QUANTUM AFFINE ALGEBRAS AT ROOTS OF UNITY 0. Introduction Let Uq(g)

49 Pages·1997·0.62 MB·English

REPRESENTATIONTHEORY
AnElectronicJournaloftheAmericanMathematicalSociety
Volume1,Pages280–328(August14,1997)
S1088-4165(97)00030-7
QUANTUM AFFINE ALGEBRAS AT ROOTS OF UNITY
VYJAYANTHI CHARI AND ANDREW PRESSLEY
Abstract. Let Uq(^g) be the quantized universal enveloping algebra of the
a(cid:14)ne Lie algebra ^g associated to a (cid:12)nite-dimensional complex simple Lie al-
gebrag,andletUqres(^g)betheC[q,q−1]-subalgebraofUq(^g)generatedbythe
q-divided powers of the Chevalley generators. Let U(cid:15)res(^g) be the Hopf alge-
bra obtained from Uqres(^g) by specialising q to a non-zero complex number (cid:15)
of oddorder. Weclassifythe(cid:12)nite-dimensional irreduciblerepresentations of
U(cid:15)res(^g) in terms of highest weights. We also give a \factorisation" theorem
for such representations: namely, any (cid:12)nite-dimensional irreducible represen-
tation of U(cid:15)res(^g) is isomorphic to a tensor product of two representations,
one factorbeingthe pull-backofarepresentation of^gbyLusztig’sFrobenius
homomorphism F^r(cid:15) : U(cid:15)res(^g) → U(^g), the other factor being an irreducible
representation of the Frobenius kernel. Finally, we give a concrete construc-
tion of all of the (cid:12)nite-dimensional irreducible representations of U(cid:15)res(s^l2).
Theproofsmakeuseofseveralinterestingnew identitiesinUq(^g).
0. Introduction
LetUq(g)be the Drinfel’d{Jimbo quantumgroupassociatedtoasymmetrizable
Kac{Moodyalgebrag. Thus,Uq(g)isaHopfalgebraoverthe (cid:12)eldC(q)ofrational
functions of an indeterminate q, and is de(cid:12)ned by certain generators and relations
(whicharewrittendowninProposition1.1belowforthecasesofinterestinthispa-
per). Roughlyspeaking,onethinksofUq(g)asafamilyofHopfalgebrasdepending
on a \parameter" q. To make this precise, one constructs a \specialisation" U(cid:15)(g)
of Uq(g), for a non-zero complex number (cid:15), as follows. If (cid:15) 2 C is transcendental,
de(cid:12)neU(cid:15)(g)=Uq(g)⊗C(q)C,viathealgebrahomomorphismC(q)!Cthattakesq
to (cid:15). If, on the other hand, (cid:15) is algebraic,the latter homomorphismdoes not exist,
and one proceeds by (cid:12)rst constructing a C[q,q−1]-form of Uq(g), i.e. a C[q,q−1]-
(Hopf)subalgebraU~q(g)ofUq(g)suchthatUq(g)=U~q(g)⊗C[q,q−1]C(q). Thenone
de(cid:12)nes U(cid:15)(g) = U~q(g)⊗C[q,q−1] C, via the algebra homomorphism C[q,q−1] ! C
that takes q to (cid:15).
Two such C[q,q−1]-forms have been studied in the literature. They lead to the
same algebra U(cid:15)(g) when (cid:15) is not a root of unity, but di(cid:11)erent algebras, with very
di(cid:11)erent representation theories, when (cid:15) is a root of unity.
In the \non-restricted" form, one takes U~q(g) to be the C[q,q−1]-subalgebra of
Uq(g)generatedbythe Chevalleygeneratorsei,fi ofUq(g). The(cid:12)nite-dimensional
representations of the non-restricted specialisation U(cid:15)(g) have been studied by
ReceivedbytheeditorsApril,30,1997.
1991 Mathematics Subject Classification. Primary17B67.
The(cid:12)rstauthorwaspartiallysupportedbyNATOandEPSRC(GR/K65812).
Thesecondauthor waspartiallysupportedbyNATOandEPSRC(GR/L26216) .
(cid:13)c1997 American Mathematical Society
280
QUANTUM AFFINE ALGEBRAS AT ROOTS OF UNITY 281
De Concini, Kac and Procesi [10], [11], when g is (cid:12)nite-dimensional, and by Beck
and Kac [2], when g is (untwisted) a(cid:14)ne.
In the \restricted"form, onetakes U~q(g)to be the C[q,q−1]-subalgebraofUq(g)
generated by the divided powers eri/[r]q! and fir/[r]q!, for all r >0, where [r]q! de-
notes a q-factorial. When g is (cid:12)nite-dimensional and (cid:15) is a root of unity, the struc-
ture and representation theory of the restricted specialisation Ures(g) was worked
(cid:15)
out by Lusztig (see [5], [15] and the references there). It is the purpose of the
present work to study the (cid:12)nite-dimensional representations of Ures(g) when (cid:15) is a
(cid:15)
root of unity and g is (untwisted) a(cid:14)ne.
Part of our work may be regarded as the quantum analogue of Garland’s pa-
per [13]. A crucial role is played in [13] by a certain identity (Lemma 7.5) that
is needed to prove a suitable triangular decomposition of the restricted form of
U(^g) (analogous to Ures(^g)). The proof of the analogue of Garland’s lemma in
q
the quantum case (Lemma 5.1) is, however, more di(cid:14)cult than in the classical sit-
uation because the generators of the \positive part" of Ures(^g) do not commute,
(cid:15)
whereas their classical analogues do commute. Moreover, Garland makes use of a
natural derivation of U(^g) which turns out to have no straightforwardanalogue in
the quantum situation. Lemma 5.1 is one of several interesting new identities in
Ures(^g)thatweobtainbelow. Oneofthese (see(19))showsanunexpected (andas
q
yetunexplained)connectionbetweenUres(s^l )andYoungdiagrams. (Thisrelation
q 2
is invisible in the classical situation considered by Garland.)
Once the triangular decomposition of Ures(^g) is available, we are able to give,
(cid:15)
in Theorem 8.2, an abstract highest weight description of its (cid:12)nite-dimensional
irreducible representations. It turns out that there is a natural one-to-one cor-
respondence between the (cid:12)nite-dimensional irreducible representations of Ures(^g)
(cid:15)
when(cid:15) is a rootofunity, andthose ofU(cid:15)(^g)when (cid:15) is nota rootofunity, although
corresponding representations have di(cid:11)erent dimensions, in general (the represen-
tations of U(cid:15)(^g) when (cid:15) is not a root of unity were classi(cid:12)ed in [4], [5] and [6]).
This is exactly parallel to the situation for Ures(g) when dim(g) < 1, where the
(cid:15)
(cid:12)nite-dimensional irreducible representations are parametrised by dominant inte-
gral weights whether or not (cid:15) is a root of unity (see [5], Chapter 11, and [15]).
Oneofthemostimportantresultsaboutthe(cid:12)nite-dimensionalirreduciblerepre-
sentationsV ofUres(g)whendim(g)<1and(cid:15) is a rootofunity oforder(cid:96) asserts
(cid:15)
that V factorises into the tensor product of a representation whose highest weight
is divisible by (cid:96) and one whose highest weight is \less than (cid:96)", in the sense that its
value on every simple coroot of g is less than (cid:96) (see [5] and [15]). In Section 9, we
prove an analogue of this result for Ures(^g) (Theorem 9.1).
(cid:15)
It is well known that there is a close relationship between (cid:12)nite-dimensional
representationsofquantuma(cid:14)nealgebrasanda(cid:14)neTodatheories(see[7]andthe
references there). The value of (cid:15) is determined by the \coupling constant" of the
associated theory. Since the representation theory of Ures(^g) depends crucially on
(cid:15)
whether (cid:15) is a root of unity or not, one would expect that the structure of a(cid:14)ne
Toda theories will be di(cid:11)erent at certain special values of the coupling constant.
This does indeed appear to be the case (T. J. Hollowood, private communication),
but we shall leave further discussion of this point to another place.
282 VYJAYANTHI CHARI AND ANDREW PRESSLEY
1. Preliminaries
In this section and the next, we recall certain facts about g and ^g and their
associated quantum groups that will be needed later. See [1], [5] and [15] for
further details.
Let (aij)i,j2I be th‘e Cartanmatrix of the (cid:12)nite-dimensionalcomplex simple Lie
algebra g, let I^= I f0g, and let (aij)i,j2Iˆ be the generalised Cartan matrix of
the untwisteda(cid:14)ne Lie algebrag^ of g. Let (di)i2Iˆbe the coprime positive integers
such that the matrix (diaij)i,j2Iˆ is symmetric. P
LePtP(cid:20) bethelatticeoverZwithbasis(λ(cid:20)i)i2I,letα(cid:20)j = i2Iajiλ(cid:20)i (j 2I),andlet
Q(cid:20) = i2IZα(cid:20)i (cid:18) P(cid:20). The root lattice Q = HomZ(P(cid:20),Z) has basis the simple roots
(αi)i2I of g, where αi(λ(cid:20)j) = δij. Similarly, the weight lattice P = HomZ(Q(cid:20),Z)
has basis the fundamental weights (λi)i2I of g,Pwhere λi(α(cid:20)j) = δij. Let P+ =
fλ2P j λ(α(cid:20)i)(cid:21)0 for all i2Ig, and let Q+ = i2IN.αi.
For i2I, de(cid:12)ne si :P(cid:20) !P(cid:20) by si(x)=x−αi(x)α(cid:20)i, and let W be the group of
automorphismsofP(cid:20) generatedbythesi. Then,W actsonQbysi(ξ)=ξ−ξ(α(cid:20)i)αi,
fSor ξ 2 Q. TheSroot and coroot systems of g are given, respectively, by R =
i2IWαi, R(cid:20) = i2IWα(cid:20)i. There is a partial order on R such that α (cid:20) β if and
only if β −α is a linear combination of the αi (i2I) with non-negative integer
coe(cid:14)cients. Thecorrespondenceαi $α(cid:20)i extendsto aW-invariantcorrespondence
R $ R(cid:20), written α $ α(cid:20), such that α(α(cid:20)) = 2 for all α 2 R. De(cid:12)ne sα : P(cid:20) ! P(cid:20) by
sα(x)=x−α(x)α(cid:20), for all α2R.
LetW^ =W(cid:2)~P(cid:20) be the semi-directproductgroupde(cid:12)nedusingthe W actionon
P(cid:20). Fors2W writes for (s,0)2W^,and forx2P(cid:20) writex for(1,x)2W^, where 1
is the identity element of W. Let θ be the highest root of g with respect to (cid:20), and
write s0 for (sθ,θ(cid:20))2 W^. Let W~ be the (normal) subgroup of W^ generated by the
si fori2I^,andletT =W^/W~. Then,T isa(cid:12)nite groupisomorphictoasubgroup
of the group of diagram automorphisms of g^, i.e. the bijections τ : I^ ! I^ such
that aτ(i)τ(j) = aij for all i,j 2 I^. Moreover, there is an isomorphism of groups
W^ (cid:24)= T(cid:2)~W~, where the semi-direct product is de(cid:12)ned using the action of T in W~
givenbyτ.si =sτ(i) (see[1]). Ifw2W^,areducedexpressionforwisanexpression
w=τsi1si2...sin with τ 2T, i1,i2,... ,in 2I^and n minimal.
The universal enveloping algebra U(g) (resp. U(^g)) is the associative algebra
over C with generators e(cid:6)i , hi for i2I (resp. i2I^) and de(cid:12)ning relations
hihj =hjhi, hie(cid:6)j −e(cid:6)j hi =(cid:6)aije(cid:6)j ,
e+i e−j −e−j e+i =δijhi,
(cid:18) (cid:19)
1X−aij(−1)r 1−aij (e(cid:6))re(cid:6)(e(cid:6))1−aij−r =0 if i6=j,
i j i
r
r=0
where i,j 2I (resp. i,j 2I^).
Let q be an indeterminate, let C(q) be the (cid:12)eld of rational functions of q with
complex coe(cid:14)cients, and let C[q,q−1] be the ring of Laurent polynomials. For
QUANTUM AFFINE ALGEBRAS AT ROOTS OF UNITY 283
r,n2N, n(cid:21)r, de(cid:12)ne
qn−q−n
[n]q = q−q−1 ,
[n]q!=[n]q[n−1]q...[2]q[1]q,
(cid:20) (cid:21)
n [n]q!
= .
r q [r]q![n−r]q!
(cid:20) (cid:21)
n
Then 2C[q,q−1] for all n(cid:21)r (cid:21)0. One has the identity
r
q
(cid:20) (cid:21)
Xn
(1) (−1)rqr(n−1) n =0
r
r=0 q
for all n(cid:21)0. Let qi =qdi for i2I^.
Proposition 1.1. There is a Hopf algebra Uq(^g) over C(q) which is generated as
an algebra by elements e(cid:6), k(cid:6)1 (i2I^), with the following defining relations:
i i
kiki−1 =ki−1ki =1, kikj =kjki,
kie(cid:6)j ki−1 =qi(cid:6)aije(cid:6)j ,
[e+i ,e−j ]=δijkqii−−kqi−i−11,
(cid:20) (cid:21)
1X−aij(−1)r 1−aij (e(cid:6))re(cid:6)(e(cid:6))1−aij−r =0 if i6=j.
r i j i
r=0 qi
The comultiplication of Uq(^g) is given on generators by
(cid:1)(e+i )=e+i ⊗ki+1⊗e+i , (cid:1)(e−i )=e−i ⊗1+ki−1⊗e−i , (cid:1)(ki)=ki⊗ki,
for i2I^.
Restricting the indices i,j to I, one obtains a Hopf algebra Uq(g).
De(cid:12)ne Uq(g)(cid:6) (resp. Uq(g)0) to be the C(q)-subalgebra of Uq(g) generated by
the e(cid:6) (resp. by the k(cid:6)1) for all i2I.
i i
ThereisanaturalinjectivehomomorphismofC(q)-HopfalgebrasUq(g)!Uq(^g)
that takes e(cid:6)i to e(cid:6)i and ki to ki, for all i2I. In particular, any representation of
Uq(^g) can be regarded as a representation of Uq(g).
It is convenient to use the following notation:
(e(cid:6))r
(e(cid:6))(r) = i .
i
[r]qi!
284 VYJAYANTHI CHARI AND ANDREW PRESSLEY
Let Ti (i 2 I^) be the C(q)-algebra automorphisms of Uq(^g) de(cid:12)ned by Lusztig
[15] (our notational conventions di(cid:11)er slightly from his):
Ti(e+i )=−e−i ki, Ti(e−i )=−ki−1e+i , Ti(kj)=ki−aijkj,
−Xaij
Ti(e+j )= (−1)r−aijqi−r(e+i )(−aij−r)e+j (e+i )(r) if i6=j,
r=0
−Xaij
Ti(e−j )= (−1)r−aijqir(e−i )(r)e−j (e−i )(−aij−r) if i6=j.
r=0
Ifi2I,Ti inducesanautomorphismofthesubalgebraUq(g)ofUq(^g),alsodenoted
by Ti.
There is, of course,a classicalanalogueTi ofTi for alli2I^, givenby the above
formulaswith ei replacedby ei andki replacedby 1,andwith Ti(hj)=hj−aijhi.
The Ti are Hopf algebra automorphisms of U(^g).
The (cid:12)nite group T acts as C(q)-Hopf algebra automorphisms of Uq(^g) by
τ.e(cid:6)i =e(cid:6)τ(i), τ.ki =kτ(i), for all i2I^.
Similarly, T acts as Hopf algebra automorphisms of U(^g) by
τ.e(cid:6)i =e(cid:6)τ(i), τ.hi =hτ(i), for all i2I^.
If w 2 W has a reduced expression w = τsi1...sin, let Tw be the C(q)-algebra
automorphism of Uq(^g) given by Tw = τTi1...Tin. Then, Tw depends only on w,
and not on the choice of its reduced expression [1]. In particular, for any i 2 I,
we havea wellde(cid:12)ned C(q)-algebraautomorphismTλˇi ofUq(^g). Similarly, one has
Hopf algebra automorphisms T of U(^g).
λˇi
Let Uqres(g) (resp. Uqres(^g)) be the C[q,q−1]-subalgebra of Uq(g) (resp. Uq(^g))
generatedbythe k(cid:6)1 andthe(e(cid:6))(r) foralli2I,r 2N(resp. foralli2I^,r 2N).
i i
Then, Uqres(g) (resp. Uqres(^g)) is aC[q,q−1]-Hopf subalgebraofUq(g) (resp. Uq(^g)).
Moreover,Uqres(g) (resp. Uqres(^g)) is preservedby the automorphismTi for alli2I
(resp. for all i2I^). In fact [15], if i,j 2I^, n2N, then
Ti((e+i )(n))=(−1)nqi−n(n−1)(e−i )(n)kin, Ti((e−i )(n))=(−1)nqn(n−1)ki−n(e+i )(n),
−Xnaij
Ti((e+j )(n))= (−1)r−naijqi−r(e+i )(−naij−r)(e+j )(n)(e+i )(r) if i6=j,
r=0
−Xnaij
Ti((e−j )(n))= (−1)r−naijqir(e−i )(r)(e−j )(n)(e−i )(−naij−r) if i6=j.
r=0
The action of T obviously preserves Ures(^g).
q
For r 2N, n2Z, i2I, de(cid:12)ne
(cid:20) (cid:21)
ki;n Yr kiqin−s+1−ki−1qi−n+s−1
= .
r qs−q−s
s=1 i i
The importance of these elements stems from the identity
(cid:20) (cid:21)
(2) (e+)(r)(e−)(s) =miXn(r,s)(e−)(s−t) ki;2t−r−s (e+)(r−t),
i i i t i
t=0
QUANTUM AFFINE ALGEBRAS AT ROOTS OF UNITY 285
for all r,s 2 N, i 2 I. It follows from this identity that [kir;n] 2 Uqres(g) for all
r2N, n2Z, i2I.
Let Ures(g)(cid:6) be the subalgebra of Ures(g) generated by the (e(cid:6))(r) for all i2I,
q q i
r 2N, and let Uqres(g)0 be the subalgebra generated by the ki(cid:6)1 and the [kir;n], for
allr2N,n2Z,i2I. Multiplicationde(cid:12)nesanisomorphismofC(q)-vectorspaces
Ures(g)−⊗Ures(g)0⊗Ures(g)+ !Ures(g).
q q q q
The appropriate de(cid:12)nitions of Ures(^g)(cid:6) and Ures(^g)0 will be given at the end of
q q
Section 3.
Let (cid:96) be an odd integer with (cid:96) (cid:21) 3, and assume that (cid:96) is not divisible by 3 if g
is of type G . Let (cid:15)2C be a primitive (cid:96)th root of unity, and set
2
U(cid:15)res(g)=Uqres(g)⊗C[q,q−1]C, U(cid:15)res(^g)=Uqres(^g)⊗C[q,q−1]C,
viathe algebrahomomorphismC[q,q−1]!C that takesq to (cid:15). If xis anyelement
of Ures(g) (resp. Ures(^g)), we denote the corresponding element of Ures(g) (resp.
q q (cid:15)
Ures(^g)) also by x.
(cid:15)
De(cid:12)neU(cid:15)res(g)(cid:6) andU(cid:15)res(cid:2)(g)0i(cid:3)ntheobviousway. ItisknownthatU(cid:15)res(g)0isgen-
erated by the k(cid:6)1 and the ki;0 , for i2I. Multiplication de(cid:12)nes an isomorphism
i (cid:96)
of C-vector spaces
Ures(g)−⊗Ures(g)0⊗Ures(g)+ !Ures(g).
(cid:15) (cid:15) (cid:15) (cid:15)
LetUfin(g)bethesubalgebraofUres(g)generatedbythee(cid:6) fori2I andUres(g)0.
(cid:15) (cid:15) i (cid:15)
We shall make use of the characteristic zero Frobenius homomorphisms de(cid:12)ned
by Lusztig (see [15], Chapter 35). Namely, there exist homomorphisms of Hopf
algebras
Fr(cid:15) :U(cid:15)res(g)!U(g), F^r(cid:15) :U(cid:15)res(^g)!U(^g)
such that, for i2I,
(
Fr(cid:15)(ki)=1, Fr(cid:15)((e(cid:6)i )(r))= (e(r(cid:6)i/)(cid:96)r)/!(cid:96) if (cid:96) divides r,
0 otherwise,
and for i2I^,
(
F^r(cid:15)(ki)=1, F^r(cid:15)((e(cid:6)i )(r))= (e(r(cid:6)i/)(cid:96)r)/!(cid:96) if (cid:96) divides r,
0 otherwise.
To discuss (cid:12)nite-dimensional representations of Ures(^g), or of its specialization
q
U(cid:15)res(^g),itisadvantageoustouseanotherrealizationofUq(^g),duetoDrinfel’d[12],
Beck [1] and Jing [14].
Theorem 1.2. There is an isomorphism of C(q)-Hopf algebras from Uq(^g) to the
algebra with generators x(cid:6)i,r (i2I, r2Z), ki(cid:6)1 (i2I), hi,r (i2I, r 2Znf0g) and
286 VYJAYANTHI CHARI AND ANDREW PRESSLEY
c(cid:6)1/2, and the following defining relations:
c(cid:6)1/2 are central,
kiki−1 =ki−1ki =1, c1/2c−1/2 =c−1/2c1/2 =1,
kikj =kjki, kihj,r =hj,rki,
kix(cid:6)j,rki−1 =qi(cid:6)aijx(cid:6)j,r,
1 cr−c−r
[hi,r,hj,s]=δr,−sr[raij]qiqj −qj−1,
(3) 1
[hi,r,x(cid:6)j,s]=(cid:6)r[raij]qic(cid:7)jrj/2x(cid:6)j,r+s,
x(cid:6) x(cid:6) −q(cid:6)aijx(cid:6) x(cid:6) =q(cid:6)aijx(cid:6) x(cid:6) −x(cid:6) x(cid:6) ,
i,r+1 j,s i j,s i,r+1 i i,r j,s+1 j,s+1 i,r
c(r−s)/2ψ+ −c−(r−s)/2ψ−
[x+i,r,x−j,s]=δij i,r+qsi−qi−1 i,r+s,
(cid:20) (cid:21)
X Xm
(−1)k m x(cid:6) ...x(cid:6) x(cid:6) x(cid:6) ...x(cid:6) =0, if i6=j,
k i,rπ(1) i,rπ(k) j,s i,rπ(k+1) i,rπ(m)
π2Σmk=0 qi
for all sequences of integers r1,... ,rm, where m = 1−aij, (cid:6)m is the symmetric
(cid:6)
group on m letters, and the ψ are determined by equating powers of u in the
i,r
formal power series
!
X1 X1
ψi(cid:6),(cid:6)ru(cid:6)r =ki(cid:6)1exp (cid:6)(qi−qi−1) hi,(cid:6)su(cid:6)s .
r=0 s=1
The isomorphism is given by
(cid:6) r (cid:7)r (cid:6)
x =o(i) T (e ),
i,r λˇi i
where o : I ! f(cid:6)1g is a map such that o(i) = −o(j) whenever aij < 0 (it is clear
that there are exactly two possible choices for o).
(cid:6)
The classical analogues of the generators xi,r and hi,r are given by
x(cid:6)i,r =o(i)rT(cid:7)λˇir(e(cid:6)i ), hi,r =[x+i,r,x−i,0].
The latter elements, together with a central element c, generate U(^g) and the
de(cid:12)ning relations are obtainedfrom those in 1.2 by interpreting c as qc, ki as qihi,0,
ψi+,r (resp. ψi−,r)as (qi−qi−1)hi,r if r>0 (resp. ifr <0),andletting q tend to one.
For each i 2 I, there is an injective homomorphism of C(q)-algebras ϕi :
Uqi(s^l2) ! Uq(^g) that takes x(cid:6)r to x(cid:6)i,r and ψr(cid:6) to ψi(cid:6),r, for all r 2 Z (as usual,
we drop the (unique) Dynkin diagram subscript when dealing with Uq(s^l2)) (see
[1], Proposition 3.8). It follows immediately from 1.2 that Ures(s^l ) is gener-
qi 2
ated as a C[q,q−1]-algebra by c(cid:6)1/2, k(cid:6)1 and the (x(cid:6))(r) for n 2 Z, r 2 N (for
n
e+ = x− and e− = c−1kx+ ). Since the T preserve Ures(^g), it is also clear that
0 1 0 −1 λˇi q
(x(cid:6)i,n)(r) 2Uqres(^g)foralli2I,n2Z,r 2N. Itfollowsthatϕi(Uqries(s^l2))(cid:18)Uqres(^g),
and hence that ϕi induces a C-algebra homomorphism ϕi : U(cid:15)ries(s^l2) ! U(cid:15)res(^g).
We have that Tλˇj (cid:12)xes ϕi(Uqi(s^l2)) pointwise if i,j 2 I, i 6= j ([1], Corollary 3.2).
We denote the obvious classical analogue of ϕi by ϕi :U(s^l2)!U(^g).
Finally, we shall need the following automorphisms of Uq(^g):
QUANTUM AFFINE ALGEBRAS AT ROOTS OF UNITY 287
Proposition 1.3. (a) There exists a C(q)-algebra anti-automorphism (cid:8) of Uq(^g)
such that, for all i2I,r 2Z,
(cid:8)(x(cid:6)i,r)=x(cid:7)i,r, (cid:8)(c1/2)=c−1/2, (cid:8)(ψi(cid:6),r)=ψi(cid:6),r, (cid:8)(hi,r)=hi,r.
(b) There exists a C-algebra and coalgebra anti-automorphism Ω of Uq(^g) such
that Ω(q)=q−1 and, for all i2I,r2Z,
Ω(x(cid:6)i,r)=x(cid:7)i,−r, Ω(ψi(cid:6),r)=ψi(cid:7),−r, Ω(hi,r)=hi,−r, Ω(c1/2)=c−1/2.
Moreover, Ω and (cid:8) each preserve Ures(^g), commute with T, and we have, for all
q
i2I,
ΩTi =TiΩ, (cid:8)Ti =Ti−1(cid:8), (cid:8)Ω=Ω(cid:8).
Proof. The existence of(cid:8) andΩ canbe veri(cid:12)eddirectly by using 1.2. Itis obvious
that (cid:8) and Ω commute with T. The relations between Ω, (cid:8) and Ti are checkedby
verifying agreement on the generators e(cid:6)j , kj(cid:6)1 (j 2I^) of Uq(^g), on which they are
given by
(cid:6) (cid:7)
(cid:8)(ej )=ej , (cid:8)(kj)=kj,
Ω(e(cid:6)j )=e(cid:7)j , Ω(kj)=kj−1.
These formulas obviously imply that (cid:8) and Ω preserve Ures(^g). The fact that Ω
q
is a coalgebra anti-automorphism can also be checked on the generators e(cid:6),k(cid:6)1
j j
(j 2I^).
2. Representation Theory of Uq
A representation V of Uq(g) is said to be of type I if, for all i 2 I, ki acts
semisimply on V with eigenvalues in qZ. Any (cid:12)nite-dimensional irreducible repre-
i
sentation of Uq(g) can be obtained from a type I representation by tensoring with
a one-dimensional representation that takes each e(cid:6)i to zero and each ki to (cid:6)1.
If λ2P, the weight space Vλ of a representation V of Uq(g) is de(cid:12)ned by
(cid:8) (cid:12) (cid:9)
Vλ = v 2V (cid:12)ki.v =qiniv ,
where ni =λ(α(cid:20)i). We have
e(cid:6)i .Vλ (cid:18)Vλ(cid:6)αi.
A vector v in a type I representation V of Uq(g) is said to be a highest weight
vector if there exists λ2P such that
v 2Vλ and e+i .v =0 for all r 2N,i2I.
If, in addition, V = Uq(g).v, then V is said to be a highest weight representation
with highest weight λ.
For any λ 2 P, there exists, up to isomorphism, a unique irreducible represen-
tation V(λ) of Uq(g) with highest weight λ. We have
M
V(λ)= V(λ)µ.
µ(cid:20)λ
Every (cid:12)nite-dimensional irreducible type I representation of Uq(g) is isomorphic
to V(λ) for some (unique) λ 2 P+. The character of V(λ), and in particular its
dimension, is the same as that of the irreducible representationV(λ) of g with the
same highest weight λ.
288 VYJAYANTHI CHARI AND ANDREW PRESSLEY
Turning now to Uq(^g), we say that a representation V of Uq(^g) is type I if it
is type I as a representation of Uq(g) and c1/2 acts as one on V. A vector v in
a type I representation V of Uq(^g) is said to be a highest weight vector if v is
annihilatedbyx+ ,andisaneigenvectorofψ(cid:6) ,foralli2I,r2Z. If,inaddition,
i,r i,r
V = Uq(^g).v, then V is said to be a highest weight representation of Uq(^g). One
(cid:6)
shows by the usual Verma module arguments that, for any set of scalars Ψ such
i,r
that Ψ+i,0Ψ−i,0 =1, there is an irreducible highest weight representation V of Uq(^g)
with highest weight vector v such that ψ(cid:6) .v = Ψ(cid:6) v for all i2 I,r 2Z (compare
i,r i,r
(cid:6)
the proof of 7.3). Moreover, V is uniquely determined by the scalars Ψ , up to
i,r
isomorphism.
Theorem 2.1. The irreducible highest weight representation of Uq(^g) determined
by the scalars Ψ(cid:6)i,r is finite-dimensional if and only if there exist polynomials Pi 2
C(q)[u], for i2I, such that Pi(0)6=0 and
X1 Ψ+ ur =qdeg(Pi)Pi(qi−2u) =X1 Ψ− u−r,
i,r i i,−r
Pi(u)
r=0 r=0
in the sense that the left- and right-hand sides are the Laurent expansions of the
middle term about u=0 and u=1, respectively.
This is proved in [4], [5], [6] and [9]. From now on, we denote by (cid:5)q the set
of polynomials P 2 C(q)[u] such that P(0) 6= 0, and by (cid:5)I the set of I-tuples of
q
such polynomials. If P = (Pi)i2I 2 (cid:5)Iq, we denote by V(P) the (cid:12)nite-dimensional
irreducible representationof Uq(^g) determined by P as in 2.1.
If P = (Pi)i2I, Q = (Qi)i2I 2 (cid:5)Iq, de(cid:12)ne P⊗Q = (PiQi)i2I. The following
result is proved in [9] (see 7.4 for the proof of an analogous result):
Proposition 2.2. Let P,Q2(cid:5)I. If V(P)⊗V(Q) is irreducible as a representa-
q
tion of Uq(^g), then it is isomorphic to V(P⊗Q).
We shallalso needthe classicalanalogueof 2.1. Let (cid:5) be the set ofpolynomials
P 2C[u]suchthatP(0)6=0,andlet(cid:5)I be the setofI-tuples ofsuchpolynomials.
Then, the classical analogue of 2.1 states that every (cid:12)nite-dimensional irreducible
representation V of U(^g) is generated by a vector v that is annihilated by c and
the x(cid:6)i,r, and such that hi,r.v =Hi,rv, where the scalars Hi,r 2C satisfy
X1 P0(u) X1 P0(u)
Hi,rur =deg(Pi)−u i , Hi,−ru−r =u i ,
Pi(u) Pi(u)
r=0 r=0
for some P = (Pi)i2I 2 (cid:5)I. We write V as V(P). The obvious classical analogue
of 2.2 holds.
Finite-dimensional irreducible representations of Uq(^g) can be constructed ex-
plicitly when g = sl , by means of the following result. We take I^= f0,1g when
2
g=sl .
2
Proposition 2.3. For any non-zero a 2 C(q), there is an algebra homomorphism
eva :Uq(s^l2)!Uq(sl2) such that
(4) eva(e(cid:6)0)=q(cid:6)1a(cid:6)1e(cid:7), eva(k0)=k−1, eva(e(cid:6)1)=e(cid:6), eva(k1)=k.
Moreover, we have, for all r 2Z,
(5) eva(x+r )=q−rarkre+, eva(x−r )=q−rare−kr.
QUANTUM AFFINE ALGEBRAS AT ROOTS OF UNITY 289
See [4], Proposition 5.1, for the proof.
If V is a representation of Uq(sl2), we denote by Va the \evaluation represen-
tation" of Uq(s^l2) obtained by pulling back V by eva. Let V(n) be the (n+1)-
dimensional irreducible type I representation of Uq(sl2).
Proposition 2.4. Let r,n1,... ,nr 2 N, and let a1,... ,ar 2 C(q) be non-zero.
Then, the tensor product
V(n1)a1 ⊗(cid:1)(cid:1)(cid:1)⊗V(nr)ar
is irreducible as a representation of Uq(s^l2) if and only if, for all 1(cid:20)s6=t(cid:20)r,
as/at 6=q(cid:6)(ns+nt−2p) for all 0(cid:20)p<min(ns,nt).
This is proved by exactly the same argument as Theorem 5.8 in [4].
(cid:24)
It is easy to compute that V(n)a =V(Pn,a), where
Yn
(6) Pn,a(u)= (1−aqn−2m+1u)
m=1
(see [4], Corollary 4.2). It follows from 2.2 that, when the conditions in 2.4 are
satis(cid:12)ed,
V(n1)a1 ⊗(cid:1)(cid:1)(cid:1)⊗V(nr)ar (cid:24)=V(P),
Q
r
where P = s=1Pns,as.
The classical analogues of the homomorphisms eva exist for all g. They are the
homomorphisms eva :U(^g)!U(g), de(cid:12)ned for any non-zero a2C, such that
eva(e(cid:6)0)=a(cid:6)1e(cid:7)θ, eva(e(cid:6)i )=e(cid:6)i , eva(h0)=−θ(cid:20), eva(hi)=hi,
for i 2 I. Here, e(cid:6) are root vectors corresponding to (cid:6)θ, normalised so that
θ
[e+θ,e−θ]=θ(cid:20). EvaluationrepresentationsVa ofU(^g)arede(cid:12)nedintheobviousway.
Using this construction, one can describe explicitly the representationV(P) for all
P2(cid:5)I.
Theorem 2.5. Let P=(Pi)i2I 2(cid:5)I, and let fa−11,a−21,... ,a−r1g be the union of
the set of roots of Pi for all i2I. Write
Yr
Pi(u)=(const.) (1−asu)ni,s
s=1
P
for some integers ni,s (cid:21)0, and let µs = i2Ini,sλi 2P+. Then,
V(P)(cid:24)=V(µ1)a1 ⊗V(µ2)a2 ⊗(cid:1)(cid:1)(cid:1)⊗V(µr)ar.
Conversely, if µ1,µ2,... ,µr 2P+ and a1,a2,... ,ar 2C are non-zero, the tensor
product V(µ1)a1⊗V(µ2)a2⊗(cid:1)(cid:1)(cid:1)⊗V(µr)ar is irreducible as a representation of ^g if
and only if a1,a2,... ,ar are distinct.
Proof. By using the methods in [3], one shows that every (cid:12)nite-dimensional irre-
ducible representationof U(^g) is isomorphic to a tensor product
V(µ1)a1 ⊗V(µ2)a2 ⊗(cid:1)(cid:1)(cid:1)⊗V(µr)ar