⭈⭈⭈ q⺓ d k is closed under ⌬. From SŽ d1 . q ⭈⭈⭈ qSŽ d k . s SŽ1. s 1 s d1 q ⭈⭈⭈ qd k and positivity, we see that ⺓ d1 q ⭈⭈⭈ q⺓ d k is closed under S. Therefore ⺓ d1 q ⭈⭈⭈ q⺓ d k is a commutative Hopf subalgebra of H. Commutative Hopf algebras have been well classified: the Hopf algebra ⺓ d1 q ⭈⭈⭈ q⺓ d k is isomorphic to the dual Ž⺓Gq .* of the group algebra of a finite group Gq. More precisely, we have

 d1 , . . . , d k 4 s GUqs  d gq: gqg Gq 4 and d gq d hqs ␦ gq , hq d gq , S Ž d gq . s d gy1 , q

1s

⌬ Ž d gq . s

Ý

gqgG q

d gq,

Ý

hqkqsgq

d hqm d kq,

⑀ Ž d gq . s ␦ gq , e .

HOPF ALGEBRAS WITH POSITIVE BASES

LEMMA 3.

427

For any b g B, there are unique gq, gqg Gq such that

d hq b s ␦ gq , hq b,

bd hqs ␦ gq , hq b,

᭙hqg Gq .

Proof. From b s 1 ⭈ b s Ý hq g G q d hq b and positivity, we have, for any hqg Gq, d hq b s ␭ hq b for some nonnegative number ␭ hq, and not all the ␭ hq ’s vanish. Then from d gq d hqs ␦ gq , hq d gq, we see that ␭ gq ␭ hqs ␦ gq , hq ␭ gq. Therefore, all except one ␭# vanish. Moreover, the nonvanishing ␭# must be 1. Denote the unique elements in the last lemma by

␣q Ž b . s gq ,

␤q Ž b . s gq .

It follows from the lemma that

␣q Ž b . s gqm d gq b s b m d gq b / 0,

Ž 7.

␤q Ž b . s gqm bd gqs b m bd gq/ 0.

Ž 8.

Set Bgq , gqs  b g B: ␣q Ž b . s gq , ␤q Ž b . s gq 4 , Bgq , ⴢ s  b g B: ␣q Ž b . s gq 4 , B ⴢ, gqs  b g B: ␤q Ž b . s gq 4 . Then B is the disjoint union of all the Bgq , gq ’s. LEMMA 4.

If b1 b 2 contains b, then

␣q Ž b1 . s ␣q Ž b . ,

␤q Ž b 2 . s ␤q Ž b . .

Ž 9.

If ⌬Ž b . contains b1 m b 2 , then

␣q Ž b . s ␣q Ž b1 . ␣q Ž b 2 . ,

␤q Ž b . s ␤q Ž b1 . ␤q Ž b 2 . .

Ž 10 .

Proof. Suppose b1 b 2 contains b. Set gqs ␣q Ž b ., gqs ␤q Ž b .. Then b1 b 2 d gq and d gq b1 b 2 still contain d gq b s b s bd gq. Therefore, b 2 d gq/ 0 and d gq b1 / 0. By Ž7. and Ž8., this implies Ž9.. Let ⌬Ž b . s Ý b 1 , b 2 g B ␭ b 1 , b 2 b1 m b 2 . Then ⌬ Ž b . s ⌬ Ž d gq b . s s

Ý

b1 , b 2gB , h q kqsgq

Ý

␣ qŽ b 1 .sh q, ␣ q Ž b 2 .skq , h q kqsgq

␭ b 1 , b 2 d hq b1 m d kq b 2 ␭ b 1 , b 2 b1 m b 2 .

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Therefore, ⌬Ž b . contains only the terms b1 m b 2 satisfying ␣q Ž b1 . ␣q Ž b 2 . s hq kqs gqs ␣q Ž b .. The proof for ␤q Ž b . s ␤q Ž b1 . ␤q Ž b 2 . is similar. We now dualize the discussions above. The set B* is a positive basis of the dual Hopf algebra H *. In particular, we have ⑀ s ␭1 eU1 q ⭈⭈⭈ q␭ l eUl , where e i g B and ␭ i ) 0 for i s 1, . . . , l. Clearly, b* appears in the sum if and only if ⑀ Ž b . ) 0. Since d e g GUq is the only element in GUq with positive ⑀-value, the intersection between  e1 , . . . , e l 4 and GUq is exactly d e . Moreover, by 1 s ⑀ Ž1. s ⑀ Ž d e ., we see that the coefficient of dUe in the summation for ⑀ is 1. Thus we may rescale the e i ’s to get ⑀ s eU1 q ⭈⭈⭈ qeUl , without affecting GUq. As in Lemma 2, ⺓ eU1 q ⭈⭈⭈ q⺓ eUl is a commutative Hopf subalgebra of H *. Therefore,  e1 , . . . , e l 4 form a group Gy under the product in H, with d e as the unit. To indicate the group structure, we denote

 e1 , . . . , e l 4 s Gys  e gy: gyg Gy 4 . Then e gy e hys e gy hy,

⌬ Ž e gy . s e gym e gyq no Gym Gy-terms,

S Ž e gy . s e gy1 q no Gy-terms, y

⑀s

Ý

gygG y

eUgy.

Since d e g GUq is also the identity of Gy, we will abuse the notation and also denote e s d e . The dual of Lemma 3 is the following: For any b* g B*, we have unique gy, gyg Gy such that eUhy b* s ␦ gy , hy b*,

b*eUhys ␦ gy , hy b*,

᭙hyg Gy .

Ž 11 .

This enables us to define the maps ␣y, ␤y: B ª Gy:

␣y Ž b . s gy ,

␤y Ž b . s gy .

The following lemma is a reinterpretation of Ž11.. We note that Ž12., being equivalent to Ž11., also serves as the characterizations of ␣y, ␤y. LEMMA 5.

Suppose ␣y Ž b . s gy and ␤y Ž b . s gy. Then

⌬ Ž b . s e gym b q b m e gyq no B m Gy-terms and no Gym B-terms. Ž 12 . Proof. By Lemma 1, the equalities eUhy c* s ␦␣y Ž c., hy c* mean that ⌬Ž b . contains e hym c if and only if c s b and hys ␣y Ž c .. Moreover, when the containment happens, the coefficient is 1. This means exactly that the only

HOPF ALGEBRAS WITH POSITIVE BASES

429

Gym B-term of ⌬Ž b . is e␣y Ž b. m b, and the coefficient is 1. Similarly, the only B m Gy term of ⌬Ž b . is b m e␤y Ž b. , also with coefficient 1. With the help of Lemma 1, we may also dualize Lemma 4. LEMMA 6.

If ⌬Ž b . contains b1 m b 2 , then

␣y Ž b1 . s ␣y Ž b . ,

␤y Ž b 2 . s ␤y Ž b . .

Ž 13 .

␤y Ž b . s ␤y Ž b1 . ␤y Ž b 2 . .

Ž 14 .

If b1 b 2 contains b, then

␣y Ž b . s ␣y Ž b1 . ␣y Ž b 2 . ,

Having found Gq and Gy, we need to further construct G s Gq Gy. One approach is to make use of the Drinfel’d double DŽ H ., which has B* m B as a positive basis. From DŽ H . we can also construct a ‘‘positive ˜q and a ‘‘negative group’’ G˜y. The group G˜y consists of exactly group’’ G the pairs bU1 m b 2 g B* m B such that

⑀ Ž bU1 m b 2 . s ² bU1 , 1:⑀ Ž b 2 . ) 0.

˜ys Gqm Gy and this is a unique factorization. It remains to Thus G ˜y; Gq, Gy . is isomorphic to the Hopf algebra H. verify that H Ž G Instead of making use of the Drinfel’d double, we will continue with a more elementary approach. The approach will also be applicable to Hopf algebras in the correspondence category, where the notion of Drinfel’d double is yet to be defined. LEMMA 7. The following are equi¨ alent: Ž1. Ž2. Ž3.

␣q Ž b . s e; ␤q Ž b . s e; b g Gy.

In particular, Be, gqs Bgq , e s ⭋ for gq/ e. Proof. Since d e s e is also the multiplicative unit for Gy, we see that Ž3. implies Ž1.. Conversely, from Lemma 5, ⌬Ž b . contains e g m b, where y gys ␣y Ž b .. Then by ⑀ s mŽ S m 1. ⌬, we know that ⑀ Ž b . contains SŽ e gy . b, which further contains e gy1 b. If b satisfies Ž1., then e gy ⑀ Ž b . contains y e gy e gyy1 b s d e b s b, so that ⑀ Ž b . / 0. This implies b g Gy. Similarly, by using ⑀ s mŽ1 m S . ⌬, we can prove that Ž2. and Ž3. are equivalent. LEMMA 8. Gi¨ en gqg Gq and gyg Gy, there is a unique Bgy1 m q , ⴢ Bgq , ⴢ-term c m b contained in ⌬Ž e gy .. Moreo¨ er, b is the unique element such that ␣q Ž b . s gq and ␤y Ž b . s gy.

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Proof. We begin by finding a Bgy1 m Bgq , ⴢ-term in ⌬Ž e gy .. Since q , ⴢ mŽ1 m S . ⌬Ž e gy . s ⑀ Ž e gy . s 1 s Ý hq g G q d hq, ⌬Ž e gy . must contain a term c m b such that cSŽ b . s Ý hq g G q ␭ hq d hq with ␭ gqy1 ) 0. Then we have d gy1 cSŽ b . s ␭ gyq 1 d gyq 1 . Thus d gqy 1 c / 0, and by Ž7., ␣q Ž c . s gy1 q . Then by q Lemmas 4, 6, and 7, we have ␣q Ž b . s ␣q Ž c .y1␣q Ž e gy . s gq. Next we would like to compare the Bgq , ⴢ m Bgqy1 , ⴢ m Bgq , ⴢ -terms in Ž ⌬ m 1. ⌬Ž b . and Ž1 m ⌬ . ⌬Ž b .. For Ž ⌬ m 1. ⌬Ž b ., such terms can only be obtained from applying ⌬ m 1 to B m Bgq , ⴢ -terms in ⌬Ž b .. Now for any B m Bgq , ⴢ -term b1 m b 2 contained in ⌬Ž b ., we apply Lemma 4 and find gqs ␣q Ž b . s ␣q Ž b1 . ␣q Ž b 2 . s ␣q Ž b1 . gq. Therefore, ␣q Ž b1 . s e, so that by Lemma 7, we have b1 g Gy. Thus b1 m b 2 is really a Gym Bgq , ⴢ -term contained in ⌬Ž b .. It then follows from Lemma 5 that b1 m b 2 s e gym b, where gys ␣y Ž b ., and the Bgq , ⴢ m Bgy1 m Bgq , ⴢ -terms in Ž ⌬ m 1. ⌬Ž b . q , ⴢ come from ⌬Ž e gy . m b. Similarly, the Bgq , ⴢ m Bgy1 m Bgq , ⴢ -terms in q , ⴢ Ž1 m ⌬ . ⌬Ž b . come from b m ⌬Ž e g .. y Let Ý i ␭ i c i m bi be all the Bgy1 m Bgq , ⴢ -terms in ⌬Ž e gy .. Let Ý j ␭Xj bXj m q , ⴢ X c j be all the Bgq , ⴢ m Bgy1 -terms in ⌬Ž e gy .. Then by comparing the q , ⴢ Bgq , ⴢ m Bgqy1 , ⴢ m Bgq , ⴢ -terms in ⌬Ž e gy . m b and b m ⌬Ž e gy ., we get

Ý ␭i b m c i m bi s Ý ␭Xj bXj m cXj m b. i

j

bXj

Consequently, all bi s b, all

ž Ý ␭ c / m b q no B . s b m ž Ý ␭ c / q no B

⌬ Ž e gy . s ⌬ Ž e gy

s b, and

i i

i

i i

i

m g y1 q , ⴢ gq , ⴢ

Bgq , ⴢ-terms,

Ž 15 .

m Bgy1 -terms. q , ⴢ

Ž 16 .

The equality Ž15. shows that for any gqg Gq and gyg Gy, all the Bgy1 m Bgq , ⴢ -terms in ⌬Ž e gy . have the same right component. Now apply q , ⴢ this fact to gy1 q g Gq and gyg Gy; we see that all the B g q , ⴢ m B gy1 q , ⴢ terms in ⌬Ž e gy . have the same right component. Then Ž16. tells us that Ý i ␭ i c i is really only one term. We thus conclude that for any gqg Gq and gyg Gy, the Bgy1 m Bgq , ⴢ -term in ⌬Ž e gy . is unique. q , ⴢ Now let us prove the uniqueness of the element b satisfying ␣q Ž b . s gq and ␤y Ž b . s gy. We already have one such element appearing in ⌬ Ž e gy . s c m b q no Bgqy1 , ⴢ m Bgq , ⴢ -terms. Now suppose b⬘ is another basis element satisfying ␣q Ž b⬘. s gq and X ␤y Ž b⬘. s gy. Let ␣y Ž b⬘. s gy . As before, we write Žwe do not know a X priori that gys gy . ⌬ Ž b⬘ . s e gyX m b⬘ q b⬘ m e gyq no B m Gy-terms and no Gym B-terms,

HOPF ALGEBRAS WITH POSITIVE BASES

431

and compare the Bgq , ⴢ m Bgqy1 , ⴢ m Bgq , ⴢ -terms in Ž ⌬ m 1. ⌬Ž b⬘. and Ž1 m ⌬ . ⌬Ž b⬘.. We get Bgq , ⴢ m Bgqy 1 , ⴢ- term in ⌬ Ž e gyX . m b⬘ s b⬘ m c m b. Comparing the rightmost components, we conclude that b s b⬘. We make some remarks about the lemma above. First of all, observe that c m b is also the unique B m Bgq , ⴢ -term Žas well as the unique Bgy1 m B-term. in ⌬Ž e gy .. As a matter of fact, by q , ⴢ Lemma 4, any B m Bgq , ⴢ -term in ⌬Ž e gy . must be a Bgy1 m Bgq , ⴢ -term. q , ⴢ Second, if we start with gqg Gq and gyg Gy, similar proof also tells us that there is a unique B ⴢ, gqm B ⴢ, gy1 -term b m c contained in ⌬Ž e gy .. q Moreover, b is the unique element such that ␤q Ž b . s gq and ␣y Ž b . s gy. Briefly speaking, one may start with mŽ S m 1. ⌬Ž e gy . s 1 and conclude ⌬Ž e gy . contains a B ⴢ, gqm B ⴢ, gqy1 -term b m c. Then by comparing the B ⴢ, gq m B ⴢ, gy1 m B ⴢ, gq-terms in Ž ⌬ m 1. ⌬Ž b . and Ž1 m ⌬ . ⌬Ž b ., our claim follows. q In particular, we also see that b ª Ž ␤q Ž b ., ␣y Ž b .. is a one-to-one correspondence between B and Gq= Gy. Third, we have the one-to-one correspondences Ž ␣ q, ␤ y.

Ž ␤ q, ␣ y.

Ž gq , gy . ¤ b ª Ž gq , gy . ,

Ž 17 .

⌬ Ž e gy . s ␭ c m b q ⭈⭈⭈ ,

Ž 18 .

⌬ Ž e gy . s ␭ b m c q ⭈⭈⭈ .

Ž 19 .

and the equalities

The one-to-one correspondences in Ž17. enable us to define a right action of Gy on Gq Gq= Gyª Gq : Ž gq , gy . ¬ gqgys gq ,

Ž 20 .

and a left action of Gq on Gy Gq= Gyª Gy : Ž gq , gy . ¬ gq gys gy .

Ž 21 .

In Lemma 12, we will show that they are the actions Ž4. and Ž5. in our standard model and that they satisfy the matching relations Ž6.. The action Ž20. has the following interpretation: Given gq and gy, we use Ž18. to find the unique b g Bgq , ⴢ . Then gqs ␤q Ž b . is gy gq Ži.e., b g Bgq , gq .. The action Ž21. has the following interpretation: Given gq and gy, we establish Ž18. and Ž19. with b g Bg , ⴢ . Then the element gy in Ž19. is gygq. q

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The reason for the interpretations above follows from Ž15., Ž16., and the uniqueness. We may also find gq and gy from gq and gy in a similar way. These are the other two actions in the standard model, and we will not use them. LEMMA 9. For any b g Bgq , gq, there is b⬘ g Bgq , gq such that bb⬘ s ␭ d gq and b⬘b s ␭ d gq for some ␭ ) 0. Proof. Let ␤y Ž b . s gy. From Lemma 8 and the first two of the subsequent remarks, we have a unique B m B ⴢ, gq-term c m b in ⌬Ž e gy .. Then by mŽ S m 1. ⌬Ž e gy . s ⑀ Ž e gy . s 1, we have SŽ c . b s ␮ d gq for some ␮ ) 0. By positivity, there is some term b⬘ in SŽ c . such that b⬘b s ␭ d gq for some ␭ ) 0. We claim that b⬘ g Bgq , gq. From d gq b⬘b s ␭ d g2q s ␭ d gq/ 0, we have d gq b⬘ / 0, so that ␣q Ž b⬘. s gq. From Ž b⬘d gq . b s b⬘Ž d gq b . s b⬘b s ␭ d gq / 0, we have b⬘d gq/ 0, so that ␤q Ž b⬘. s gq. We have shown that for any b g Bgq , gq, there is b⬘ g Bgq , gq such that b⬘b s ␭ d gq for some ␭ ) 0. Applying this conclusion to b⬘, there is b⬙ g Bgq , gq such that b⬙ b⬘ s ␭⬘d gq for some ␭⬘ ) 0. Then we have b⬙ b⬘b s Ž b⬙ b⬘ . b s ␭⬘d gq b s ␭⬘b,

b⬙ b⬘b s b⬙ Ž b⬘b . s ␭ b⬙ d gqs ␭ b⬙ .

This implies that ␭ s ␭⬘ and b⬙ s b. The next lemma shows that the antipode S is a permutation on B up to rescaling. We give a proof that is also valid in the corresponding category. For Hopf algebras over ⺓, it can also be proved more directly by using the positivity of S and the fact that S N s id for some positive integer N. LEMMA 10. Gi¨ en any b g B, SŽ b . s ␭ b⬘ for some b⬘ g B and ␭ ) 0. Moreo¨ er, we ha¨ e ␣y Ž b⬘. s ␤y Ž b .y1 , ␤q Ž b⬘. s ␣q Ž b .y1 , and these uniquely determine b⬘. Proof. First we show that SŽ e gy . s e gyy1 for any gyg Gy. We already know SŽ e gy . s e gyy1 q no Gy-terms. Suppose gy/ e, and S Ž e gy . contains b f Gy. Then by Lemma 7, hqs ␣q Ž b . / e, so that d hq SŽ e gy . s . s 0. On the other hand, d h SŽ e g . still contains d h b s b and SŽ e gy d hy1 q q y q cannot vanish. The contradiction shows that SŽ e gy . s e gyy1 . On the other hand, suppose gys e. Then we already know e gys d e and SŽ d e . s d e . Let g " and g " be associated to b as in Ž17.. Then from Lemma 8 we have ⌬ Ž e gy . s ␭ c m b q ⭈⭈⭈ . Applying S, we have ⌬ Ž e gyy1 . s ␭ S Ž b . m S Ž c . q ⭈⭈⭈ .

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HOPF ALGEBRAS WITH POSITIVE BASES

The proof of the last lemma implies that S Ž c . contains a term c⬘ g Bgq , gq. In fact, we may take c⬘ to be the element b⬘ in that proof. Therefore, if SŽ b . contains b⬘, then b⬘ m c⬘ must be contained in ⌬Ž e gyy1 .. By the first . must be unique. remark after Lemma 8, the B m Bgq , gq-term in ⌬Ž e gy1 y Therefore, for the fixed c⬘, there is only one term like b⬘ m c⬘ in ⌬Ž e gyy1 .. This implies that there is only one term, a constant multiple of b⬘, contained in SŽ b .. Thus we conclude the first part of the lemma, and have S Ž b . s ␮ b⬘,

S Ž c . s ␯ c⬘,

⌬ Ž e gyy1 . s ␭␮␯ b⬘ m c⬘ q ⭈⭈⭈ .

Moreover, we have also assumed that c⬘ g Bgq , gq. By Lemmas 4 and 6, we know that ␣y Ž b⬘. s ␤y Ž b .y1 and ␤q Ž b⬘. s ␣q Ž b .y1 . By the second remark after Lemma 8, this uniquely determines b⬘. LEMMA 11. bc: Ž1. Ž2.

For b g Bgq , gq and c g Bhq , hq, there are two possibilities for

If gqs hq, then bc is a positi¨ e multiple of an element in Bgq , hq; If gq/ hq, then bc s 0.

Proof. If gq/ hq, then bc s Ž bd gq .Ž d hq c . s bŽ d gq d hq . c s 0. If gqs hq, then Lemma 9 immediately implies bc / 0. Suppose bc contains two different terms d, d⬘ g B. Then by d gq bc s bc s bcd hq and positivity, we see that d, d⬘ g Bgq , hq. By Lemma 9, there is b⬘ g Bgq , gq such that bb⬘ s ␭ d gq and b⬘b s ␭ d gq for some ␭ ) 0. Then b⬘bc s ␭ c contains b⬘d and b⬘d⬘. By positivity, we must have b⬘d s ␮ c and b⬘d⬘ s ␮⬘c for some ␮ ) 0 and ␮⬘ ) 0. Thus ␭ d s ␭ d gq d s bb⬘d s ␮ bc, ␭ d⬘ s ␭ d gq d⬘ s bb⬘d⬘ s ␮⬘bc. This is in contradiction with the assumption that d and d⬘ are distinct. Lemma 10 implies that Sy1 exists, and for any b g B, the coordinates of S b . are nonnegative. Therefore, B* is also a positive basis for Ž H *. o p. In particular, all the proofs we have done so far are value in Ž H *. o p. Note that the two actions Ž20. and Ž21. are related by duality in the following way, y1 Ž

op

H ␣q Ž b . s gq

m l

H* ␣y Ž b* . s gy

m l

␤q Ž b . s gq

l

␤y Ž b* . s gy

l

␣y Ž b . s gy

l

␣q Ž b* . s gq

l

␤y Ž b . s gy

l

␤q Ž b* . s gq

l

Ž H *. op ␣y Ž b* . s gy , ␤yo p Ž b* . s gy , ␣qo p Ž b* . s gq , ␤qo p Ž b* . s gq ,

gqs gqgy

l

gys gygq

l

gys gq gy ,

gys gq gy

l

gqs gy gq

l

gqs gqgy.

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Therefore, by restating the results we have proved in Ž H *. o p , we may exchange the two actions in these results. LEMMA 12. Relations Ž20. and Ž21. are indeed actions. Moreo¨ er, the two actions satisfy the matching relations Ž6.. hy Proof. Given gqgys hq and hq s kq, we may find Žby the third remark after Lemma 8. b g Bgq , hq and b⬘ g Bhq , kq, with gys ␤y Ž b . and hys ␤y Ž b⬘.. Then by Lemma 11, we have bb⬘ s ␭ c, with ␭ ) 0 and c g Bgq , kq. hy By Lemma 6, we have ␤y Ž c . s gy hy. Therefore, gqgyhys kqs hq s g h Ž gqy. y. This proves Ž20. is a right action. As explained in the remark above, we may exchange the two actions and still get an equality. Applying the principle to gqgyhys Ž gqgy. hy, we get hqgq gys hqŽgq gy .. This proves Ž21. is a left action. Now we prove the matching relations. By the third remark after Lemma 8, we interpret the actions gqs gqgy and gys gq gy as the following equalities,

⌬ Ž e gy . s ␭ c m b q ⭈⭈⭈ , ⌬ Ž e gy . s ␭ b m c q ⭈⭈⭈ , We also interpret the action hys

gq

b g Bgq, gq.

Ž 22 .

hy as the following equalities,

⌬ Ž e hy . s ␭⬘c⬘ m b⬘ q ⭈⭈⭈ , ⌬ Ž e hy . s ␭⬘b⬘ m c⬘ q ⭈⭈⭈ ,

b g Bgq , ⴢ .

Ž 23 .

Multiplying Ž22. and Ž23. together, we have Žusing Lemma 11 so that cc⬘ and bb⬘ are elements of B up positive rescaling. ⌬ Ž e gy hy . s ⌬ Ž e gy . ⌬ Ž e hy . s ␭␭⬘cc⬘ m bb⬘ q ⭈⭈⭈ ,

bb⬘ g Bgq , ⴢ . Ž 24 .

⌬ Ž e gy hy . s ⌬ Ž e gy . ⌬ Ž e hy . s ␭␭⬘bb⬘ m cc⬘ q ⭈⭈⭈ ,

By the third remark after Lemma 8, Ž24. means gy hys gq Ž gy hy .. Therefore, gq

gy

Ž gy hy . s gy hys gq gy gq hys gq gy Ž gq . hy .

This is the matching relation. Applying the remark before the lemma, we get the other matching relation g

g

Ž q gy . gy gq . Ž hq gq . y s hq

HOPF ALGEBRAS WITH POSITIVE BASES

435

With a matching pair of groups Gq, Gy, we may construct a group G s Gq Gy and projections ␣ ", ␤ ": G ª G ". Furthermore, we have a standard Hopf algebra H Ž G; Gq, Gy . with G as a positive basis. The next lemma shows that up to positive scaling, the product between the basis elements of H is the same as the product between the basis elements of H Ž G; Gq, Gy .. LEMMA 13.

bb⬘ is a positi¨ e multiple of c if and only if

␣q Ž b . s ␣q Ž c . ,

␤q Ž b . s ␣q Ž b⬘ . ,

␣y Ž b . ␣y Ž b⬘ . s ␣y Ž c . .

Moreo¨ er, for fixed c, Ž b, b⬘. ¬ ␣y Ž b . is a one-to-one correspondence between the pairs Ž b, b⬘. and Gy. Proof. The necessity follows from Ž9., Ž14., and Lemma 11. For sufficiency, the condition ␤q Ž b . s ␣q Ž b⬘. implies bb⬘ s ␭ c⬘ for some ␭ ) 0 and c⬘ g B. Then we have ␣q Ž c . s ␣q Ž b . s ␣q Ž c⬘. and ␣y Ž c . s ␣y Ž b . ␣y Ž b⬘. s ␣y Ž c⬘.. If we can show Ž ␣q Ž c ., ␣y Ž c .. are in one-to-one correspondence with Ž ␣q Ž c ., ␤y Ž c .., then it follows from Lemma 8 that c s c⬘. By the definition of the action of Gq on Gy, from Ž ␣q Ž c ., ␤y Ž c .. we have ␣y Ž c . s␣qŽ c.␤y Ž c .. Since this is a group action, from Ž ␣q Ž c ., ␤y Ž c .. ␣qŽ c.y1 Ž . Ž ␣y c .. This proves the sufficiency. we also have ␤y c s For the one-to-one correspondence, we fix ␣y Ž b . g Gy. The condition ␣q Ž b . s ␣q Ž c . then uniquely determines b. Moreover, the conditions ␤q Ž b . s ␣q Ž b⬘., ␣y Ž b . ␣y Ž b⬘. s ␣y Ž c . give us ␣q Ž b⬘. and ␣y Ž b⬘., which also have a unique corresponding b⬘. This proves the one-to-one correspondence between pairs Ž b, b⬘. and Gy. By making use of Lemma 1 and the remark before Lemma 12, we get the following dual of Lemma 13. It shows that up to positive scaling, the coproduct of basis elements of H is the same as the coproduct of basis elements of H Ž G; Gq, Gy .. LEMMA 14.

For any b g B, ⌬Ž b . contains exactly those c m c⬘ satisfying

␣y Ž c . s ␣y Ž b . ,

␤y Ž c . s ␣y Ž c⬘ . ,

␣q Ž c . ␣q Ž c⬘ . s ␣q Ž b . .

Moreo¨ er, for fixed b, c m c⬘ ¬ ␣q Ž c . is a one-to-one correspondence between c m c⬘ and Gq. 4. RESCALING Given a finite dimensional Hopf algebra H with a positive basis B, we have identified H with a standard Hopf algebra H Ž G; Gq, Gy . Žas well as

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B with G . up to positive scaling. In this section, we will rescale the elements of G by positive scalars so that the Hopf algebra structure in H and H Ž G; Gq, Gy . are exactly the same. The problem may be viewed in the following way. On the vector space V s ⺓G, we have the standard Hopf algebra structure H Ž G; Gq, Gy .. We also have the given Hopf algebra structure H. The problem is to find a function ␶ : G ª ⺢q such that g ¬ ␶ Ž g . g is a Hopf algebra isomorphism from H to H Ž G; Gq, Gy .. The Hopf algebra structure H is determined by

 g 4  h4 s ⌬ g 4 s

½

␭Ž g , h .  ghy 4 0

Ý

in case gqs hq , in case gq/ hq ,

␮ Ž h, k .  h4 m  k 4 ,

Ž 25 .

hqksg , kysh y

S g 4 s ␯ Ž g .  gy1 4 . The structure constants ␭Ž g, h., ␮ Ž h, k ., ␯ Ž g . are positive and defined exactly when the conditions indicated above are satisfied. We assume H is already well scaled on GUq and Gy, i.e., with canonical products, coproducts, and antipodes, Ž⺓Gq .* is a sub-Hopf algebra of H and Ž⺓Gy .* is a sub-Hopf algebra of H *. Then the Hopf algebra condition means the following compatibility equations Žin cases both sides are defined.:

␭Ž g , h . ␭Ž ghy , k . s ␭Ž g , hky . ␭Ž h, k . , ␮ Ž g , h . ␮ Ž gq h, k . s ␮ Ž g , hq k . ␮ Ž h, k . , X ␭ Ž hq k, hXq k⬘ . ␮ Ž hhXy , kky . s ␭Ž h, h⬘ . ␭Ž k, k⬘ . ␮ Ž h, k . ␮ Ž h⬘, k⬘ . ,

␭Ž hq , h . s 1, ␭ Ž h, hq . s 1, ␮ Ž h, hy . s 1, ␮ Ž hy , h . s 1, ␯ Ž h . ␭ Ž hy1 , hy1 q hy . s 1, ␯ Ž h . ␭ Ž hy1 q hy , h . s 1. It is clear from the equations above that for any real number ⑀ , ␭⑀ , ␮ ⑀ , ␯ ⑀ still satisfy the compatibility equations. Therefore, if we replace ␭, ␮ , ␯ in Ž25. with ␭⑀ , ␮ ⑀ , ␯ ⑀ , we still get a Hopf algebra Žwhich is also well scaled on GUq and Gy ., which we denote by H⑀ . Then we have H s H1 and H Ž G; Gq, Gy . s H0 .

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HOPF ALGEBRAS WITH POSITIVE BASES

LEMMA 15. The Hopf algebras H⑀ are semisimple and cosemisimple. Proof. Let ⌳ s Ý gy g G y  gy 4 . Because H⑀ is already well scaled on Gy, we have x⌳ s ⑀ Ž x . ⌳. Therefore, ⌳ is a left integral of the Hopf algebra. Since ⑀ Ž ⌳ . s < Gy < / 0, it follows from wLSx that H⑀ is semisimple. Similarly, the integral Ý gq g G q  gq 4 * tells us that H⑀U is also semisimple. Consider H0 as a point in the space X of all bialgebra structures on V. We have natural action of the general linear group GLŽ V . on X. The orbit GLŽ V . H0 consists of all the bialgebra structures linearly isomorphic to H0 . According to Corollary 1.5 and Theorem 2.1 of wSx, the semisimplicity and the cosemisimplicity of H0 implies that the orbit GLŽ V . H0 is a Zariski open subset of X. Therefore, for sufficiently small ⑀ , H⑀ belongs to the orbit GLŽ V . H0 . In particular, for any ␦ ) 0, we can find an ⑀ ) 0 and a linear transformation T, such that T Ž H⑀ . s H0 and T g4 s

Ý

<␶g , g y 1 < - ␦ ,

␶g , h h4 ,

<␶g , h < - ␦

for h / g . Ž 26 .

hgG

LEMMA 16.

For sufficiently small ⑀ , T preser¨ es ⺓Gy.

Proof. Consider T gy 4 s Ý h g G ␶gy , h h4 . We would like to show ␶gy , h s 0 for h f Gy. If not, we find a term ␶gy , k k 4 , k f Gy, with <␶g

y,

k

< s max  <␶g

y,

h

< : h f Gy 4 .

Since k f Gy, we have kq/ e. Therefore,  kq 4 gy 4 s 0 in H⑀ . Since T preserves products, we have T kq 4T gy 4 s 0. On the other hand, we will show that for sufficiently small ⑀ , the coefficient of  k 4 in T kq 4T gy 4 cannot be zero. The contradiction proves the lemma. Write T  kq 4 s ␶ kq , kq  kq 4 q T  gy 4 s ␶gy , k  k 4 q

Ý h/kq

Ý hgG y

␶ kq , h h4 s P q Q,

␶gy , h h4 q

Ý hfG y, h/k

␶gy , h h4

s L q M q N. Suppose Ž26. is satisfied for some small ␦ ) 0. Then we have Ž1. Ž2. Ž3. number

PL s ␶ kq , kq ␶gy , k k 4 , with <␶ kq , kq ␶gy , k < G Ž1 y ␦ .<␶gy , k <; P Ž M q N . can never produce  k 4 ; Q Ž L q N . may produce some  k 4 -terms. However, the total of product terms is no more than Ždim H . 2 , and the coefficients

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LU, YAN, AND ZHU

are all smaller than ␦ <␶ gy , k <. Therefore, the k-coefficient of this part is bounded by ␦ <␶gy , k <Ždim H . 2 ; Ž4. QM is a sum of Gy-terms, so that it can never produce  k 4 . Now for sufficiently small ⑀ , we can find T satisfying Ž26., with ␦ small enough so that Ž1 y ␦ . ) ␦ Ždim H . 2 . Then we conclude that the  k 4 -term in T kq 4T gy 4 is nontrivial. LEMMA 17.

For sufficiently small ⑀ , T Ž⺓Ž Gqy e4.. ; ⺓Ž G y Gy ..

Proof. For any gqg Gq, gq/ e, write T  gq 4 s

Ý

hygG y

␶gq , hy  hy 4 q

Ý hfG y

␶gq , h h4 s P q Q.

If P is not zero, we have some kyg Gy such that <␶g

q,

ky

< s max  <␶g

q,

hy

< : hyg Gy 4 .

Since gq/ e, we have  gq 4 e4 s 0 in H⑀ , which implies T gq 4T e4 s 0. On the other hand, we will show that for sufficiently small ⑀ , the coefficient of  ky 4 in T gq 4T e4 cannot be zero. The contradiction proves the lemma. For sufficiently small ⑀ , we have T  e 4 s ␶e, e e 4 q

Ý

hygG y, h y/e

␶e, hy  hy 4 s L q M

from Lemma 16. Suppose Ž26. is further satisfied for some small ␦ ) 0. Then we have Ž1. The ky-term in PL is ␶g , k ␶e, e ky 4 , with <␶g , k ␶e, e < ) Ž1 y q y q y ␦ .<␶gq , ky <; Ž2. PM may have some  k 4 -terms, but the total number of product terms is no more than < Gy < 2 , and the coefficients are all smaller than ␦ <␶gq , ky <. Thus the k-coefficient of this part is bounded by ␦ <␶gq , ky < < Gy < 2 ; Ž3. Q Ž L q M . contains no Gy-terms, so that it can never produce  ky 4 . Now for sufficiently small ⑀ , we can find T satisfying Lemma 16 and Ž26., with ␦ small enough so that Ž1 y ␦ . ) ␦ < Gy < 2 . Then we conclude that the  ky 4 -term in T gq 4T e4 is nontrivial. LEMMA 18.

For sufficiently small ⑀ , T preser¨ es ⺓Ž G y Gy ..

Proof. Let g f Gy. Then gq/ e, so that by Lemma 17, T gq 4 contains no Gy-terms.

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In H⑀ , we have  gq 4 g 4 s  g 4 , which implies T g 4 s T gq 4T g 4 . Since T gq 4 contains no Gy-terms, T gq 4T g 4 also contains no Gy-terms. Consequently, T g 4 contains no Gy-terms. Lemmas 16 and 18 show that if we decompose ⺓G into ⺓Gy[ ⺓Ž G y Gy ., then T : H⑀ ª H0 takes the form Ts

ž

0 ¤ Gy . ) ¤ G y Gy

) 0

/

Ž 27 .

Now we apply this to the dual ŽT *.y1 : H⑀U ª H0U . Of course, to do so may require the choice of an even smaller ⑀ , so that all the lemmas work for ŽT *.y1 . Since the groups Gq and Gy derived from H * are the same as the groups Gy and Gq derived from H, we conclude that with regard to the decomposition Ž⺓G .* s Ž⺓Gq .* [ Ž⺓Ž G y Gq ..*,

Ž T *.

y1

s

ž

U 0 ¤ Gq . ) ¤ G* y GUq

) 0

/

Consequently, we get Ts

ž

0 ¤ Gq . ) ¤ G y Gq

) 0

/

In particular, T preserves Ž⺓Gq .*. LEMMA 19.

For sufficiently small ⑀ , T is identity on Gq.

Proof. Let gqg Gq. For sufficiently small ⑀ , from the discussion above we have T  gq 4 s

Ý

hqgG q

␶gq , hq  hq 4 ,

where ␶gq , gq is close to 1 and ␶gq , hq is small for hq/ gq. Moreover, since H⑀ is already well scaled on GUq, we have  gq 4 2 s  gq 4 in H⑀ . Therefore in H0 , we have

Ý

hqgG q

2

␶gq , hq  hq 4 s T  gq 4 s Ž T  gq 4 . s

Ý

h qgG q

␶g2q , hq  hq 4 .

␶g2q , hq .

Comparing the two sides, we have ␶gq , hqs Since ␶gq , gq is the only coefficient close to 1, we conclude that ␶gq , hqs ␦ gq , hq. The lemma also applies to the dual Hopf algebras. Taking the adjoint and then the inverse of Ž27., we have

Ž T *.

y1

s

ž

) 0

U 0 ¤ Gy . ) ¤ G* y GUy

/

Ž 28 .

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LU, YAN, AND ZHU

Since GUy in H * plays the role of Gq in H, the dual of Lemma 19 then shows that ŽT *.y1 is identity on GUy, so that

Ž T *.

y1

s

ž

U 0 ¤ Gy . ) ¤ G* y GUy

I 0

/

Ž 29 .

By taking the inverse and then the adjoint of Ž29., we see that T is also identity on Gy. Having proved T is identity on G ", we are ready to show T is a scaling everywhere. LEMMA 20.

For sufficiently small ⑀ , T is a scaling on G.

Proof. For any g s gq gyg G, we have  gq 4 g 4 s  g 4 in H⑀ and T gq 4 s  gq 4 . Then T g 4 s T Ž gq 4 g 4. s  gq 4T g 4 implies that T g 4 is a linear combination of terms  h4 with hqs gq. This is equivalent to the fact that, with regard to the disjoint union decomposition @gq g G q  h g G: ␣q Ž h. s gq 4 , the matrix of T is a block diagonal one. Similarly, the matrix of ŽT *.y1 is a block diagonal one with regard to the decomposition @gy g G y  h* g G*: ␤y Ž h. s gy 4 . Therefore, the matrix of T is also block diagonal with regard to the decomposition @gy g G y  h g G: ␤y Ž h. s gy 4 . This is equivalent to T g 4 is a linear combination of terms  h4 with hys gy. Thus we conclude that T g 4 is a linear combination of terms  h4 with hqs gq and hys gy. There is only one such term, namely,  g 4 . Therefore, T g 4 is a scalar multiple of  g 4 . The fact that g ¬ ␶ Ž g . g is a Hopf algebra isomorphism from H⑀ to H0 means exactly the following equalities ⑀

␭Ž g , h . s ⑀

␮ Ž g , h. s ⑀

␯ Ž g. s

␶ Ž g . ␶ Ž h. ␶ Ž ghy . ␶ Ž gq h . ␶ Ž g . ␶ Ž h. ␶ Ž g. ␶ Ž gy1 .

in case gqs hq , in case gys hy ,

.

The scaling function in Lemma 20 provides us with a function ␶ satisfying such equalities. Since all the structure constants ␭, ␮ , ␯ are positive, this

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implies that the positive scaling g ¬ <␶ Ž g .< 1r ⑀ satisfies the similar equalities without ⑀ power. In other words, g ¬ ␶ Ž g.

1r ⑀

g : H1 ª H 0

is an isomorphism of Hopf algebras.

5. HOPF ALGEBRAS IN THE CORRESPONDENCE CATEGORY In this section, we study the set-theoretical version of Theorem 1. We will assume all the sets are finite. As pointed out in the introduction, if we restrict ourselves to the usual category of finite sets, with maps as morphisms, then we will end up with only finite groups. Therefore, we enlarge the category to include correspondences, so that we may get a more interesting class of Hopf algebras. A correspondence from a set X to a set Y is a subset F of X = Y. Given a correspondence F from X to Y and a correspondence G from Y to Z, we have the composition F (G s  Ž x, z . : there is y g Y such that Ž x, y . g F and Ž y, z . g G 4 . With correspondences as morphisms, the finite sets form a category, which we call the correspondence category. With the usual product of sets as tensor product, a one-point set as an identity object, and the diagonal I X s Ž x, x .: x g X 4 ; X = X as the identity morphism, the correspondence category becomes a monoidal category as defined in wMacx. One may introduce the concept of an algebra Žcalled monoid in wMacx. in any symmetric monoidal category. By reversing the directions of arrows in all diagrams used in defining an algebra, we have the concept of a coalgebra. In the correspondence category, the product of two algebras is also an algebra. A correspondence bialgebra consists of an algebra structure and a coalgebra structure, such that the coalgebra map is an algebra morphism. A correspondence Hopf algebra is a correspondence bialgebra with an antipode making the usual diagrams commutative. The purpose of this section is to classify correspondence Hopf algebras. It is not convenient to work with correspondences directly. So we take the following Boolean viewpoint. First we observe that a correspondence F ; X = Y from X to Y gives rise to a map on the power sets PF : P Ž X . ª P Ž Y . , PF Ž A . s  b g Y : there is a g A such that Ž a, b . g F 4 .

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The map PF preserves union and sends the empty set to the empty set. Conversely, if a map P: P Ž X . ª P Ž Y . preserves union and sends the empty set to the empty set, then we have P s PF for the unique F given by F s Ž a, b .: b g P Ž a4.4 . Power sets can be considered as modules over the simplest Boolean algebra ⺒. ⺒ consists of two elements 0 and 1, and has the commutative sum and product 0 q 0 s 0,

1 q 0 s 1 q 1 s 1,

0 ⭈ 0 s 1 ⭈ 0 s 0,

1 ⭈ 1 s 1.

⺒ satisfies all the axioms of a ring except the existence of additive inverse. A ⺒-module is a set M with addition q, a special element 0, and a scalar multiplication ⺒ = M ª M, such that Ž1. q is commutative, associative, and 0 q x s x for all x g M; Ž2. 1 ⭈ x s x, 0 ⭈ x s 0; Ž3. Ž b1 q b 2 . ⭈ x s b1 ⭈ x q b 2 ⭈ x, b ⭈ Ž x 1 q x 2 . s b ⭈ x 1 q b ⭈ x 2 . It is straightforward to check that, under the first two conditions, the third condition means exactly x q x s x,

for all x g M.

Ž 30 .

In fact, a ⺒-module is equivalent to a lattice. In view of our assumption that all sets are finite, we will also assume all ⺒-modules are finite. With the obvious notion of homomorphisms, Žfinite. ⺒-modules form a category. The power set P Ž X . is a ⺒-module by A q B s A j B,

0 s ⭋.

It is easy to see that P: X ¬ PŽ X .,

F ¬ PF ,

is a functor from the correspondence category to the category of ⺒-modules. A subset B of a ⺒-module M is called a basis if every x g M is a unique linear combination of elements in B. A ⺒-module is said to be free if it has a basis. Clearly, the collection  x 4 : x g X 4 is a basis of P Ž X .. The following lemma Žespecially the third property. implies P is a one-to-one correspondence between finite sets and finite free ⺒-modules. LEMMA 21.

Let B be a basis of a ⺒-module M. Let b g B.

Ž1. If the b-coordinate of x g M is 1, then for any y g M, the b-coordinate of x q y is also 1. Ž2. If b s x 1 q ⭈⭈⭈ qx k for some x i g M, then each x i s b or 0. Ž3. The basis of M is unique.

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Proof. In the first property, we have x s b q ␰ , where ␰ is a sum of elements in B y  b4 . We also have two possibilities for y: ysbq␩

or

y s ␩,

where ␩ is a sum of elements in B y  b4 . Then by Ž30., we have xqys

½

bqbq␰q␩ bq␰q␩

if y s b q ␩ s b q ␰ q ␩, if y s ␩

and ␰ q ␩ is still a sum of elements in B y  b4 . This proves the first property. In the second property, if x 1 is neither b nor 0, then we have b⬘ g B y  b4 such that the b⬘-coordinate of x 1 is 1. By taking x s x 1 , y s x 2 q ⭈⭈⭈ qx k , and b s b⬘ in the first property, we see that the b⬘-coordinate of b is also 1. The contradiction shows that x 1 s b or 0. The proof for x i is similar. If we take the expression b s x 1 q ⭈⭈⭈ qx k in the second property as the expansion of b Žin one basis B . in terms of another basis Žconsisting of x 1 , . . . , x k and some others., then we immediately see that B is the only basis of M. The tensor product of ⺒-modules can be defined in the same way as modules over commutative rings. With ⺒ as the unit object, ⺒-modules then form a monoidal category. Since we clearly have P Ž X . m P Ž Y . s P Ž X = Y . and P Žone point. s ⺒, P is a functor of monoidal categories. Concepts such as ⺒-algebra, ⺒-coalgebra, and ⺒-Hopf algebra can be introduced just like the corresponding concepts over commutative rings. Since P is a functor of monoidal categories, P sends correspondence Hopf algebras to free ⺒-Hopf algebras. Moreover, because of the third property in Lemma 21, P is a one-to-one correspondence between correspondence Hopf algebras and free ⺒-Hopf algebras. Motivated by the classification of Hopf algebras with positive bases, we may introduce free ⺒-Hopf algebras H Ž G; Gq, Gy . as in Section 2 and conjecture that these are all the free ⺒-Hopf algebras. This is indeed the case. THEOREM 2. Any free ⺒-Hopf algebra is isomorphic to H Ž G; Gq, Gy . for a unique group G and a unique factorization G s Gq Gy. Proof. The theorem may be proved by basically repeating the contents of Section 3. There is no need to do rescaling Žand all the positive coefficients in Section 3 can be replaced by 1. since the only nonzero element of ⺒ is 1.

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Instead of repeating all the details, we discuss some key points involved in Section 3, and prove only one lemma to illustrate that the whole section works in the category of ⺒-modules. There are two key technical points in the proofs in Section 3: One is the use of duality, opposite, and co-opposite in translating a proven lemma into another lemma in similar or dual settings. The other is the use of positivity. The dual of ⺒-modules can be defined in the same way as modules over commutative rings, and satisfies similar properties such as Ž M*.* s M. The opposite and co-opposite of ⺒-Hopf algebras can be introduced similarly, as far as the antipode is invertible. As for the positivity, the role can be replaced by the three properties in Lemma 21. Here we only illustrate how this works by proving Lemma 2. Assume B is the basis of a ⺒-Hopf algebra H. The identity element 1 of H is a sum 1 s d1 q ⭈⭈⭈ qd k of basis elements. Then we have d i s d i ⭈ 1 s d i d1 q ⭈⭈⭈ qd i d k . Because d i is a basis element, by the second property in Lemma 21 we see that d i d j s 0 or d i . Similarly, from d j s 1 ⭈ d j , we see that d i d j s 0 or d j . Then we may conclude d i d j s 0 when i / j and d i d i s d i , as before. Similarly, from ⌬Ž d1 . q ⭈⭈⭈ q⌬Ž d k . s ⌬Ž1. s 1 m 1 s Ý i, j d i m d j , we may prove ⺒ d1 q ⭈⭈⭈ q⺒ d k is closed under ⌬. From SŽ d1 . q ⭈⭈⭈ qSŽ d k . s SŽ1. s 1 s d1 q ⭈⭈⭈ qd k , we may prove ⺒ d1 q ⭈⭈⭈ q⺒ d k is closed under S. Therefore, ⺒ d1 q ⭈⭈⭈ q⺒ d k is a Hopf subalgebra of H. The standard argument for classifying commutative Hopf algebras also applies to ⺒-modules. Then we know that ⺒ d1 q ⭈⭈⭈ q⺒ d k must be the dual ⺒-Hopf algebra of the group algebra ⺒Gq. This completes the proof of Lemma 2 for ⺒-Hopf algebras. By the one-to-one correspondence between correspondence Hopf algebras and free ⺒-Hopf algebras, we may translate Theorem 2 into the following classification of correspondence Hopf algebras. THEOREM 3. Any finite correspondence Hopf algebra is a finite group G, with a unique factorization G s Gq Gy such that m s  Ž gy hq , hq ky , gy hq ky . : hqg Gq , gy , kyg Gy 4 ; Ž G = G . = G, ⌬ s  Ž gq hy kq , gq hy , hy kq . : gq , kqg Gq , hyg Gy 4 ; G = Ž G = G . , 1 s  Ž pt, gq . : gqg Gq 4 ;  pt 4 = G,

⑀ s  Ž g , pt . : gyg Gy 4 ; G =  pt 4 , S s  Ž g , gy1 . : g g G 4 ; G = G.

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