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On Semi-groups Arising from Quantum Groups
Journal of Algebra 212, 669]682 Ž1999. Article ID jabr.1998.7631, available online at http:rrwww.idealibrary.com on
On Semi-groups Arising from Quantum Groups Zhen-hua Wang* Department of Mathematics, East China Normal Uni¨ ersity, Shanghai, 200062, People’s Republic of China Communicated by Walter Feit Received July 30, 1997
Let Gq be a quantum group. In this paper, we introduce and completely characterize a semi-group and its Borel sub-semi-groups consisting of algebra homomorphisms from the coordinate algebra k w Gq x of Gq . Also, we present invariant elements for the semi-group, which are closely connected to Hopf ideals of k w Gq x. Finally, based on the theory of structures of the semi-group, we prove the existence of a Frobenius morphism over Gq whenever the parameter q is a root of unity. Q 1999 Academic Press
INTRODUCTION The approach to quantum groups includes the study of coordinate algebras related to Yang]Baxter operators. This was carried out only for the case of type A by Parshall and Wang w3x, although all Yang]Baxter operators of classical types had been found a long time before. The difficulty lies in the complexity of coordinate algebras of quantum groups. To solve this, we develop the theory of semi-groups with generators respecting structures of some closed subgroups of the quantum groups, which resorts to techniques from algebraic groups. In developing the theory, root systems emerge in the form of indices and are closely related to structures of coordinate algebras of quantum groups. As applications of the theory, aside from a success in the proof of an important theorem, i.e., the theorem of density of big cell, the existence of Frobenius morphisms for quantum groups with parameters of roots of unity is also verified. *I take this opportunity to give my special thanks to Professors Jian-pan Wang and Jian-yi Shi for their leading me into the field of quantum groups, and Jia-shen Ye for much encouragement. 669 0021-8693r99 $30.00 Copyright Q 1999 by Academic Press All rights of reproduction in any form reserved.
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Fix a field k in any characteristic. In this paper, we adopt the ‘‘naive’’ point of view of w3x, identifying the category QGrk of quantum groups over k with the dual of the category k-Hopf of finitely generated k-Hopf algebras. The Hopf algebra corresponding to a quantum group Gq , denoted by k w Gq x, is called the coordinate algebra of Gq . Moreover, let H and H1 be two quantum groups. Then H1 is called as a subgroup of H whenever there is a surjective Hopf algebra homomorphism k w H x ª k w H1 x. Let Gq be a fixed quantum group. We organize the paper as follows. Section 1 presents some concepts, e.g., generic point and root system, etc. Section 2 introduces a fundamental semi-group associated to Gq and two of the Borel sub-semi-groups. Also we develop the basic theory on the semi-group, which includes the connection of some distinguished elements, called invariant elements in the paper, of the semi-group and Hopf ideals of k w Gq x. Sections 3, 4, and 5 are devoted to determining the invariant elements of the semi-group and its Borel sub-semi-groups. As a q-analogue of the affine group, a semi-simple quantum group is defined, based on the density of ‘‘big cells’’ in Section 5. Section 6 characterizes entirely structures of the semi-group and the Borel sub-semi-groups. Finally, Section 7 proves that there is a surjective homomorphism from the semi-group associated to Gq to that associated to an algebraic group, which is equivalent to the existence of the Frobenius morphism from Gq to the algebraic group, in case q is a root of unity. In a sequel, we shall prove that quantum groups of classical types, associated to Yang]Baxter operators, are essentially determined by these semi-groups.
1. THE GENERIC POINTS AND THE ROOT SYSTEM Let K be a quasi-polynomial algebra over k with generators t, ty1 , x 4 subject to the relations tty1 s ty1 t s 1,
xt s qtx.
Then, we can make K into two Hopf algebras, denoted by K 1 and K 2 , respectively. The associated comultiplications D 1 and D 2 , and antipodes g 1 and g 2 are respectively defined as follows. D 1 sends t to t m t, and x to ty1 m x q x m t; D 2 sends t to t m t and x to t m x q x m ty1 ; g 1 sends t to ty1 and x to qy1 x, and g 2 sends t to ty1 and x to qx. DEFINITION 1.1. Let Gq be a quantum group with a maximally diagonal torus T and X ŽT . the character group of T. If there are a surjective Hopf algebra homomorphism f : k w Gq x ª K 1 Žresp. f : k w Gq x ª K 2 . and a Hopf
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671
algebra homomorphism f T : k w T x ª K sending a unique element a g X ŽT . to t 2 Žresp. ty2 . such that the following commutative diagram holds
Ki 6
fT
Ki
6
6
kwT x
f
6
k w Gq x
for i s 1 Žresp. i s 2., where two vertical maps are canonical, then f is called as a positive Žresp. negative. generic-point of Gq associated to a or simply called an a-point in the language of algebraic geometry. Denote by P Ž Gq ., PqŽ Gq ., and PyŽ Gq ., respectively, the set of generic-points, positive generic-points, and negative generic-points of Gq , respectively. Usually, an a-point of Gq is written as fa for a g X ŽT .. DEFINITION 1.2. Ž1. Let Gq be a quantum group with a maximally diagonal torus T ; let RŽ Gq ., RqŽ Gq ., and RyŽ Gq . be subsets of X ŽT . such that P Ž Gq . s fa < a g R Ž Gq . 4 , Pq Ž Gq . s fa < a g Rq Ž Gq . 4 Py Ž Gq . s fa < a g Ry Ž Gq . 4 . Then RŽ Gq . s RqŽ Gq . j RyŽ Gq . is clearly a partition; RŽ Gq ., RqŽ Gq ., and RyŽ Gq . are defined as a root system, a positive system, and a negative system of Gq , respectively. Ž2. Let a g RqŽ Gq .. a is called as a simple root of Gq if there are no a 1 , a 2 g RqŽ Gq . such that a s a 1 q a 2 ; denote by P Ž Gq . the set of simple roots of Gq . A generic point, corresponding to a simple root, is called a simply generic point. Denote by SP the set of simply generic points. Also, put SPys fya g PyŽ Gq .< a g P Ž Gq .4 . Ž3. For a g RqŽ Gq . and an algebraic group homomorphism f : k=ª T, define ² a , f : to be an element in Z such that a f Ž m. s m² a , f : for any m g K= where k= is the multiplicative group of units of k. In particular, there is an f, called the coroot of a and denoted by a ¨ , such that ² a , f : s 2. In the remainder of this paper, we shall assume that P Ž Gq . is identified with a simple laced system of simple roots.
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2. THE SEMI-GROUP AND THE BOREL SUB-SEMI-GROUPS Fix a quantum group Gq with a maximally diagonal torus T throughout this section. Let I Ž Gq . be the set of all ideals of k w Gq x and F Ž Gq . the set of all unity-preserving homomorphisms of k-algebras from k w Gq x. Then there is a map ker: F Ž Gq . ª I Ž Gq . with kerŽ f . denoting the kernel of f for f g F Ž Gq .. Define an equivalence relation Gq on F Ž Gq . such that f Gq g iff kerŽ f . s kerŽ g . for f, g g F Ž Gq . and have then a quotient set F Ž Gq .rGq , consisting of equivalence classes of elements, of F Ž Gq .. Also, ker induces a 1-1 map Žstill written as. ker: F Ž Gq .rGq ª I Ž Gq .. PROPOSITION 2.1. There is a multiplication ) on F Ž Gq .rGq such that F Ž Gq .rGq becomes a semi-group with the unity « under it. Proof. Let D and « be the comultiplication and the counit of k w Gq x, respectively. Define the multiplication ) on F Ž Gq .rGq as f 1 ) f 2 s Ž f 1X m f 2X . D
for f 1 , f 2 g F Ž Gq . rGq ,
where f 1X and f 2X are representatives for f 1 and f 2 , respectively. It is well defined, indeed, if f 1Y Gq f 1X and f 2Y Gq f 2X . Then it follows that ker Ž Ž f 1X m f 2X . D . s ker Ž Ž f 1Y m f 2Y . D . . As for the associativity of ), it follows now from the coassociativity of D, while the claim that « is the unity follows from the fact that « is the counity of k w Gq x, proving the proposition. Let BSqŽ Gq ., BSyŽ Gq ., and BSŽ Gq . be sub-semi-groups, generated by equivalence classes of elements of SP Ž Gq ., SPyŽ Gq ., and SPyŽ Gq . j SPqŽ Gq ., respectively, of F Ž Gq .rGq . DEFINITION 2.2. BSŽ Gq ., BSqŽ Gq ., and BSyŽ Gq . are called a fundamental semi-group of Gq , a positive Borel sub-semi-group, and a negative Borel sub-semi-group, respectively. 2 THEOREM 2.3. Ž1. fa ) fy b s fy b ) fa , fa2 s fa , and fy a s fy a Ž . for a / b g P Gq . Ž2. The multiplication ) satisfies braid relations on SPqŽ Gq . Ž resp. yŽ SP Gq ..; in other words, we ha¨ e for fa , fb g SPqŽ Gq . Ž resp. SPyŽ Gq ..
fa ) fb ) fa s fb ) fa ) fb , fa ) fb s fb ) fa ,
² a , b ¨ : s ² b , a ¨ : s y1; ² a , b ¨ : s ² b , a ¨ : s 0.
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673
Ž3. There is an anti-automorphism i of BSŽ Gq . such that iŽ f x . s f x for x s a , ya g P Ž Gq .. Proof. Ž1. The claims are proved easily. Indeed, the claim that fa2 s 2 . fa Žresp. fy a s fy a follows from the fact that the corresponding algebra k Gq rker Ž fa .
Ž resp. k
Gq rker Ž fy a . .
is even a Hopf algebra. To prove the first claim, let X i, j < i, j g I 4 for some index set I be the coordinates of k w Gq x, and both
pa : k Gq ª k Gq rker Ž fa .
and
pyb : k Gq ª k Gq rker Ž fy b . canonical ŽHopf algebra. homomorphisms. Put P0 s X i , j < i / j, pa Ž X i , j . s 0, py b Ž X i , j . s 0 4 , P1 s X i , i y 1
Xi , l m Xl , j
Ý lGi , jGl
for 1 F i, j F 3. We can take the a-point fa in the form of
fa Ž X 1, 1 . s ty1 , fa Ž X 1, 3 . s 0,
fa Ž X 1, 2 . s x, fa Ž X 2, 3 . s 0,
fa Ž X 2, 2 . s t , fa Ž X 3, 3 . s 1
and the b-point fb in the form of
fb Ž X 1, 1 . s 1, fb Ž X 1, 3 . s 0,
fb Ž X 1 , 2 . s 0, fb Ž X 2, 3 . s x,
fb Ž X 2, 2 . s ty1 , fb Ž X 3, 3 . s t.
Thus, we have
fa ) fb ) fa Ž X 1, 1 . s ty1 m 1 m ty1 , fa ) fb ) fa Ž X 1, 2 . s x m ty1 m t q ty1 m 1 m x, fa ) fb ) fa Ž X 2 , 2 . s t m ty1 m t , fa ) fb ) fa Ž X 2, 3 . s t m x m 1,
fa ) fb ) fa Ž X 1, 3 . s x m x m 1, fa ) fb ) fa Ž X 3, 3 . s 1 m t m 1
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ZHEN-HUA WANG
and
fb ) fa ) fb Ž X 1, 1 . s 1 m ty1 m 1,
fb ) fa ) fb Ž X 1, 2 . s 1 m x m ty1 ,
fb ) fa ) fb Ž X 2, 2 . s ty1 m t m ty1 ,
fb ) fa ) fb Ž X 1, 3 . s 1 m x m x,
fb ) fa ) fb Ž X 2, 3 . s ty1 m t m x q x m 1 m t , fb ) fa ) fb Ž X 3, 3 . s t m 1 m t. Check easily that kerŽ fb fa fb . s kerŽ fa fb fa . is generated by X s, t
Ž s - t. ,
X s, t X s, m y qy1 X s, m X s, t
Ž t - m. ,
X t , s X m , s y qy1 X m , s X t , s
Ž t - m. ,
X s, t X l , m y X l , m X s, t
Ž s - l, t ) m . ,
X 1, 1 X 2, 2 X 3, 3 y 1, X s, t X l , m y X l , m X s, t y Ž 1 y q 2 . X s, m X t , l
Ž s - l, t - m .
for s, t, l, m g 1, 2, 34 . Ž3. Let g be the antipode of k w Gq x. Then the map i, defined by iŽ f . s fg for f g BSŽ Gq ., will work, proving the theorem. Let Q be a semi-group and g g Q. We call g an invariant element of Q if ga s g and ag s g for a g Q. THEOREM 2.4. With the abo¨ e notations, then the in¨ ariant element g is unique. Consequently, if i is an anti-automorphism of Q, then iŽ g . s g. Proof. The proof is straightforward. THEOREM 2.5. Let f g BSŽ Gq .. Then kerŽ f . is a Hopf ideal iff there is a sub-semi-group SG9 of BSŽ Gq . such that f is the in¨ ariant element of SG9. Proof. First of all, establish notations as follows. For i g Z, put
fi s
~¡f
ai
¢f
g SP Ž Gq . ,
ya < i <
g SPy Ž Gq . ,
if i ) 0; if i - 0.
Assume now that f s f i1 ) ??? ) f i s is a reduced expression for f . Let SG9 be a sub-semi-group of BSŽ Gq . generated by f i1, . . . , f i s 4 . Observe that kerŽ f . is a Hopf ideal if f s f ) f and iŽ f . s f . This is satisfied if f is the invariant element of the semi-group SG9.
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675
Conversely, it is clear that kerŽ f . ; kerŽ f i j . for j s 1, 2, . . . , s. If kerŽ f . is a Hopf ideal, then either f i j g SP Ž GqX . or f i j g SPyŽ GqX . according to whether i j ) 0 or - 0 for j s 1, 2, . . . , s, and thus, f is the invariant element of the semi-group SG9. Here GqX is a quantum group with the coordinate algebra k w Gq xrkerŽ f ., completing the proof. 3. THE INVARIANT ELEMENTS OF BOREL SUB-SEMI-GROUPS In this section, we consider BSqŽ Gq ., while all results on it apply to BSyŽ Gq .. Let P Ž Gq . s a 1 , a 2 , . . . , a n4 be a simple laced system of simple roots and W the Weyl group corresponding to it. Put fa i s f i g SP Ž Gq . for i g 1, 2, . . . , n4 . THEOREM 3.1. Let w 0 be the longest word in W and F s f i1 ) f i 2 ) ??? ) f i m g BSqŽ Gq .. If w 0 s a i1 a i 2 ??? a i m , then F is the in¨ ariant element of BSqŽ Gq .. Proof. The claim follows from the following facts: Ž1. Let F9 s f j ) f j ) ??? ) f jX . Then F s F9 if w 0 s a j a j ??? a jX , 1 2 m 1 2 m Ž by 2. of Theorem 2.3; Ž2. For given f i g SP Ž Gq ., there is a F9, related to w 0 in the way as in Ž1., such that F9 s F0) f i or F9 s f i ) F0 for some F0 g BSqŽ Gq .; Ž3. It follows that f i2 s f i by Ž1. of Theorem 2.3 for i s 1, 2, . . . , n. The proof is completed. 4. THE INVARIANT ELEMENTS OF FUNDAMENTAL SEMI-GROUPS OF RANK 1 In this section, we consider a sub-semi-group BSŽ Gq Ž1.. of BSŽ Gq . which is generated by fa , fy a 4 for a g P Ž Gq .. BSŽ Gq Ž1.. is usually called a fundamental semi-group of rank 1. THEOREM 4.1. Ž1. fa ) fy a s fy a ) fa . Ž2. fa ) fy a is the in¨ ariant element of BSŽ Gq Ž1... Proof. To prove Ž1., we only need to consider such a quantum group Gq that its coordinate algebra k w Gq x is generated by X i, j <1 F j, i F 24 with the comultiplication, D Ž Xi , j . s
Ý lFlF2
Xi , l m Xl , j
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for 1 F i, j F 2. Thus, we can take the ya-point fya in the form of
fya Ž X 1, 1 . s ty1 ,
fy a Ž X 2, 1 . s x,
fya Ž X 1, 2 . s 0,
fy a Ž X 2, 2 . s t
and the a-point fa in the form of
fa Ž X 1, 1 . s ty1 ,
fa Ž X 2 , 1 . s 0,
fa Ž X 1, 2 . s x,
fa Ž X 2 , 2 . s t.
We have
fya fa Ž X 1 , 1 . s ty1 m ty1 , fya fa Ž X 1, 2 . s ty1 m x,
fy a fa Ž X 2, 1 . s x m ty1 , fa fy a Ž X 2, 2 . s t m t q x m x
and
fa fy a Ž X 1, 1 . s ty1 m ty1 q x m x, fa fy a Ž X 1, 2 . s x m t ,
fa fy a Ž X 2, 1 . s t m x,
fa fy a Ž X 2, 2 . s t m t.
Check easily that kerŽ fya ) fa . s kerŽ fa ) fy a . is spanned by X s, t X s, m y qy1 X s, m X s, t y1
Xt , s Xm , s y q
Xm , s Xt , s
X s, t X l , m y X l , m X s, t
Ž t - m. , Ž t - m. ,
Ž s - l, t ) m . ,
X s, t X l , m y X l , m X s, t y Ž 1 y q 2 . X s, m X t , l
Ž s - l, t - m . ,
X 1, 1 X 2, 2 y qy1 X 1, 2 X 2, 1 y 1 for s, t, l, m g 1, 24 , proving Ž1.. As for Ž2., we have K w Gq xrkerŽ fy a ) fa . , k w SL 2 Ž q .x, which is the coordinate algebra of a special quantum group of rank 1, as in w3x, completing the proof.
5. THE INVARIANT ELEMENT OF THE FUNDAMENTAL SEMI-GROUP AND A SEMI-SIMPLE QUANTUM GROUP Assume that P Ž Gq . s a 1 , a 2 , . . . , a n 4 , SPq Ž Gq . s fa 1 , fa 2 , . . . , fa n 4
and
SPy Ž Gq . s fy a 1 , fy a 2 , . . . , fy a n 4 .
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677
Recall in Section 3 that there are fq and fy which are invariant in BSqŽ Gq . and BSyŽ Gq ., respectively. THEOREM 5.1. in BSŽ Gq ..
fq) fys fy) fq. Consequently, fq) fy is in¨ ariant
Proof. Put fya i s fyi and fa i s f i for i s 1, 2, . . . , n. To prove the claim, it is sufficient to prove that f i ) fyj s fyj ) f i for i, j s 1, 2, . . . , n. The claim follows from Ž1. of Theorem 2.3 for i / j, while, for i s j, this follows from Theorem 4.1. COROLLARY 5.2.
BSŽ Gq . is a finite semi-group, more explicitly, we ha¨ e 2
BS Ž Gq . s BSq Ž Gq . BSy Ž Gq . s BSq Ž Gq . . Proof. The claim is a direct consequence of Theorem 5.1. DEFINITION 5.3. Let Gq be as the above. Then, we call Gq a semi-simple quantum group of rank n if fq) fys 0 where fq and fy are defined as above. Note that the above definition for q s 1 is reconciled with that in algebraic groups. COROLLARY 5.4. With the abo¨ e notations, let Gq be a semi-simple y quantum group. Then we ha¨ e two subgroups Bq q and Bq , called the positi¨ e x w x Ž q. and negati¨ e Borel subgroups, respecti¨ ely, of Gq : k w Bq q s k Gq rker f yx y. q qx y w w x Ž w x w and k Bq s k Gq rker f . Let both p : k Gq ª k Bq and p : k w Gq x x be canonical homomorphisms of Hopf algebras. Then the algebra ª k w By q x w qx is injecti¨ e, where D homomorphism Žpym pq . D: k w Gq x ª k w By q m k Bq w x is the comultiplication of k Gq . Proof. The statement follows from Definition 5.3.
6. THE STRUCTURES OF SEMI-GROUPS In this section, we give the complete relations among generators of BSŽ Gq ., BSqŽ Gq ., and BSyŽ Gq ., respectively. Assume that P Ž Gq . s a 1 , a 2 , . . . , a n 4 , SPq Ž Gq . s f 1 , f 2 , . . . , fn 4
and
SPy Ž Gq . s fy1 , fy2 , . . . , fyn 4 .
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Let FSGq be a semi-group with a unity e, generated by z1 , z 2 , . . . , z n4 subject to the relations z i2 s z i , zi z j zi s z j zi z j , zi z j s z j zi ,
1 F i F n; ² a i , a j¨ : s y1; ² a i , a j¨ : s 0.
THEOREM 6.1. There is an isomorphism of semi-groups sq: FSGqª BSqŽ Gq . such that sqŽ z i . s f i for i s 1, 2, . . . , n. Proof. sq as a surjective homomorphism of semi-groups is clear, see wTheorem 2.3x. To prove that sq is an isomorphism, we need to prove that any relation
f i1 ) f i 2 ) ??? ) f i s s f j1 ) f j 2 ) ??? ) f j
t
in BSqŽ Gq . implies the relation z i1 ) z i 2 ) ??? ) z i s s z j1 ) z j 2 ) ??? ) z j t in FSGq. z i , z j 4 for i, j g Let FSGq i, j be a sub-semi-group, generated by 1, 2, . . . , n4 , of FSGq. Then there is a canonically surjective semi-group homomorphism p i, j : FSGqª FSGq i, j , sending z m to z m for m s i, j, z m to e for m other than i and j. An important observation is that FSGq is isomorphic to BSqŽ Gq . in case of n s 2. Thus, if a relation
f i1 ) f i 2 ) ??? ) f i s s f j1 ) f j 2 ) ??? ) f j t follows in BSqŽ Gq ., then the relation
p i , j Ž z i1 ) z i 2 ) ??? ) z i s . s p i , j Ž z j1 ) z j 2 ) ??? ) z j t . follows in FSGq for i, j g 1, 2, . . . , n4 . This enables us to conclude that z i1 ) z i 2 ) ??? ) z i s s z j1 ) z j 2 ) ??? ) z j t , proving the theorem. COROLLARY 6.2. There is an isomorphism of the semi-group sy: FSGq ª BSyŽ Gq . such that syŽ z i . s fyi g SPyŽ Gq . for i s 1, 2, . . . , n. Proof. The proof is entirely similar to the above.
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679
THEOREM 6.3. Let FBG be a semi-group generated by z1 , z 2 , . . . , z n4 and zX1 , zX2 , . . . , zXn4 subject to the relations z i zXj s zXj z i ,
z i2 s z i ,
zi z j zi s z j zi z j , zi z j s z j zi , zXi zXj zXi s zXj zXi zXj , zXi zXj s zXj zXi ,
2 Ž zXi . s zXi ,
1 F i , j F n;
² a i , a j¨ : s y1, ² a i , a j¨ : s 0; ² a i , a j¨ : s y1, ² a i , a j¨ : s 0.
Then there is an isomorphism of semi-groups s : FBG ª BSŽ Gq . such that s Ž z i . s f i and s Ž zXi . s fyi for 1 F i F n. Proof. The statement follows from Theorems 6.1 and 6.2, together with relations f i fyj s fyj f i for 1 F i, j F n. 7. THE FROBENIUS MORPHISM In this section, let q be a Nth primitive root of unity and Gq a semi-simple quantum group of rank n with a maximally diagonal torus T. Also, let G be a semi-simple algebraic group with the same maximally diagonal torus as that of Gq and choose the positive subsystem of the root system of G related to T determined by the upper triangle Borel subgroup. Here, we consider G a subgroup of GLŽ V . for some k-vector space V. x Let K 0 s k w t 0 , ty1 0 , x 0 be a polynomial algebra. Then there is a homomorphism of algebras f : K 0 ª K Žcf. Section 1., sending t 0 to t N and x 0 to x N. Moreover, K 0 can be endowed with two of the structures of Hopf algebras, denoted by K 01 and K 02 the corresponding Hopf algebras, such that both f : K 01 ª K 1 and f : K 02 ª K 2 are homomorphisms of Hopf algebras. LEMMA 7.1. There are surjecti¨ e Hopf algebra homomorphisms,
fa1 : k w Tg a x ª K 01
Ž resp. fy1 a : k w Tgy a x ª K 02 . . restrictions of which to k w T x send the unique a Ž resp. ya . to t 02 Ž resp. ty2 0 for a g P Ž G .. Here these g a are subgroups of G associated to roots a .
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ZHEN-HUA WANG
Proof. It is well known that k w Tg a x Žresp. k w Tgy a x. is generated by elements, say t 4 and x a Žresp. xya .. Moreover, among t 4 , there are ta and taX such that D Ž x a . s x a m ta q taX m x a
Ž resp. D Ž xya . s xya m taX q ta m xya . . Thus, we can define homomorphisms of Hopf algebras
fa1 : k w Tg a x ª K 01
Ž resp. fy1 a : k w Tgy a x ª K 02 . as
fa1 Ž taX . s ty1 0 ,
fa1 Ž ta . s t 0 , fa1 Ž t . s 1,
fa1 Ž xa . s x 0
Žresp. X 1 y1 fy a Ž ta . s t 0 ,
1 fy a Ž ta . s t 0 , 1 fy a Ž t . s 1,
1 fy a Ž xy a . s x 0 .
for t g t 4 other than ta and taX , proving the lemma. THEOREM 7.2. With the abo¨ e setup and the assumption that P Ž Gq . s P Ž G ., then there is a homomorphism F: k w G x ª k w Gq x of Hopf algebras such that the following commutati¨ e diagram follows: k w Gq x
F
k w T x,
6
6
kwT x
F
6
kw G x
6
Ž1.
where ¨ ertical maps are canonical and F < T Ž t . s t N for coordinates t 4 of T. Ž2. F is injecti¨ e and its images are central. Proof. Let G9 be a quantum matrix space the coordinate of which is a freely generated k-bialgebra k w G9x over coordinates X i,X j < i, j g J 4 corresponding to coordinates X i, j < i, j g J 4 of k w Gq x. Define a homomorphism of algebras F9: k w G9x ª k w Gq x sending X i,X j to X i,Nj for i, j g J. Again, let
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G be the algebraic group as the above with coordinates x i, j < i, j g J 4 in the same index set J and p : k w G9x ª k w G x a homomorphism of bialgebras sending X i,X j to x i, j for i, j g J. As in Section 2, F Ž G9.rG is the set of equivalence classes of algebra homomorphisms from k w G9x: f Gg iff kerŽ f . s kerŽ g . for f, g g k w G9x. 1 < Recall that BSŽ Gq . s ² fa , fy a < a g P Ž Gq .: and BSŽ G . s ² fa1 , fy a ag 1 1 P Ž G .:. Put BSŽ G9. s ² fa p 9, fy a p 9 < a g P Ž G .: ; F Ž G9.rG. Then, there is a map 1 < H : fa , fy a < a g P Ž G . 4 ª fa1 p 9, fy a p 9 a g P Ž G. 4 1 Ž . Ž . such that H Ž fa . s fa1 p 9 and, H Ž fy a . s fy a p 9 for a g P Gq s P G . By Theorem 6.3, H can be extended to a surjective homomorphism of semi-groups from BSŽ Gq . to BSŽ G9., denoted by the same symbol H. Note that H Ž f . s f F9 for f g BSŽ Gq .. Let fq and fy be the invariant elements of sub-semi-groups BSq Ž Gq . and BSyŽ Gq ., respectively. fq ) fy and hence H Ž fq. ) H Ž fy. are invariant elements of BSŽ Gq . and BSŽ G9., respectively. Thus, there is a homomorphism of Hopf algebras
F : k w G x , k w G9 x rker Ž H Ž fq . ) H Ž fy . . ª k Gq rker Ž fq ) fy . , k Gq , where the first and third isomorphisms are induced by p and the canonical map, respectively, the second map is induced by F9. Clearly, the following commutative diagram results, k w Gq x 6
F
kwT x
6
6
kwT x
F
6
kw G x
where vertical maps are canonical and F < T Ž t . s t N for coordinates t 4 of T. The injectivity of F follows now from that of F < T by the virtue of the related result of algebraic groups, see, e.g., Anderson w1x. To complete the proof, that images of F in k w Gq x are central remains to be checked. As images of the map, K w G9 x rker Ž f x F9 . s k w G9 x rker Ž H Ž f x . . ª k Gq rker Ž f x . , induced by F9, are central for x s a , ya g P Ž Gq . s P Ž G ., it is sufficient to prove the following for our claim: if images of both maps
u 1 : k w G9 x rker Ž H Ž f . . ª k Gq rker Ž f . u 2 : k w G9 x rker Ž H Ž f i . . ª k Gq rker Ž f i .
and
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induced by F9, are central for a i g P Ž Gq . and f g BSŽ Gq ., then images of the map
u 1 m u 2 : k w G9 x rker Ž H Ž f . . m k w G9 x rker Ž H Ž f i . . ª k Gq rker Ž f . m k Gq rker Ž f i . are central. But, this is trivial, proving the theorem.
REFERENCES 1. A. A. Anderson, ‘‘Algebraic Groups,’’ Lecture Notes in Mathematics, Springer-Verlag, New YorkrBerlin, 1980. 2. A. A. Beilinson, G. Lusztig, and R. MacPherson, A geometric setting for the quantum deformation of GLn , Duke Math. J. 61 Ž1990., 655]677. 3. B. Parshall and J. P. Wang, Quantum linear groups, Mem. Amer. Math. Soc. 439 Ž1991.. 4. D. Kazhdan and G. Lusztig, Affine Lie algebras and quantum groups, Internat. Math. Res. Notices 2 Ž1991., 21]29. 5. D. Kazhdan and G. Lusztig, Tensoring structure arising from quantized enveloping algebras, preprint. 6. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, In¨ ent. Math. 53 Ž1979., 165]184. 7. G. Lusztig, Modular representation and quantum groups, classical groups and related topics, Contemp. Math. 82 Ž1989., 59]77. 8. G. Lusztig, Finite dimensional algebras arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 Ž1990., 257]328. 9. G. Lusztig, Introduction to quantum groups, Progr. Math. 110 Ž1993.. 10. G. Lusztig, Canonical bases arising from quantized enveloping algebra, Progr. Theoret. Phys. Suppl. 102 Ž1990., 175]201. 11. G. Lusztig, Canonical bases in tensors products, Proc. Nat. Acad. Sci. U.S. A. 89 Ž1992.. 12. L. Woronowicz, Tannaka]Klein duality for compact metric pseudogroups, Twisted SUŽ N . groups, In¨ ent. Math. 93 Ž1988., 35]76. 13. M. Hashimoto and T. Tayshi, Quantum multilinear group, preprint. 14. M. Jimbo, R matrix related to the generalized Toda system, in Lecture Notes in Physics, Vol. 264, pp. 335]361, Springer-Verlag, New YorkrBerlin. 15. M. Kashiwara, Global crystal bases of quantum groups, preprint, 1992. 16. M. Sweedler, ‘‘Hopf-Algebra,’’ Benjamin, Elmsford, New York, 1969. 17. M. Takeuchi, Matric bialgebra and quantum groups, Israel J. Math., in press. 18. M. Takeuchi, Some topics on GL q Ž m., J. Algebra, 147 Ž1992., 379]410.
On Semi-groups Arising from Quantum Groups Zhen-hua Wang* Department of Mathematics, East China Normal Uni¨ ersity, Shanghai, 200062, People’s Republic of China Communicated by Walter Feit Received July 30, 1997
Let Gq be a quantum group. In this paper, we introduce and completely characterize a semi-group and its Borel sub-semi-groups consisting of algebra homomorphisms from the coordinate algebra k w Gq x of Gq . Also, we present invariant elements for the semi-group, which are closely connected to Hopf ideals of k w Gq x. Finally, based on the theory of structures of the semi-group, we prove the existence of a Frobenius morphism over Gq whenever the parameter q is a root of unity. Q 1999 Academic Press
INTRODUCTION The approach to quantum groups includes the study of coordinate algebras related to Yang]Baxter operators. This was carried out only for the case of type A by Parshall and Wang w3x, although all Yang]Baxter operators of classical types had been found a long time before. The difficulty lies in the complexity of coordinate algebras of quantum groups. To solve this, we develop the theory of semi-groups with generators respecting structures of some closed subgroups of the quantum groups, which resorts to techniques from algebraic groups. In developing the theory, root systems emerge in the form of indices and are closely related to structures of coordinate algebras of quantum groups. As applications of the theory, aside from a success in the proof of an important theorem, i.e., the theorem of density of big cell, the existence of Frobenius morphisms for quantum groups with parameters of roots of unity is also verified. *I take this opportunity to give my special thanks to Professors Jian-pan Wang and Jian-yi Shi for their leading me into the field of quantum groups, and Jia-shen Ye for much encouragement. 669 0021-8693r99 $30.00 Copyright Q 1999 by Academic Press All rights of reproduction in any form reserved.
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Fix a field k in any characteristic. In this paper, we adopt the ‘‘naive’’ point of view of w3x, identifying the category QGrk of quantum groups over k with the dual of the category k-Hopf of finitely generated k-Hopf algebras. The Hopf algebra corresponding to a quantum group Gq , denoted by k w Gq x, is called the coordinate algebra of Gq . Moreover, let H and H1 be two quantum groups. Then H1 is called as a subgroup of H whenever there is a surjective Hopf algebra homomorphism k w H x ª k w H1 x. Let Gq be a fixed quantum group. We organize the paper as follows. Section 1 presents some concepts, e.g., generic point and root system, etc. Section 2 introduces a fundamental semi-group associated to Gq and two of the Borel sub-semi-groups. Also we develop the basic theory on the semi-group, which includes the connection of some distinguished elements, called invariant elements in the paper, of the semi-group and Hopf ideals of k w Gq x. Sections 3, 4, and 5 are devoted to determining the invariant elements of the semi-group and its Borel sub-semi-groups. As a q-analogue of the affine group, a semi-simple quantum group is defined, based on the density of ‘‘big cells’’ in Section 5. Section 6 characterizes entirely structures of the semi-group and the Borel sub-semi-groups. Finally, Section 7 proves that there is a surjective homomorphism from the semi-group associated to Gq to that associated to an algebraic group, which is equivalent to the existence of the Frobenius morphism from Gq to the algebraic group, in case q is a root of unity. In a sequel, we shall prove that quantum groups of classical types, associated to Yang]Baxter operators, are essentially determined by these semi-groups.
1. THE GENERIC POINTS AND THE ROOT SYSTEM Let K be a quasi-polynomial algebra over k with generators t, ty1 , x 4 subject to the relations tty1 s ty1 t s 1,
xt s qtx.
Then, we can make K into two Hopf algebras, denoted by K 1 and K 2 , respectively. The associated comultiplications D 1 and D 2 , and antipodes g 1 and g 2 are respectively defined as follows. D 1 sends t to t m t, and x to ty1 m x q x m t; D 2 sends t to t m t and x to t m x q x m ty1 ; g 1 sends t to ty1 and x to qy1 x, and g 2 sends t to ty1 and x to qx. DEFINITION 1.1. Let Gq be a quantum group with a maximally diagonal torus T and X ŽT . the character group of T. If there are a surjective Hopf algebra homomorphism f : k w Gq x ª K 1 Žresp. f : k w Gq x ª K 2 . and a Hopf
SEMI-GROUPS FROM QUANTUM GROUPS
671
algebra homomorphism f T : k w T x ª K sending a unique element a g X ŽT . to t 2 Žresp. ty2 . such that the following commutative diagram holds
Ki 6
fT
Ki
6
6
kwT x
f
6
k w Gq x
for i s 1 Žresp. i s 2., where two vertical maps are canonical, then f is called as a positive Žresp. negative. generic-point of Gq associated to a or simply called an a-point in the language of algebraic geometry. Denote by P Ž Gq ., PqŽ Gq ., and PyŽ Gq ., respectively, the set of generic-points, positive generic-points, and negative generic-points of Gq , respectively. Usually, an a-point of Gq is written as fa for a g X ŽT .. DEFINITION 1.2. Ž1. Let Gq be a quantum group with a maximally diagonal torus T ; let RŽ Gq ., RqŽ Gq ., and RyŽ Gq . be subsets of X ŽT . such that P Ž Gq . s fa < a g R Ž Gq . 4 , Pq Ž Gq . s fa < a g Rq Ž Gq . 4 Py Ž Gq . s fa < a g Ry Ž Gq . 4 . Then RŽ Gq . s RqŽ Gq . j RyŽ Gq . is clearly a partition; RŽ Gq ., RqŽ Gq ., and RyŽ Gq . are defined as a root system, a positive system, and a negative system of Gq , respectively. Ž2. Let a g RqŽ Gq .. a is called as a simple root of Gq if there are no a 1 , a 2 g RqŽ Gq . such that a s a 1 q a 2 ; denote by P Ž Gq . the set of simple roots of Gq . A generic point, corresponding to a simple root, is called a simply generic point. Denote by SP the set of simply generic points. Also, put SPys fya g PyŽ Gq .< a g P Ž Gq .4 . Ž3. For a g RqŽ Gq . and an algebraic group homomorphism f : k=ª T, define ² a , f : to be an element in Z such that a f Ž m. s m² a , f : for any m g K= where k= is the multiplicative group of units of k. In particular, there is an f, called the coroot of a and denoted by a ¨ , such that ² a , f : s 2. In the remainder of this paper, we shall assume that P Ž Gq . is identified with a simple laced system of simple roots.
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2. THE SEMI-GROUP AND THE BOREL SUB-SEMI-GROUPS Fix a quantum group Gq with a maximally diagonal torus T throughout this section. Let I Ž Gq . be the set of all ideals of k w Gq x and F Ž Gq . the set of all unity-preserving homomorphisms of k-algebras from k w Gq x. Then there is a map ker: F Ž Gq . ª I Ž Gq . with kerŽ f . denoting the kernel of f for f g F Ž Gq .. Define an equivalence relation Gq on F Ž Gq . such that f Gq g iff kerŽ f . s kerŽ g . for f, g g F Ž Gq . and have then a quotient set F Ž Gq .rGq , consisting of equivalence classes of elements, of F Ž Gq .. Also, ker induces a 1-1 map Žstill written as. ker: F Ž Gq .rGq ª I Ž Gq .. PROPOSITION 2.1. There is a multiplication ) on F Ž Gq .rGq such that F Ž Gq .rGq becomes a semi-group with the unity « under it. Proof. Let D and « be the comultiplication and the counit of k w Gq x, respectively. Define the multiplication ) on F Ž Gq .rGq as f 1 ) f 2 s Ž f 1X m f 2X . D
for f 1 , f 2 g F Ž Gq . rGq ,
where f 1X and f 2X are representatives for f 1 and f 2 , respectively. It is well defined, indeed, if f 1Y Gq f 1X and f 2Y Gq f 2X . Then it follows that ker Ž Ž f 1X m f 2X . D . s ker Ž Ž f 1Y m f 2Y . D . . As for the associativity of ), it follows now from the coassociativity of D, while the claim that « is the unity follows from the fact that « is the counity of k w Gq x, proving the proposition. Let BSqŽ Gq ., BSyŽ Gq ., and BSŽ Gq . be sub-semi-groups, generated by equivalence classes of elements of SP Ž Gq ., SPyŽ Gq ., and SPyŽ Gq . j SPqŽ Gq ., respectively, of F Ž Gq .rGq . DEFINITION 2.2. BSŽ Gq ., BSqŽ Gq ., and BSyŽ Gq . are called a fundamental semi-group of Gq , a positive Borel sub-semi-group, and a negative Borel sub-semi-group, respectively. 2 THEOREM 2.3. Ž1. fa ) fy b s fy b ) fa , fa2 s fa , and fy a s fy a Ž . for a / b g P Gq . Ž2. The multiplication ) satisfies braid relations on SPqŽ Gq . Ž resp. yŽ SP Gq ..; in other words, we ha¨ e for fa , fb g SPqŽ Gq . Ž resp. SPyŽ Gq ..
fa ) fb ) fa s fb ) fa ) fb , fa ) fb s fb ) fa ,
² a , b ¨ : s ² b , a ¨ : s y1; ² a , b ¨ : s ² b , a ¨ : s 0.
SEMI-GROUPS FROM QUANTUM GROUPS
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Ž3. There is an anti-automorphism i of BSŽ Gq . such that iŽ f x . s f x for x s a , ya g P Ž Gq .. Proof. Ž1. The claims are proved easily. Indeed, the claim that fa2 s 2 . fa Žresp. fy a s fy a follows from the fact that the corresponding algebra k Gq rker Ž fa .
Ž resp. k
Gq rker Ž fy a . .
is even a Hopf algebra. To prove the first claim, let X i, j < i, j g I 4 for some index set I be the coordinates of k w Gq x, and both
pa : k Gq ª k Gq rker Ž fa .
and
pyb : k Gq ª k Gq rker Ž fy b . canonical ŽHopf algebra. homomorphisms. Put P0 s X i , j < i / j, pa Ž X i , j . s 0, py b Ž X i , j . s 0 4 , P1 s X i , i y 1
Xi , l m Xl , j
Ý lGi , jGl
for 1 F i, j F 3. We can take the a-point fa in the form of
fa Ž X 1, 1 . s ty1 , fa Ž X 1, 3 . s 0,
fa Ž X 1, 2 . s x, fa Ž X 2, 3 . s 0,
fa Ž X 2, 2 . s t , fa Ž X 3, 3 . s 1
and the b-point fb in the form of
fb Ž X 1, 1 . s 1, fb Ž X 1, 3 . s 0,
fb Ž X 1 , 2 . s 0, fb Ž X 2, 3 . s x,
fb Ž X 2, 2 . s ty1 , fb Ž X 3, 3 . s t.
Thus, we have
fa ) fb ) fa Ž X 1, 1 . s ty1 m 1 m ty1 , fa ) fb ) fa Ž X 1, 2 . s x m ty1 m t q ty1 m 1 m x, fa ) fb ) fa Ž X 2 , 2 . s t m ty1 m t , fa ) fb ) fa Ž X 2, 3 . s t m x m 1,
fa ) fb ) fa Ž X 1, 3 . s x m x m 1, fa ) fb ) fa Ž X 3, 3 . s 1 m t m 1
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ZHEN-HUA WANG
and
fb ) fa ) fb Ž X 1, 1 . s 1 m ty1 m 1,
fb ) fa ) fb Ž X 1, 2 . s 1 m x m ty1 ,
fb ) fa ) fb Ž X 2, 2 . s ty1 m t m ty1 ,
fb ) fa ) fb Ž X 1, 3 . s 1 m x m x,
fb ) fa ) fb Ž X 2, 3 . s ty1 m t m x q x m 1 m t , fb ) fa ) fb Ž X 3, 3 . s t m 1 m t. Check easily that kerŽ fb fa fb . s kerŽ fa fb fa . is generated by X s, t
Ž s - t. ,
X s, t X s, m y qy1 X s, m X s, t
Ž t - m. ,
X t , s X m , s y qy1 X m , s X t , s
Ž t - m. ,
X s, t X l , m y X l , m X s, t
Ž s - l, t ) m . ,
X 1, 1 X 2, 2 X 3, 3 y 1, X s, t X l , m y X l , m X s, t y Ž 1 y q 2 . X s, m X t , l
Ž s - l, t - m .
for s, t, l, m g 1, 2, 34 . Ž3. Let g be the antipode of k w Gq x. Then the map i, defined by iŽ f . s fg for f g BSŽ Gq ., will work, proving the theorem. Let Q be a semi-group and g g Q. We call g an invariant element of Q if ga s g and ag s g for a g Q. THEOREM 2.4. With the abo¨ e notations, then the in¨ ariant element g is unique. Consequently, if i is an anti-automorphism of Q, then iŽ g . s g. Proof. The proof is straightforward. THEOREM 2.5. Let f g BSŽ Gq .. Then kerŽ f . is a Hopf ideal iff there is a sub-semi-group SG9 of BSŽ Gq . such that f is the in¨ ariant element of SG9. Proof. First of all, establish notations as follows. For i g Z, put
fi s
~¡f
ai
¢f
g SP Ž Gq . ,
ya < i <
g SPy Ž Gq . ,
if i ) 0; if i - 0.
Assume now that f s f i1 ) ??? ) f i s is a reduced expression for f . Let SG9 be a sub-semi-group of BSŽ Gq . generated by f i1, . . . , f i s 4 . Observe that kerŽ f . is a Hopf ideal if f s f ) f and iŽ f . s f . This is satisfied if f is the invariant element of the semi-group SG9.
SEMI-GROUPS FROM QUANTUM GROUPS
675
Conversely, it is clear that kerŽ f . ; kerŽ f i j . for j s 1, 2, . . . , s. If kerŽ f . is a Hopf ideal, then either f i j g SP Ž GqX . or f i j g SPyŽ GqX . according to whether i j ) 0 or - 0 for j s 1, 2, . . . , s, and thus, f is the invariant element of the semi-group SG9. Here GqX is a quantum group with the coordinate algebra k w Gq xrkerŽ f ., completing the proof. 3. THE INVARIANT ELEMENTS OF BOREL SUB-SEMI-GROUPS In this section, we consider BSqŽ Gq ., while all results on it apply to BSyŽ Gq .. Let P Ž Gq . s a 1 , a 2 , . . . , a n4 be a simple laced system of simple roots and W the Weyl group corresponding to it. Put fa i s f i g SP Ž Gq . for i g 1, 2, . . . , n4 . THEOREM 3.1. Let w 0 be the longest word in W and F s f i1 ) f i 2 ) ??? ) f i m g BSqŽ Gq .. If w 0 s a i1 a i 2 ??? a i m , then F is the in¨ ariant element of BSqŽ Gq .. Proof. The claim follows from the following facts: Ž1. Let F9 s f j ) f j ) ??? ) f jX . Then F s F9 if w 0 s a j a j ??? a jX , 1 2 m 1 2 m Ž by 2. of Theorem 2.3; Ž2. For given f i g SP Ž Gq ., there is a F9, related to w 0 in the way as in Ž1., such that F9 s F0) f i or F9 s f i ) F0 for some F0 g BSqŽ Gq .; Ž3. It follows that f i2 s f i by Ž1. of Theorem 2.3 for i s 1, 2, . . . , n. The proof is completed. 4. THE INVARIANT ELEMENTS OF FUNDAMENTAL SEMI-GROUPS OF RANK 1 In this section, we consider a sub-semi-group BSŽ Gq Ž1.. of BSŽ Gq . which is generated by fa , fy a 4 for a g P Ž Gq .. BSŽ Gq Ž1.. is usually called a fundamental semi-group of rank 1. THEOREM 4.1. Ž1. fa ) fy a s fy a ) fa . Ž2. fa ) fy a is the in¨ ariant element of BSŽ Gq Ž1... Proof. To prove Ž1., we only need to consider such a quantum group Gq that its coordinate algebra k w Gq x is generated by X i, j <1 F j, i F 24 with the comultiplication, D Ž Xi , j . s
Ý lFlF2
Xi , l m Xl , j
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ZHEN-HUA WANG
for 1 F i, j F 2. Thus, we can take the ya-point fya in the form of
fya Ž X 1, 1 . s ty1 ,
fy a Ž X 2, 1 . s x,
fya Ž X 1, 2 . s 0,
fy a Ž X 2, 2 . s t
and the a-point fa in the form of
fa Ž X 1, 1 . s ty1 ,
fa Ž X 2 , 1 . s 0,
fa Ž X 1, 2 . s x,
fa Ž X 2 , 2 . s t.
We have
fya fa Ž X 1 , 1 . s ty1 m ty1 , fya fa Ž X 1, 2 . s ty1 m x,
fy a fa Ž X 2, 1 . s x m ty1 , fa fy a Ž X 2, 2 . s t m t q x m x
and
fa fy a Ž X 1, 1 . s ty1 m ty1 q x m x, fa fy a Ž X 1, 2 . s x m t ,
fa fy a Ž X 2, 1 . s t m x,
fa fy a Ž X 2, 2 . s t m t.
Check easily that kerŽ fya ) fa . s kerŽ fa ) fy a . is spanned by X s, t X s, m y qy1 X s, m X s, t y1
Xt , s Xm , s y q
Xm , s Xt , s
X s, t X l , m y X l , m X s, t
Ž t - m. , Ž t - m. ,
Ž s - l, t ) m . ,
X s, t X l , m y X l , m X s, t y Ž 1 y q 2 . X s, m X t , l
Ž s - l, t - m . ,
X 1, 1 X 2, 2 y qy1 X 1, 2 X 2, 1 y 1 for s, t, l, m g 1, 24 , proving Ž1.. As for Ž2., we have K w Gq xrkerŽ fy a ) fa . , k w SL 2 Ž q .x, which is the coordinate algebra of a special quantum group of rank 1, as in w3x, completing the proof.
5. THE INVARIANT ELEMENT OF THE FUNDAMENTAL SEMI-GROUP AND A SEMI-SIMPLE QUANTUM GROUP Assume that P Ž Gq . s a 1 , a 2 , . . . , a n 4 , SPq Ž Gq . s fa 1 , fa 2 , . . . , fa n 4
and
SPy Ž Gq . s fy a 1 , fy a 2 , . . . , fy a n 4 .
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677
Recall in Section 3 that there are fq and fy which are invariant in BSqŽ Gq . and BSyŽ Gq ., respectively. THEOREM 5.1. in BSŽ Gq ..
fq) fys fy) fq. Consequently, fq) fy is in¨ ariant
Proof. Put fya i s fyi and fa i s f i for i s 1, 2, . . . , n. To prove the claim, it is sufficient to prove that f i ) fyj s fyj ) f i for i, j s 1, 2, . . . , n. The claim follows from Ž1. of Theorem 2.3 for i / j, while, for i s j, this follows from Theorem 4.1. COROLLARY 5.2.
BSŽ Gq . is a finite semi-group, more explicitly, we ha¨ e 2
BS Ž Gq . s BSq Ž Gq . BSy Ž Gq . s BSq Ž Gq . . Proof. The claim is a direct consequence of Theorem 5.1. DEFINITION 5.3. Let Gq be as the above. Then, we call Gq a semi-simple quantum group of rank n if fq) fys 0 where fq and fy are defined as above. Note that the above definition for q s 1 is reconciled with that in algebraic groups. COROLLARY 5.4. With the abo¨ e notations, let Gq be a semi-simple y quantum group. Then we ha¨ e two subgroups Bq q and Bq , called the positi¨ e x w x Ž q. and negati¨ e Borel subgroups, respecti¨ ely, of Gq : k w Bq q s k Gq rker f yx y. q qx y w w x Ž w x w and k Bq s k Gq rker f . Let both p : k Gq ª k Bq and p : k w Gq x x be canonical homomorphisms of Hopf algebras. Then the algebra ª k w By q x w qx is injecti¨ e, where D homomorphism Žpym pq . D: k w Gq x ª k w By q m k Bq w x is the comultiplication of k Gq . Proof. The statement follows from Definition 5.3.
6. THE STRUCTURES OF SEMI-GROUPS In this section, we give the complete relations among generators of BSŽ Gq ., BSqŽ Gq ., and BSyŽ Gq ., respectively. Assume that P Ž Gq . s a 1 , a 2 , . . . , a n 4 , SPq Ž Gq . s f 1 , f 2 , . . . , fn 4
and
SPy Ž Gq . s fy1 , fy2 , . . . , fyn 4 .
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ZHEN-HUA WANG
Let FSGq be a semi-group with a unity e, generated by z1 , z 2 , . . . , z n4 subject to the relations z i2 s z i , zi z j zi s z j zi z j , zi z j s z j zi ,
1 F i F n; ² a i , a j¨ : s y1; ² a i , a j¨ : s 0.
THEOREM 6.1. There is an isomorphism of semi-groups sq: FSGqª BSqŽ Gq . such that sqŽ z i . s f i for i s 1, 2, . . . , n. Proof. sq as a surjective homomorphism of semi-groups is clear, see wTheorem 2.3x. To prove that sq is an isomorphism, we need to prove that any relation
f i1 ) f i 2 ) ??? ) f i s s f j1 ) f j 2 ) ??? ) f j
t
in BSqŽ Gq . implies the relation z i1 ) z i 2 ) ??? ) z i s s z j1 ) z j 2 ) ??? ) z j t in FSGq. z i , z j 4 for i, j g Let FSGq i, j be a sub-semi-group, generated by 1, 2, . . . , n4 , of FSGq. Then there is a canonically surjective semi-group homomorphism p i, j : FSGqª FSGq i, j , sending z m to z m for m s i, j, z m to e for m other than i and j. An important observation is that FSGq is isomorphic to BSqŽ Gq . in case of n s 2. Thus, if a relation
f i1 ) f i 2 ) ??? ) f i s s f j1 ) f j 2 ) ??? ) f j t follows in BSqŽ Gq ., then the relation
p i , j Ž z i1 ) z i 2 ) ??? ) z i s . s p i , j Ž z j1 ) z j 2 ) ??? ) z j t . follows in FSGq for i, j g 1, 2, . . . , n4 . This enables us to conclude that z i1 ) z i 2 ) ??? ) z i s s z j1 ) z j 2 ) ??? ) z j t , proving the theorem. COROLLARY 6.2. There is an isomorphism of the semi-group sy: FSGq ª BSyŽ Gq . such that syŽ z i . s fyi g SPyŽ Gq . for i s 1, 2, . . . , n. Proof. The proof is entirely similar to the above.
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THEOREM 6.3. Let FBG be a semi-group generated by z1 , z 2 , . . . , z n4 and zX1 , zX2 , . . . , zXn4 subject to the relations z i zXj s zXj z i ,
z i2 s z i ,
zi z j zi s z j zi z j , zi z j s z j zi , zXi zXj zXi s zXj zXi zXj , zXi zXj s zXj zXi ,
2 Ž zXi . s zXi ,
1 F i , j F n;
² a i , a j¨ : s y1, ² a i , a j¨ : s 0; ² a i , a j¨ : s y1, ² a i , a j¨ : s 0.
Then there is an isomorphism of semi-groups s : FBG ª BSŽ Gq . such that s Ž z i . s f i and s Ž zXi . s fyi for 1 F i F n. Proof. The statement follows from Theorems 6.1 and 6.2, together with relations f i fyj s fyj f i for 1 F i, j F n. 7. THE FROBENIUS MORPHISM In this section, let q be a Nth primitive root of unity and Gq a semi-simple quantum group of rank n with a maximally diagonal torus T. Also, let G be a semi-simple algebraic group with the same maximally diagonal torus as that of Gq and choose the positive subsystem of the root system of G related to T determined by the upper triangle Borel subgroup. Here, we consider G a subgroup of GLŽ V . for some k-vector space V. x Let K 0 s k w t 0 , ty1 0 , x 0 be a polynomial algebra. Then there is a homomorphism of algebras f : K 0 ª K Žcf. Section 1., sending t 0 to t N and x 0 to x N. Moreover, K 0 can be endowed with two of the structures of Hopf algebras, denoted by K 01 and K 02 the corresponding Hopf algebras, such that both f : K 01 ª K 1 and f : K 02 ª K 2 are homomorphisms of Hopf algebras. LEMMA 7.1. There are surjecti¨ e Hopf algebra homomorphisms,
fa1 : k w Tg a x ª K 01
Ž resp. fy1 a : k w Tgy a x ª K 02 . . restrictions of which to k w T x send the unique a Ž resp. ya . to t 02 Ž resp. ty2 0 for a g P Ž G .. Here these g a are subgroups of G associated to roots a .
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Proof. It is well known that k w Tg a x Žresp. k w Tgy a x. is generated by elements, say t 4 and x a Žresp. xya .. Moreover, among t 4 , there are ta and taX such that D Ž x a . s x a m ta q taX m x a
Ž resp. D Ž xya . s xya m taX q ta m xya . . Thus, we can define homomorphisms of Hopf algebras
fa1 : k w Tg a x ª K 01
Ž resp. fy1 a : k w Tgy a x ª K 02 . as
fa1 Ž taX . s ty1 0 ,
fa1 Ž ta . s t 0 , fa1 Ž t . s 1,
fa1 Ž xa . s x 0
Žresp. X 1 y1 fy a Ž ta . s t 0 ,
1 fy a Ž ta . s t 0 , 1 fy a Ž t . s 1,
1 fy a Ž xy a . s x 0 .
for t g t 4 other than ta and taX , proving the lemma. THEOREM 7.2. With the abo¨ e setup and the assumption that P Ž Gq . s P Ž G ., then there is a homomorphism F: k w G x ª k w Gq x of Hopf algebras such that the following commutati¨ e diagram follows: k w Gq x
F
k w T x,
6
6
kwT x
F
6
kw G x
6
Ž1.
where ¨ ertical maps are canonical and F < T Ž t . s t N for coordinates t 4 of T. Ž2. F is injecti¨ e and its images are central. Proof. Let G9 be a quantum matrix space the coordinate of which is a freely generated k-bialgebra k w G9x over coordinates X i,X j < i, j g J 4 corresponding to coordinates X i, j < i, j g J 4 of k w Gq x. Define a homomorphism of algebras F9: k w G9x ª k w Gq x sending X i,X j to X i,Nj for i, j g J. Again, let
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681
G be the algebraic group as the above with coordinates x i, j < i, j g J 4 in the same index set J and p : k w G9x ª k w G x a homomorphism of bialgebras sending X i,X j to x i, j for i, j g J. As in Section 2, F Ž G9.rG is the set of equivalence classes of algebra homomorphisms from k w G9x: f Gg iff kerŽ f . s kerŽ g . for f, g g k w G9x. 1 < Recall that BSŽ Gq . s ² fa , fy a < a g P Ž Gq .: and BSŽ G . s ² fa1 , fy a ag 1 1 P Ž G .:. Put BSŽ G9. s ² fa p 9, fy a p 9 < a g P Ž G .: ; F Ž G9.rG. Then, there is a map 1 < H : fa , fy a < a g P Ž G . 4 ª fa1 p 9, fy a p 9 a g P Ž G. 4 1 Ž . Ž . such that H Ž fa . s fa1 p 9 and, H Ž fy a . s fy a p 9 for a g P Gq s P G . By Theorem 6.3, H can be extended to a surjective homomorphism of semi-groups from BSŽ Gq . to BSŽ G9., denoted by the same symbol H. Note that H Ž f . s f F9 for f g BSŽ Gq .. Let fq and fy be the invariant elements of sub-semi-groups BSq Ž Gq . and BSyŽ Gq ., respectively. fq ) fy and hence H Ž fq. ) H Ž fy. are invariant elements of BSŽ Gq . and BSŽ G9., respectively. Thus, there is a homomorphism of Hopf algebras
F : k w G x , k w G9 x rker Ž H Ž fq . ) H Ž fy . . ª k Gq rker Ž fq ) fy . , k Gq , where the first and third isomorphisms are induced by p and the canonical map, respectively, the second map is induced by F9. Clearly, the following commutative diagram results, k w Gq x 6
F
kwT x
6
6
kwT x
F
6
kw G x
where vertical maps are canonical and F < T Ž t . s t N for coordinates t 4 of T. The injectivity of F follows now from that of F < T by the virtue of the related result of algebraic groups, see, e.g., Anderson w1x. To complete the proof, that images of F in k w Gq x are central remains to be checked. As images of the map, K w G9 x rker Ž f x F9 . s k w G9 x rker Ž H Ž f x . . ª k Gq rker Ž f x . , induced by F9, are central for x s a , ya g P Ž Gq . s P Ž G ., it is sufficient to prove the following for our claim: if images of both maps
u 1 : k w G9 x rker Ž H Ž f . . ª k Gq rker Ž f . u 2 : k w G9 x rker Ž H Ž f i . . ª k Gq rker Ž f i .
and
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induced by F9, are central for a i g P Ž Gq . and f g BSŽ Gq ., then images of the map
u 1 m u 2 : k w G9 x rker Ž H Ž f . . m k w G9 x rker Ž H Ž f i . . ª k Gq rker Ž f . m k Gq rker Ž f i . are central. But, this is trivial, proving the theorem.
REFERENCES 1. A. A. Anderson, ‘‘Algebraic Groups,’’ Lecture Notes in Mathematics, Springer-Verlag, New YorkrBerlin, 1980. 2. A. A. Beilinson, G. Lusztig, and R. MacPherson, A geometric setting for the quantum deformation of GLn , Duke Math. J. 61 Ž1990., 655]677. 3. B. Parshall and J. P. Wang, Quantum linear groups, Mem. Amer. Math. Soc. 439 Ž1991.. 4. D. Kazhdan and G. Lusztig, Affine Lie algebras and quantum groups, Internat. Math. Res. Notices 2 Ž1991., 21]29. 5. D. Kazhdan and G. Lusztig, Tensoring structure arising from quantized enveloping algebras, preprint. 6. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, In¨ ent. Math. 53 Ž1979., 165]184. 7. G. Lusztig, Modular representation and quantum groups, classical groups and related topics, Contemp. Math. 82 Ž1989., 59]77. 8. G. Lusztig, Finite dimensional algebras arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 Ž1990., 257]328. 9. G. Lusztig, Introduction to quantum groups, Progr. Math. 110 Ž1993.. 10. G. Lusztig, Canonical bases arising from quantized enveloping algebra, Progr. Theoret. Phys. Suppl. 102 Ž1990., 175]201. 11. G. Lusztig, Canonical bases in tensors products, Proc. Nat. Acad. Sci. U.S. A. 89 Ž1992.. 12. L. Woronowicz, Tannaka]Klein duality for compact metric pseudogroups, Twisted SUŽ N . groups, In¨ ent. Math. 93 Ž1988., 35]76. 13. M. Hashimoto and T. Tayshi, Quantum multilinear group, preprint. 14. M. Jimbo, R matrix related to the generalized Toda system, in Lecture Notes in Physics, Vol. 264, pp. 335]361, Springer-Verlag, New YorkrBerlin. 15. M. Kashiwara, Global crystal bases of quantum groups, preprint, 1992. 16. M. Sweedler, ‘‘Hopf-Algebra,’’ Benjamin, Elmsford, New York, 1969. 17. M. Takeuchi, Matric bialgebra and quantum groups, Israel J. Math., in press. 18. M. Takeuchi, Some topics on GL q Ž m., J. Algebra, 147 Ž1992., 379]410.