Quantum affine algebras and their representations

arXiv:hep-th/9411145v1 18 Nov 1994 Canadian Mathematical Society Conference Proceeding Volume 00, 0000 Quantum Affine Algebras and their Representations VYJAYANTHI CHARI AND ANDREW PRESSLEY Abstract. We prove a highest weight classification of the finite-dimensional irreducible representations of a quantum affine algebra, in the spirit of Cartan’s classification of the finite-dimensional irreducible representations of complex simple Lie algebras in terms of dominant integral weights. We also survey what is currently known about the structure of these representations. 1. Introduction Around 1985, V.G. Drinfel’d and M. Jimbo showed, independently, how to associate to any symmetrizable Kac–Moody algebra g over C a family Uq (g) of Hopf algebras, depending on a parameter q ∈ C× , and reducing (essentially) to the classical universal enveloping algebra U (g) when q = 1. The introduction of quantum groups has opened up a fascinating new chapter in representation theory; in addition, quantum groups have turned out to have surprising connections with several areas of mathematics (algebraic groups in characteristic p, knot theory, . . . ) and physics (two–dimensional integrable systems, conformal field theories, . . . ). Many of the applications of quantum groups (such as those in knot theory, for example) depend on the fact that, if g is finite–dimensional and q is not a root of unity, one can associate to any finite–dimensional representation V of Uq (g) an operator R ∈ End (V ⊗V ) which satisfies the quantum Yang–Baxter equation (QYBE): (1) R12 R13 R23 = R23 R13 R12 (here, R12 means R⊗id ∈ End(V ⊗V ⊗V ), etc.). In fact, if W is another finite– dimensional representation of Uq (g), it turns out that the tensor products V ⊗W and W ⊗V are isomorphic as representations of Uq (g), and further that there is a 1991 Mathematics Subject Classification. Primary 17B37, 81R50; Secondary 16W30, 82B23. The first author was partially supported by NSF Grant #9207701 1 2 CHARI AND PRESSLEY canonical choice of isomorphism IV,W : V ⊗W → W ⊗V . If V = W and σ is the flip map V ⊗V → V ⊗V , the matrix R = σI satisfies (1). In some situations, however, it is important to have a solution of the ‘QYBE with spectral parameters’: (2) R12 (u, v)R13 (u, w)R23 (v, w) = R23 (v, w)R13 (u, w)R12 (u, v). Here, R(u, v) is a family of operators in End (V ⊗V ), for some finite–dimensional vector space V , depending on a pair of complex parameters u, v. In many cases, possibly after making a change of variable u 7→ f (u), v 7→ f (v), R(u, v) becomes a function of u − v, which we write as R(u − v). In the theory of two–dimensional lattice models in statistical mechanics, for example, R(u) is a matrix whose entries are the ‘interaction’ energies of the atoms in the lattice, and u is a parameter on which the properties of the model depend, such as the values of external electric or magnetic fields. From R(u) one constructs the ‘transfer matrices’ T (u) = R01 (u)R02 (u) . . . R0N (u) ∈ End (V ⊗V ⊗N ) (the first copy of V in V ⊗V ⊗N is numbered 0, the others 1, . . . , N ) and from these the partition function Z = traceV ⊗N (traceV (T )N ) (we assume that the lattice is N atoms wide in each direction and that periodic boundary conditions are imposed). It is explained in [1], for example, that the physical properties of the model may be deduced from Z. If R(u) is invertible and satisfies (2), it is easy to show that traceV (T (u)) commutes with traceV (T (v)) for all u, v: for this reason, such models are called ‘integrable’. One can hope to construct solutions of (2) whenever one has a Hopf algebra A equipped with a family of automorphisms τu . For, if V is a finite–dimensional (complex) representation of A, pulling back V by τu gives a 1-parameter family of representations V (u). Assume that, for all parameters u, v, w, and for some representation V of A, (i) V (u)⊗V (v) is isomorphic to V (v)⊗V (u), and (ii) V (u)⊗V (v)⊗V (w) is irreducible, and let I(u, v) : V (u)⊗V (v) → V (v)⊗V (u) be an intertwiner (which, by (i), is well–defined up to a scalar multiple). If R = σI, equation (2) is the condition for the equality of the two composites of intertwiners V (u)⊗V (v)⊗V (w) → V (v)⊗V (u)⊗V (w) → V (v)⊗V (w)⊗V (u) → V (w)⊗V (v)⊗V (u) and V (u)⊗V (v)⊗V (w) → V (u)⊗V (w)⊗V (v) → V (w)⊗V (u)⊗V (v) → V (w)⊗V (v)⊗V (u). Thus, condition (ii) guarantees that (2) is satisfied up to a scalar multiple. QUANTUM AFFINE ALGEBRAS AND THEIR REPRESENTATIONS 3 Let ĝ be the (untwisted) affine Lie algebra associated to a finite–dimensional complex simple Lie algebra g. Recall that ĝ is a central extension, with 1–dimensional centre, of the Lie algebra g[t, t−1 ] of Laurent polynomial maps C× → g, under pointwise operations. There is an obvious multiplicative 1-parameter group of automorphisms of ĝ, given by rescaling t, which fixes each element of the centre. On the other hand, ĝ is a symmetrizable Kac–Moody algebra, so one can define the Hopf algebra Uq (ĝ). We shall assume from now on that q is transcendental. It turns out that Uq (ĝ) also has a multiplicative 1-parameter group of automorphisms τu , which reduce, in the limit q → 1, to the rescaling automorphisms of ĝ. According to Drinfel’d [11], property (i) holds for generic values of u, v, and there exists a canonical choice of isomorphism I(u, v) such that R(u, v) = σI(u, v) satisfies (2). Moreover, the multiplicative property of τu implies that R(u, v) depends only on u/v; reparametrizing by u 7→ eu , v 7→ ev , we get a solution of (2) which depends only on u − v. Thus, it is of considerable interest to describe the finite–dimensional irreducible representations of Uq (ĝ). The main result proved in this paper (Theorem 3.3) gives a parametrization of the finite–dimensional irreducible representations of Uq (ĝ) analogous to Cartan’s highest weight classification of the finite–dimensional irreducible representations of g. The role of dominant integral weights in the representation theory of g is played for Uq (ĝ) by the set of rank(g)–tuples P of polynomials in one variable with constant coefficient 1; let V (P) be the representation of Uq (ĝ) associated to P. To construct explicit solutions of (2), one needs to understand the structure of the representations V (P). Now, there is a canonical embedding of Hopf algebras Uq (g) ֒→ Uq (ĝ) which, in the limit q → 1, becomes the embedding g ֒→ ĝ given by regarding elelments of g as constant maps C× → g. Thus, representations of Uq (ĝ) can be regarded as representations of Uq (g). Since finite-dimensional representations of Uq (g) are completely reducible, a first step in understanding V (P) would be to describe its decomposition under Uq (g). We shall say that two representations of Uq (ĝ) are equivalent if they are isomorphic as representations of Uq (g). Unfortunately, the problem of describing the structure of V (P) as a representation of Uq (g) appears to be intractable for general P. However, it is still interesting to understand the representations V (P) of some special type. Any V (P) has a unique irreducible Uq (g)–subrepresentation of maximal highest weight. Conversely, given a finite–dimensional irreducible representation V of Uq (g), one can consider the representations V (P) of Uq (ĝ) which have V as their top Uq (g)–component – V (P) is then called an affinization of V . (Thus, every V (P) is an affinization of its top Uq (g)-component.) Classically, every finite–dimensional representation V of g has an affinization (in the obvious sense) which is irreducible under g. For, there is an algebra homomorphism evu : ĝ → g, for any u ∈ C× , which annihilates the centre of ĝ and evaluates maps C× → g at u; note that evu is the identity on g. Pulling back V by evu gives a family of representations V (u) of ĝ, which are obviously isomorphic to V as representations of g. In the quantum case, however, there are simple examples 4 CHARI AND PRESSLEY of irreducible representations of Uq (g) which have no affinization that is irreducible under Uq (g). Thus, it is natural to look for the ‘smallest’ affinization(s). In [4], a natural partial ordering was defined on the set of equivalence classes of finite–dimensional representations of Uq (g). One can show that a given irreducible representation V of Uq (g) has only finitely many affinizations, up to equivalence, so it makes sense to look for the minimal one(s). In Section 6, we give necessary and sufficient conditions on P for V (P) to be a minimal affinization of its top Uq (g)– component, summarizing results in [7], [9], [4], and [10]. We use these results to describe the Uq (g)–structure of the minimal affinizations in some cases. 2. Quantum affine algebras We begin by recalling the definition of the Hopf algebras Uq (g). Let g be a finite–dimensional complex simple Lie algebra with Cartan subalgebra h and Cartan matrix A = (aij )i,j∈I . Fix coprime positive integers (di )i∈I such that (di aij ) is symmetric. Let P = ZI and let P + = {λ ∈ P | λ(i) ≥ 0 for all i ∈ I}. For i ∈ I, define λi ∈ P + by λi (j) = δij . Let R (resp. R+ ) be the set of roots (resp. positive roots) of g. Let αi (i ∈ I) be the simple roots and let θ be the highest P root. Let Q = ⊕i∈I Z.αi ⊂ h∗ be the root lattice, and set Q+ = i∈I N.αi . Define P a partial order ≥ on P by λ ≥ µ iff λ − µ ∈ Q+ . If η = i∈I mi αi ∈ Q+ , define P height(η) = i∈I mi . Define a non-degenerate symmetric bilinear form ( , ) on h∗ by (αi , αj ) = di aij , and set d0 = 21 (θ, θ). Let q ∈ C× be transcendental, and, for r, n ∈ N, n ≥ r, define q n − q −n , q − q −1 [n]q ! = [n]q [n − 1]q . . . [2]q [1]q , hni [n]q ! . = [r]q ![n − r]q ! r q [n]q = Set qi = q di . Proposition 2.1. There is a Hopf algebra Uq (g) over C which is generated as ±1 an algebra by elements x± (i ∈ I), with the following defining relations: i , ki ki ki−1 = ki−1 ki = 1, ki kj = kj ki , ±aij ± xj , −1 ki x± = qi j ki − [x+ i , xj ] = δij ki − ki−1 , qi − qi−1 1−aij X  1 − aij  r ± ± 1−aij −r (x± = 0, i ) xj (xi ) r q i r=0 i 6= j. QUANTUM AFFINE ALGEBRAS AND THEIR REPRESENTATIONS 5 The comultiplication ∆, counit ǫ, and antipode S of Uq (g) are given by + + ∆(x+ i ) = xi ⊗ki + 1⊗xi , − −1 − ∆(x− i ) = xi ⊗1 + ki ⊗xi , ∆(ki±1 ) = ki±1 ⊗ki±1 , ±1 ǫ(x± i ) = 0, ǫ(ki ) = 1, + −1 − − ±1 ∓1 S(x+ i ) = −xi ki , S(xi ) = −ki xi , S(ki ) = ki , for all i ∈ I. The generators and relations in 2.1 serve, in fact, to define a Hopf algebra Uq (g) when g is an arbitrary symmetrizable Kac–Moody algebra. In particular, if ĝ is the (untwisted) affine Lie algebra associated to g, one can define the Hopf algebra Uq (ĝ) as in 2.1, but replacing I by Iˆ = I ∐ {0} and A by the extended Cartan matrix  = (aij )i,j∈Iˆ of g; we let q0 = q d0 . ± Note that there is a canonical homomorphism Uq (g) → Uq (ĝ) such that x± i 7→ xi , ±1 ±1 ki 7→ ki for all i ∈ I. Thus, any representation of Uq (ĝ) may be regarded as a representation of Uq (g). Now ĝ is better understood than an arbitrary infinite–dimensional Kac–Moody Lie algebra because it has another realization as (a central extension of) a space of maps C× → g, as we mentioned in Section 1. In [12], Drinfel’d stated (in a slightly different form) a realization of Uq (ĝ) which, although still in terms of generators and relations, more closely resembles the description of ĝ as a space of maps. In the following form, the result was proved by Beck [2]: ±1 Theorem 2.2. Let Aq be the algebra with generators x± i,r (i ∈ I, r ∈ Z), ki (i ∈ I), hi,r (i ∈ I, r ∈ Z\{0}) and c±1/2 , and the following defining relations: c±1/2 are central, ki ki−1 = ki−1 ki = 1, c1/2 c−1/2 = c−1/2 c1/2 = 1, ki kj = kj ki , ki hj,r = hj,r ki , ±aij ± xj,r , ki xj,r ki−1 = qi 1 ∓|r|/2 ± xj,r+s , [hi,r , x± j,s ] = ± [raij ]qi c r ±a ±aij ± ± ± ± ± ± xj,s xi,r+1 = qi ij x± x± i,r xj,s+1 − xj,s+1 xi,r , i,r+1 xj,s − qi (3) − [x+ i,r , xj,s ] = δij (4) P π∈Σm Pm k k=0 (−1) m k qi −(r−s)/2 − c(r−s)/2 φ+ φi,r+s i,r+s − c qi − qi−1 , ± ± ± ± x± i,rπ(1) . . . xi,rπ(k) xj,s xi,rπ(k+1) . . . xi,rπ(m) = 0, if i 6= j, for all sequences of integers r1 , . . . , rm , where m = 1 − aij , Σm is the symmetric group on m letters, and the φ± i,r are determined by equating powers of u 6 CHARI AND PRESSLEY in the formal power series ∞ X ±r φ± i,±r u = ki±1 exp ±(qi − qi−1 ) r=0 ∞ X hi,±s u s=1 ±s ! . Q P mi + If θ = i∈I ki . Suppose that the root vector xθ of g i∈I mi αi , set kθ = + corresponding to θ is expressed in terms of the simple root vectors xi (i ∈ I) of g as + + + + x+ θ = λ[xi1 , [xi2 , · · · , [xik , xj ] · · · ]] for some λ ∈ C× . Define maps wi± : Uq (ĝ) → Uq (ĝ) by ±1 ∓1 ± wi± (a) = x± i,0 a − ki aki xi,0 . Then, there is an isomorphism f : Uq (ĝ) → Aq defined on generators by ± f (k0 ) = kθ−1 , f (ki ) = ki , f (x± i ) = xi,0 , f (x+ 0) − f (x0 ) = = (i ∈ I), µwi−1 −1 · · · wi−k (x− j,1 )kθ , λkθ wi+1 · · · wi+k (x+ j,−1 ), where µ ∈ C× is determined by the condition − [x+ 0 , x0 ] = k0 − k0−1 . q0 − q0−1 Let Û ± (resp. Û 0 ) be the subalgebra of Uq (ĝ) generated by the x± i,r (resp. by ± ± 0 the φi,r ) for all i ∈ I, r ∈ Z. Similarly, let U (resp. U ) be the subalgebra of ±1 Uq (g) generated by the x± i (resp. by the ki ) for all i ∈ I. We have the following weak version of the Poincaré–Birkhoff–Witt theorem: Proposition 2.3. (a) Uq (g) = U − .U 0 .U + . (b) Uq (ĝ) = Û − .Û 0 .Û + . See [8] or [14] for details. 3. Representation theory of Uq (g) and Uq (ĝ) We begin by summarizing the relevant facts about the representation theory of Uq (g) (we continue to assume that q is transcendental). For further details, see [8] or [14], for example. Let W be a representation of Uq (g). One says that λ ∈ P is a weight of W if the weight space Wλ = {w ∈ W |ki .w = q λ(i) w} is non–zero. We say that W is of type 1 if M W = Wµ . µ∈P QUANTUM AFFINE ALGEBRAS AND THEIR REPRESENTATIONS 7 A non–zero vector w ∈ Wλ is called a highest weight vector if x+ i .w = 0 for all i ∈ I, and W is called a highest weight representation with highest weight λ if W = Uq (g).w for some highest weight vector w ∈ Wλ . Any highest weight representation is of type 1. For any λ ∈ P , let M (λ) be the quotient of Uq (g) by the left ideal generated λ(i) by {x+ .1}i∈I . Then, M (λ) is a highest weight representation of Uq (g) i , ki − q with highest weight λ, and it follows from 2.3(a) that M (λ)λ is one–dimensional. The standard argument implies that M (λ) has a unique irreducible quotient V (λ), and that every irreducible highest weight representation with highest weight λ is isomorphic to V (λ). For any i ∈ I, let σi be the algebra automorphism of Uq (g) such that − − δij δij + σi (x+ j ) = (−1) xj , σi (kj ) = (−1) kj , σi (xj ) = xj (5) for all j ∈ I. Proposition 3.1. (a) Every finite–dimensional representation of Uq (g) is completely reducible. (b) Every finite–dimensional irreducible representation of Uq (g) can be obtained from a type 1 representation by twisting with a product of the automorphisms σi . (c) Every finite–dimensional irreducible representation of Uq (g) of type 1 is highest weight. (d) The representation V (λ) is finite–dimensional iff λ ∈ P + . (e) If λ ∈ P + , V (λ) has the same character as the irreducible representation of g of the same highest weight. (f) The multiplicity mν (V (λ)⊗V (µ)) of V (ν) in the tensor product V (λ)⊗V (µ), where λ, µ, ν ∈ P + , is the same as in the tensor product of the irreducible representations of g of the same highest weight (this statement makes sense in view of parts (a), (c) and (d)). We now turn to the representation theory of Uq (ĝ). A representation V of Uq (ĝ) is of type 1 if c1/2 acts as the identity on V , and if V is of type 1 as a representation of Uq (g). A vector v ∈ V is a highest weight vector if ± ± x+ i,r .v = 0, φi,r .v = Φi,r v, c1/2 .v = v, for some complex numbers Φ± i,r . A type 1 representation V is a highest weight representation if V = Uq (ĝ).v, for some highest weight vector v, and the pair of (I × Z)–tuples (Φ± i,r )i∈I,r∈Z is its highest weight. (In [8], highest weight representations of Uq (ĝ) are called ‘pseudo-highest weight’.) − Note that Φ+ i,r = 0 (resp. Φi,r = 0) if r < 0 (resp. if r > 0), and that − + Φi,0 Φi,0 = 1. Conversely, if Φ = (Φ± i,r )i∈I,r∈Z is a set of complex numbers satisfying Φ) be the quotient of Uq (ĝ) by the left ideal generated by these conditions, let M (Φ ± ± ±1/2 Φ) is a highest weight represen{x+ − 1}. Then, M (Φ i,r , φi,r − Φi,r .1}i∈I,r∈Z ∪ {c Φ) as a representation of tation of Uq (ĝ). It follows from 2.3(b) that, regarding M (Φ 8 CHARI AND PRESSLEY Φ))λ = 1, and hence that M (Φ Φ) has a unique irreducible Uq (g), we have dim(M (Φ Φ). Clearly, every irreducible highest quotient (as a representation of Uq (ĝ)), say V (Φ Φ). weight representation of Uq (ĝ) is isomorphic to some V (Φ Let σi (i ∈ I) be the algebra automorphisms of Uq (ĝ) defined by the formulas in ˆ Also, let σ be the algebra automorphism of Uq (ĝ) (5), but with the indices i, j ∈ I. given, in terms of the presentation 2.2, by σ(c1/2 ) = −c1/2 , σ(ki ) = ki , r ± σ(x± i,r ) = (−1) xi,r , σ(hi,r ) = hi,r . Proposition 3.2. Let V be a finite-dimensional irreducible representation of Uq (ĝ). (a) V can be obtained from a type 1 representation by twisting with a product ˆ σ. of some of the automorphisms σi (i ∈ I), (b) If V is of type 1 (as a representation of Uq (ĝ)), then V is highest weight. See Section 12.2 of [8] for the proof. Thus, to classify the finite-dimensional irreducible representations of Uq (ĝ), we Φ) is finite-dimensional. have only to determine for which Φ the representation V (Φ The answer to this question is the main result of this paper. If λ ∈ P + , let P λ be the set of all I-tuples (Pi )i∈I of polynomials Pi ∈ C[u], with constant term 1, such that deg(Pi ) = λ(i) for all i ∈ I. Set P = ∪λ∈P + P λ . Theorem 3.3. Let Φ = (Φi,r )i∈I,r∈Z be a pair of (I × Z)-tuples of complex Φ) of Uq (ĝ) is finitenumbers, as above. Then, the irreducible representation V (Φ dimensional iff there exists P = (Pi )i∈I ∈ P such that (6) ∞ X r=0 −2 deg(Pi ) Pi (qi u) r Φ+ i,r u = qi Pi (u) = ∞ X −r Φ− , i,−r u r=0 in the sense that the left- and right-hand terms are the Laurent expansions of the middle term about 0 and ∞, respectively. By abuse of notation, we denote the finite-dimensional irreducible representation of Uq (ĝ) associated to P by V (P), and say that P is its highest weight. The ‘only if’ part of 3.3 is proved in [8], and we shall say no more about it in this paper. The ‘if’ part is proved in the next two sections. To conclude the present section, however, we describe the behaviour of the Ituples P under tensor products. If P = (Pi )i∈I , Q = (Qi )i∈I ∈ P, let P⊗Q ∈ P be the I–tuple (Pi Qi )i∈I . Proposition 3.4. Let P, Q ∈ P, and let vP and vQ be Uq (ĝ)-highest weight vectors in V (P) and V (Q), respectively. Then, in V (P)⊗V (Q), we have ± ± x+ i,r .(vP ⊗vQ ) = 0, φi,r .(vP ⊗vQ ) = Ψi,r (vP ⊗vQ ) QUANTUM AFFINE ALGEBRAS AND THEIR REPRESENTATIONS 9 for all i ∈ I, r ∈ Z, where the complex numbers Ψ± i,r are related to P⊗Q as the Φ± are related to P in (6). i,r See [8] for the proof. The following result is an immediate consequence: Corollary 3.5. Let P, Q ∈ P. Then, V (P⊗Q) is isomorphic, as a representation of Uq (ĝ), to a quotient of the subrepresentation of V (P)⊗V (Q) generated by the tensor product of the highest weight vectors in V (P) and V (Q). Since every polynomial is a product of linear polynomials, the last result suggests that we define a representation V (P) of Uq (ĝ) to be fundamental if, for some i ∈ I, Pj = 1 if j 6= i and deg(Pi ) = 1. Then, iterating 3.5, we obtain Corollary 3.6. For any P ∈ P, the representation V (P) of Uq (ĝ) is isomorphic to a subquotient of a tensor product of fundamental representations. This suggests a method of proving the ‘if’ part of Theorem 3.3. For, in view of 3.6, it clearly suffices to prove that the fundamental representations of Uq (ĝ) are all finite-dimensional. Since the fundamentals are the ‘simplest’ representations of Uq (ĝ), it should be possible to describe them ‘explicitly’, and, in particular, to prove that they are finite-dimensional. We shall use this approach in the sl2 case in the next section, and, although we have no doubt that it can be carried through in the general case, we shall use a different, more abstract, approach to complete the proof of 3.3 in Section 5. 4. Proof of the main theorem: sl2 case It is easy to construct finite–dimensional representations of the classical affine Lie algebra ĝ thanks to the existence of the family of homomorphisms eva : ĝ → g which annihilate the centre of ĝ and evaluate maps C× → g at a ∈ C× . If V is a representation of g, the pull–back of V by eva is a representation Va of ĝ. Jimbo ˆ 2 ): [13] defined an analogue of eva for Uq (sl ˆ 2) → Proposition 4.1. There is a family of algebra homomorphisms eva : Uq (sl × 1/2 Uq (sl2 ), defined for all a ∈ C , such that eva (c ) = 1 and −r −r r + eva (x+ a k1 x1 , 1,r ) = q −r −r − r eva (x− a x1 k1 , 1,r ) = q for all r ∈ Z. See [6], Proposition 4.1, for the proof. ˆ n ), for all n ≥ 2 (strictly Remark. Jimbo defined an analogue of eva for Uq (sl speaking, if n > 2 Jimbo’s homomorphism takes values in an ‘enlargement’ of Uq (sln )). If g is not of type A, there is no homomorphism Uq (ĝ) → Uq (g) which is the identity on Uq (g) ⊂ Uq (ĝ) (see [8]). If V is a type 1 representation of Uq (sl2 ), its pull–back Va by eva is obviously a ˆ 2 ); we call Va an evaluation representation of Uq (sl ˆ 2 ). type 1 representation of Uq (sl 10 CHARI AND PRESSLEY Since eva is the identity on Uq (sl2 ), Va is isomorphic to V as a representation of Uq (sl2 ); in particular, Va is irreducible if V is. The finite-dimensional irreducible type 1 representations of Uq (sl2 ) are easy to describe. We know that there is exactly one such representation V (r) of each dimension r + 1 ≥ 1, since the same is true for sl2 . It is easy to check that, if {v0 , v1 , . . . , vr } is a basis of V (r), the formulas − k1 .vk = q r−2k vk , x+ 1 .vk = [r − k + 1]q vk−1 , x1 .vk = [k + 1]q vk+1 define the required representation (we set v−1 = vr+1 = 0). Using the relations in 2.2, it follows that v0 is a Uq (ĝ)–highest weight vector of V (r)a , and that −k k(r−1) r φ± q (q − q −r )v0 1,k .v0 = a [rk]q v0 . k Using these formulas, one finds that V (r)a ∼ = V (Pr,a ), where h1,k .v0 = q −k a−k Pr,a (u) = r Y (1 − a−1 q r−2k+1 u). k=1 −r+1 −r+3 The set Σr,a = {aq , aq , . . . , aq r−1 } of roots of Pr,a is called the q-segment of length r and centre a. At this point, it is easy to complete the proof of 3.3 in the sl2 case. As we noted at the end of Section 3, it suffices to prove that the fundamental representations are finite-dimensional. But, since P1,a (u) = 1 − a−1 u, it follows that the fundaˆ 2 ) are precisely the V (1)a , for arbitrary a ∈ C× . In mental representations of Uq (sl particular, they all have dimension 2. Before turning to the general case of 3.3, however, we shall describe the structure ˆ 2 ) in more detail: of the representations V (P ) of Uq (sl Proposition 4.2. Let r1 , r2 , . . . , rk ∈ N, a1 , a2 , . . . , ak ∈ C× , k ∈ N. Then, the tensor product V (r1 )a1 ⊗V (r2 )a2 ⊗ · · · ⊗V (rk )ak is reducible as a representation of ˆ 2 ) iff at least one pair of q-segments Σr ,a , Σr ,a , 1 ≤ i, j ≤ k, are in special Uq (sl j j i i position, in the sense that their union is a q-segment which properly contains them both. This is proved in [6]. It is now easy to describe the representation V (P ), for any polynomial P ∈ C[u] with constant coefficient 1. The roots of P form a multiset, i.e. a finite set of nonzero complex numbers (the roots of P ), with a positive integer attached to each element of the set (its multiplicity as a root of P ). It is not difficult to show that every multiset can be written uniquely as a union of q-segments, no two of which are in special position. (The union is in the sense of multisets: the multiplicity of a complex number in a union of multisets is the sum of its multiplicities in each of them.) We can thus write multiset of roots of P = Σr1 ,a1 ∪ Σr2 ,a2 ∪ · · · ∪ Σrk ,ak QUANTUM AFFINE ALGEBRAS AND THEIR REPRESENTATIONS 11 for some r1 , r2 , . . . , rk ∈ N, a1 , a2 , . . . , ak ∈ C× , k ∈ N, and where no pair Σri ,ai , Σrj ,aj is in special position. By 3.5 and 4.2, there is an isomorphism of representaˆ 2) tions of Uq (sl V (r1 )a1 ⊗V (r2 )a2 ⊗ · · · ⊗V (rk )ak ∼ = V (Pr1 ,a1 Pr2 ,a2 . . . Prk ,ak ). But, the polynomial Pr1 ,a1 Pr2 ,a2 . . . Prk ,ak has the same roots as P , with the same multiplicities, and hence is equal to P (both polynomials having constant coefficient 1). Thus, V (P ) ∼ = V (r1 )a ⊗V (r2 )a ⊗ · · · ⊗V (rk )a . 2 1 k We have proved ˆ 2 ) of Theorem 4.3. Every finite-dimensional irreducible representation of Uq (sl type 1 is isomorphic to a tensor product of evaluation representations. There is an amusing interpretation of q-segments in terms of ‘q-derivatives’, which will allow us to give a kind of Weyl dimension formula for V (P ). We recall that, if P ∈ C[u], its q-derivative is (Dq P )(u) = P (q 2 u) − P (u) . q2 u − u It is obvious that Dq P is a polynomial in u (and q), and that lim Dq P = q→1 dP . du The interpretation we have in mind is based on the following elementary result, whose proof we leave to the reader. Proposition 4.4. Let P ∈ C[u] have non-zero constant coefficient, and let ΣP be its multiset of roots. Then, for each integer k ≥ 2, the number of q-segments of length k in ΣP is equal to the number of common roots of the polynomials P, Dq P, . . . , Dqk−1 P . To clarify the meaning of 4.4, suppose that, in the canonical decomposition of ΣP into a union of q-segments, no two of which are in special position, there is one segment of length 2 and one of length 3. Then, the number of q-segments of length 2 in ΣP is 3: ◦| {z }◦ ◦| {z }◦| {z }◦ Of course, there is one q-segment of length 3 in ΣP , and none of length > 3. In general, suppose that, for each k ≥ 1, there are nk q-segments of length k in the canonical decomposition of ΣP . Then, there are Nk q-segments of length k 12 CHARI AND PRESSLEY altogether, where N1 = n1 + 2n2 + 3n3 + · · · + rnr , N2 = n2 + 2n3 + 3n4 + · · · + (r − 1)nr , .. . N r = nr , and r = deg(P ). Hence, nk = Nk − 2Nk+1 + Nk+2 (we set Nk = 0 if k > r). By 4.3, and the discussion preceding it, it is clear that dim(V (P )) = r Y (k + 1)nk . k=1 A little rearrangement now gives Proposition 4.5. For any P ∈ C[u] with constant coefficient 1, dim(V (P )) = 2 deg(P ) deg(P )  2 Y k=2 k −1 k2 Nk , where, for each integer k ≥ 2, Nk is the number of common roots of P, Dq P, . . . . . . , Dqk−1 P . It would be interesting to find an analogue of this result for the dimensions of the representations V (P) of Uq (ĝ), for arbitrary g. 5. Proof of the main theorem: general case Let P = (Pi )i∈I ∈ P and let vP be a Uq (ĝ)-highest weight vector in V (P). Since ±1 + φ± by λ(i) = deg(Pi ), then i,0 = ki , it follows from (6) that, if we define λ ∈ P λ(i) ki .vP = qi (7) vP , so vP ∈ V (P)λ . By 2.3(b), V (P)λ = CvP and M V (P)λ−η . V (P) = η∈Q+ Thus, to prove the ‘if’ part of 3.3, it is enough to prove the following assertions: (a) V (P)λ−η = 0 for all except finitely many η ∈ Q+ . (b) For all η ∈ Q+ , dim(V (P)λ−η ) < ∞. PROOF OF (a). Let 0 6= v ∈ V (P)µ , where µ = λ − η, η ∈ Q+ . Let Ui ±1 be the subalgebra of Uq (ĝ) generated by x± (i ∈ I), and let Vi = Ui .v. i and ki Note that there is an obvious homomorphism of algebras (actually an isomorphism) ± ±1 Uqi (sl2 ) → Ui which takes x± 7→ ki±1 , so Vi may be regarded as a 1 7→ xi , k1 representation of Uqi (sl2 ). We claim that, to prove (a), it suffices to prove (c) If 0 6= v ∈ V (P)µ and Vi = Ui .v, then dim(Vi ) < ∞. QUANTUM AFFINE ALGEBRAS AND THEIR REPRESENTATIONS 13 To see that (c) implies (a), note that, if si is the ith fundamental reflection in the Weyl group W of g, the finite-dimensionality of Vi implies that its set of weights is stable under the action of si (this follows from 3.1(e) and the analogous classical statement). Hence, V (P)µ 6= 0 implies V (P)si (µ) 6= 0 for all i ∈ I. It follows that, if w ∈ W is arbitrary, then V (P)w(µ) 6= 0. Since one can choose w so that w(µ) ∈ P + , it follows that any µ ∈ P such that V (P)µ 6= 0 belongs to the finite set W.{ν ∈ P + | ν ≤ λ}. Thus, we are reduced to proving (c). Now (c) is clearly a consequence of (d) If V (P)µ 6= 0, there exists N > 0 such that V (P)µ−rαi = V (P)µ+rαi = 0 if r > N. r Indeed, assuming (d), it is clear that Vi is spanned by {(x± i ) .v | 0 ≤ r ≤ N }. To prove (d), note that it is obvious that V (P)µ+rαi = 0 for r >> 0, since µ + rαi ≤ λ only for finitely many r > 0. We shall prove, on the other hand, that V (P)µ−rαi = 0 if r > 3h + λ(i), where h = height(λ − µ). Indeed, this follows from (e) For any r > 0, V (P)µ−rαi is spanned by vectors of the form − − − − − X1− x− i1 ,k1 X2 xi2 ,k2 . . . Xh xih ,kh Xh+1 .vP , (8) where λ − µ = αi1 + αi2 + · · · + αih , k1 , k2 , . . . , kh ∈ Z are arbitrary, and each Xp− , 1 ≤ p ≤ h + 1, is a product of the form − − Xp− = x− i,ℓ1,p xi,ℓ2,p . . . xi,ℓrp ,p , for some ℓ1,p , ℓ2,p , . . . , ℓrp ,p ∈ Z and r1 , r2 , . . . , rh+1 ∈ N such that r1 + r2 + · · · + rh+1 = r and (9) r1 , r2 , . . . , rh ≤ 3. To see that (e) implies that V (P)µ−rαi = 0 if r > 3h + λ(i), let Ûi be the ± subalgebra of Uq (ĝ) generated by {x± i,k , φi,k }k∈Z , and set V̂i = Ûi .vP . There is ˆ 2 ) → Ûi an obvious homomorphism of algebras (actually an isomorphism) Uqi (sl ± ± ± ± which takes x1,k 7→ xi,k , φ1,k 7→ φi,k , so V̂i may be regarded as a representation of ˆ 2) ˆ 2 ). According to Lemma 2.3 in [9], V̂i ∼ Uq (sl = V (Pi ) as a representation of Uq (sl i i (in particular, V̂i is irreducible). It follows from 4.2 that (V̂i )λ−sαi = 0 if s > λ(i). On the other hand, 2.3 implies that V (P)λ−sαi = (V̂i )λ−sαi for all s ≥ 0. Now, for any vector (8) satisfying the conditions in (e), we have rh+1 ≥ r − 3h > λ(i), so − Xh+1 .vP ∈ V (P)λ−rh+1 αi = 0 . Thus, (e) implies that V (P)λ−rαi = 0 if r > 3h + λ(i). To prove (e), note that it is obvious by 2.3(b) that V (P)µ−rαi is spanned by vectors of the form (8) satisfying all the stated conditions except possibly condition (9). Thus, it suffices to show that any vector v of the form (8) which does not 14 CHARI AND PRESSLEY satisfy (9) can be written as a linear combination of vectors of the same form which do satisfy (9). We prove this by induction on h. If h = 0, there is nothing to prove. Assume that h ≥ 1. By repeated use of relation (4) in 2.2, the product X1− x− i1 ,k1 can be expressed as a linear combination − Ỹ , where Y1− and Ỹ1− are of the same form as X1− , of terms of the form Y1− x− i1 ,k1 1 − but where Y1− is a product of ≤ 3 generators x− i,ℓ , and Ỹ1 is a product of ≥ ℓr1 − 3 such generators. (If g is simply-laced, we can assume that Y1− is a single x− i,ℓ , and − if g is of type B, C or F, that Y1 is a product of ≤ 2 such generators.) So v can be expressed as a linear combination of vectors − − − − Y1− x− i1 ,k1 Ỹ1 X2 xi2 ,k2 . . . Xh+1 .vP . By the induction hypothesis, − Ỹ1− X2− x− i2 ,k2 . . . Xh+1 .vP can be expressed as a linear combination of vectors − − − Y2− x− i2 ,k2 Y3 xi3 ,k3 . . . Yh+1 .vP , − where each of Y2− , Y3− , . . . , Yh− is a product of ≤ 3 x− i,ℓ ’s, and Yh+1 is a product of − ≥ r − 3 − 3(h − 1) = r − 3 xi,ℓ ’s. This completes the inductive step and proves (e). The proof of (a) is now complete. PROOF OF (b). We proceed by induction on h = height(η). If η = 0, there is nothing to prove. If η = αi , we have to show that the vectors x− i,k .vP (k ∈ Z) span a finite-dimensional space. But this space is obviously contained in Ûi .vP , and we have already seen that Ûi .vP is finite-dimensional. Assume now that h ≥ 2, and that (b) has been proved for η’s of height < h. The weight space V (P)λ−η is spanned, in view of 2.3(b), by vectors of the form − − x− i1 ,k1 xi2 ,k2 . . . xih ,kh .vP , (10) where η = αi1 +αi2 +· · ·+αih and k1 , k2 , . . . , kh ∈ Z. It clearly suffices to prove that the vectors (10) span a finite-dimensional space for each fixed choice of i1 , . . . , ih ; denote this space by Vi1 ,... ,ih . By the induction hypothesis, there exists M ∈ N such that, for all i ∈ {i1 , i2 , . . . , ih }, V (P)λ−η+αi is spanned by vectors of the form − − x− j2 ,ℓ2 xj3 ,ℓ3 . . . xjh ,ℓh .vP , (11) where αj2 + αj3 + · · · + αjh = η − αi and |ℓ2 |, |ℓ3 |, . . . , |ℓh | ≤ M . It suffices to prove that Vi1 ,... ,ih is contained in the space (12) W = M+1 X − x− i2 ,k2 .V (P)λ−η+αi2 + xi1 ,0 .V (P)λ−η+αi1 , k2 =−M since W is finite-dimensional by the induction hypothesis. QUANTUM AFFINE ALGEBRAS AND THEIR REPRESENTATIONS 15 For this, we prove, by induction on k1 , that the vector (10) lies in W for every k2 , . . . , kh (we assume that k1 ≥ 0, the proof for k1 ≤ 0 being essentially the same). The case k1 = 0 is obvious. For the inductive step, note that we can assume that |k2 |, |k3 |, . . . , |kh | ≤ M . Using relation (3) in 2.2, any vector (10) can be written as a linear combination of the vectors (13) − − − x− i2 ,k2 xi1 ,k1 xi3 ,k3 . . . xih ,kh .vP , (14) − − − x− i2 ,k2 +1 xi1 ,k1 −1 xi3 ,k3 . . . xih ,kh .vP , (15) − − − x− i1 ,k1 −1 xi2 ,k2 +1 xi3 ,k3 . . . xih ,kh .vP . But, vectors of types (13) and (14) obviously belong to W , and those of type (15) belong to W by the induction hypothesis on k1 . This completes the inductive step. (Note that, by the induction hypothesis again, the vector − − x− i2 ,k2 +1 xi3 ,k3 . . . xih ,kh .vP can be written as a linear combination of vectors − − x− i′ ,k′ xi′ ,k′ . . . xi′ ,k′ .vP , 2 2 3 3 h h where αi′2 + · · · + αi′h = αi2 + · · · + αih and |k2′ |, . . . , |kh′ | ≤ M .) This completes the proof of (b), and hence that of Theorem 3.3. 6. Minimal affinizations We saw at the beginning of Section 5 that, if P = (Pi )i∈I ∈ P, and λ ∈ P + is defined by λ(i) = deg(Pi ), then (16) V (P) = M V (P)λ−η and dim(V (P)λ ) = 1. η∈Q+ Since V (P) is finite-dimensional, it is completely reducible as a representations of Uq (g), and in view of (16) we have V (P) ∼ = V (λ) ⊕ M V (µ)⊕mµ µ∈P + as a representation of Uq (g), where the multiplicities mµ ∈ N are zero unless µ < λ. Thus, V (P) gives a way of extending the action of Uq (g) on V (λ) to an action of Uq (ĝ), at the expense of enlarging V (λ) by the addition of representations of Uq (g) of smaller highest weight. For this reason, we call V (P) an affinization of V (λ). We say that two affinizations are equivalent if and only if they are isomorphic as representations of Uq (g), and we denote by [V (P)] the equivalence class of V (P). There is one situation in which affinizations are unique, up to equivalence: 16 CHARI AND PRESSLEY Proposition 6.1. For any i ∈ I, V (λi ) has a unique affinization, up to equivalence. Proof. If V (P) is an affinization of V (λi ), then Pj = 1 if j 6= i and Pi (u) = 1−a−1 u, for some a ∈ C× (i.e. V (P) is a fundamental representation of Uq (ĝ)). Denoting this V (P) by V (λi , a), we have to prove that the equivalence class [V (λi , a)] is independent of a. We make use of the family of (Hopf) algebra automorphisms τt (t ∈ C× ) of Uq (ĝ) defined by ± k ± k ± 1/2 τt (x± ) = c1/2 . i,k ) = t xi,k , τt (φi,k ) = t φi,k , τt (c It is easy to see that, for any Q = (Qi )i∈I ∈ P, the pull-back τt∗ (V (Q)) of V (Q) by τt is isomorphic as a representation of Uq (ĝ) to V (Qt ), where Qt = (Qti ) and Qti (u) = Qi (tu). In particular, τa∗ (V (λi , a)) ∼ = V (λi , 1). Since τa is the identity of Uq (g), it follows that [V (λi , a)] = [V (λi , 1)]. Corollary 6.2. For any λ ∈ P + , V (λ) has, up to equivalence, only finitely many affinizations. Proof. By 3.6, any affinization V (P) of V (λ) is isomorphic as a representation of Uq (ĝ) to a subquotient of a tensor product λ(i) OO V (λi , bj,i ), i∈I j=1 for some bj,i ∈ C× (the order of the factors is unimportant). By 6.1, this tensor product is, up to Uq (g)-isomorphism, independent of the bj,i . It therefore has only finitely many subquotients, regarded as a representation of Uq (g). In general, a representation V (λ) of Uq (g) has many inequivalent affinizations, and it is natural to ask if one can make a canonical choice among them. To this end, the following partial order on the set of affinizations was introduced in [4]. Proposition 6.3. Let λ ∈ P + and let V (P) and V (Q) be affinizations of V (λ). Then, we write [V (P)]  [V (Q)] iff, for all µ ∈ P + , either (i) mµ (V (P)) ≤ mµ (V (Q)), or (ii) there exists ν > µ with mν (V (P)) < mν (V (Q)). Then,  is a partial order on the set of equivalence classes of affinizations of V (λ). An affinization V (P) of V (λ) is minimal if, whenever V (Q) is an affinization of V (λ) and [V (Q)]  [V (P)], we have [V (P)] = [V (Q)]. In view of 6.2, minimal affinizations certainly exist. ˆ 2) → If g = sl2 , we explained in Section 4 that the homomorphisms eva : Uq (sl Uq (sl2 ) enable one to extend the action of Uq (sl2 ) on any representation V (λ) to QUANTUM AFFINE ALGEBRAS AND THEIR REPRESENTATIONS 17 ˆ 2 ) on the same space. These evaluation representations obviously an action of Uq (sl provide the unique minimal affinization. We mentioned in Section 4 that there are analogues of the eva when g = sln for any n ≥ 2, so the minimal affinizations are also unique, and irreducible under Uq (g), in that case. The following result, proved in [7], gives the defining polynomials of the minimal affinizations in the type A case. Theorem 6.4. Let g = sln+1 (C) and let λ ∈ P + . Number the nodes of the Dynkin diagram of g as in [3]. Then, V (λ) has, up to equivalence, a unique minimal affinization. It is represented by V (P), where P = (Pi )i∈I ∈ P λ , iff, for all i ∈ I such that λ(i) > 0, the roots of Pi form a q-segment with centre ai ∈ C× (say) and length λ(i), where (i) either, for all i < j such that λ(i) > 0 and λ(j) > 0, ai = q λ(i)+2(λ(i+1)+···+λ(j−1))+λ(j)+j−i , aj (ii) or, for all i < j such that λ(i) > 0 and λ(j) > 0, aj = q λ(i)+2(λ(i+1)+···+λ(j−1))+λ(j)+j−i . ai To state the corresponding results when g is of type B, C or F, number the nodes of the Dynkin diagram as in [3], and define, for any λ ∈ P + , complex numbers ci (λ) as follows:  di (λ(i)+λ(i+1)+1) q if ai+1 i ai i+1 = 1, ci (λ) = di λ(i)+di+1 λ(i+1)+2di+1 −1 q if ai+1 i ai i+1 = 1. Theorem 6.5. Let g be non-simply-laced, and let λ ∈ P + . Then, V (P) is a minimal affinization of V (λ) iff P ∈ P λ satisfies the following conditions: (i) For all i ∈ I, either Pi = 1 or the roots of Pi form a qi -segment of length λ(i) and centre ai (say). (ii) Either, for all i < j such that λ(i) > 0 and λ(j) > 0, we have j−1 Y ai cs , = aj s=i or, for all i < j such that λ(i) > 0 and λ(j) > 0, we have j−1 Y ai 2dj −2di c−1 =q s . aj s=i The minimal affinization of V (λ) is unique, up to equivalence. See [10] for the proof. Note that, for any r, I\Ir defines a type A subdiagram, so 6.4 gives the precise conditions under which V (PI\Ir ) is a minimal affinization. 18 CHARI AND PRESSLEY Turning finally to the D and E cases, we introduce the following notation. If ∅ 6= J ⊆ I, and λ ∈ P + , let λJ be the restriction of λ : I → Z to J. Also, if P = (Pi )i∈I ∈ P, let PJ be the J-tuple (Pj )j∈J . Let i0 be the unique node of the Dynkin diagram of g which is linked to three other nodes. Then, I\{i0 } = I1 ∐ I2 ∐ I3 , where I1 , I2 and I3 define type A subdiagrams. Theorem 6.6. Let g be of type D or E, let λ ∈ P + , and assume that λ(i0 ) 6= 0. If λIr = 0 for some r ∈ {1, 2, 3}, then V (λ) has a unique minimal affinization, up to equivalence. It is represented by V (P) iff V (PI\Ir ) is a minimal affinization of V (λI\Ir ). If λIr 6= 0 for all r ∈ {1, 2, 3}, then V (λ) has exactly three minimal affinizations, up to equivalence. In fact, V (P) is a minimal affinization of V (λ) iff there exist r 6= s in {1, 2, 3} such that V (PI\Ir ) and V (PI\Is ) are minimal affinizations of V (λI\Ir ) and V (λI\Is ), respectively. See [9] for the proof. Remark. The result of this theorem no longer holds if we drop the assumption λ(i0 ) > 0. If g is of type D4 , for example, and λ(i0 ) = 0, the number of minimal affinizations of V (λ) increases with λ (roughly speaking), and is generally greater than three. To conclude our discussion of minimal affinizations, we consider their structure as representations of Uq (g). Except when g is of type A, when the minimal affinizations are irreducible under Uq (g), this is not well understood. We give two results. Theorem 6.7. Let g be of type B2 , let θ be the highest root of g, and assume that α2 is the short simple root. Let λ ∈ P + and let V (P) be a minimal affinization of V (λ). Then, as representations of Uq (g), V (P) ∼ = [ 12 λ(2)] M V (λ − rθ). r=0 See [5] for the proof. Our final result gives the Uq (g)-structure of most of the fundamental representations of Uq (ĝ). Theorem 6.8. Number the nodes of the Dynkin diagram of g as in [3]. (a) V (λi , 1) ∼ = V (λi ) under any of the following conditions: (i) g is of type A or C and i is arbitrary; (ii) g is of type Bn (n ≥ 2) and i = 1 or n; (iii) g is of type Dn (n ≥ 4) and i = 1, n − 1 or n. QUANTUM AFFINE ALGEBRAS AND THEIR REPRESENTATIONS 19 (b) If g is of type Bn or Dn+1 (n ≥ 3) and 1 < i < n, [i/2] V (λi , 1) ∼ = M V (λi−2j ). j=0 (c) If g is of type E6 , V (λ1 , 1) ∼ = V (λ1 ), V (λ2 , 1) ∼ = V (λ2 ) ⊕ C, ∼ V (λ3 , 1) = V (λ3 ) ⊕ V (λ6 ), V (λ4 , 1) ∼ = V (λ4 ) ⊕ V (λ1 + λ6 ) ⊕ V (λ2 ) ⊕ V (λ2 ) ⊕ C, V (λ5 , 1) ∼ = V (λ5 ) ⊕ V (λ1 ), V (λ6 , 1) ∼ = V (λ6 ). (Here and below, C denotes the 1-dimensional trivial representation.) (d) If g is of type E7 , V (λ1 , 1) ∼ = V (λ1 ) ⊕ C, V (λ2 , 1) ∼ = V (λ2 ) ⊕ V (λ7 ), ∼ V (λ3 , 1) = V (λ3 ) ⊕ V (λ6 ) ⊕ V (λ1 ) ⊕ V (λ1 ) ⊕ C, V (λ6 , 1) ∼ = V (λ7 ). = V (λ6 ) ⊕ V (λ1 ) ⊕ C, V (λ7 , 1) ∼ (e) If g is of type E8 , V (λ1 , 1) ∼ = V (λ1 ) ⊕ V (λ8 ) ⊕ C, ∼ V (λ7 , 1) = V (λ7 ) ⊕ V (λ1 ) ⊕ V (λ8 ) ⊕ V (λ8 ) ⊕ C, V (λ8 , 1) ∼ = V (λ8 ) ⊕ C. (f ) If g is of type F4 , V (λ1 , 1) ∼ = V (λ1 ) ⊕ C, ∼ V (λ2 , 1) = V (λ2 ) ⊕ V (2λ4 ) ⊕ V (λ1 ) ⊕ V (λ1 ) ⊕ C, V (λ3 , 1) ∼ = V (λ3 ) ⊕ V (λ1 ), V (λ4 , 1) ∼ = V (λ4 ). (g) If g is of type G2 , V (λ1 , 1) ∼ = V (λ2 ). = V (λ1 ) ⊕ C, V (λ2 , 1) ∼ This can be proved using the techniques of [5]. References 1. 2. 3. 4. R. J. Baxter, Exactly solved models in statistical mechanics, Academic Press, New York, 1982. J. Beck, Braid group action and quantum affine algebras, preprint, MIT (1993). N. Bourbaki, Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Hermann, Paris, 1968. V. Chari, Minimal affinizations of representations of quantum groups: the rank 2 case, preprint (1994). , Minimal affinizations of representations of quantum groups: Uq (g)-structure, preprint 5. (1994). 6. V. Chari and A. N. Pressley, Quantum affine algebras, Commun. Math. Phys. 142 (1991), 261–283. , Small representations of quantum affine algebras, Lett. Math. Phys. 30 (1994), 131– 7. 145. 20 8. 9. 10. 11. 12. 13. 14. CHARI AND PRESSLEY , A Guide to Quantum Groups, Cambridge University Press, Cambridge, 1994. , Minimal affinizations of representations of quantum groups: the simply-laced case, preprint (1994). , Minimal affinizations of representations of quantum groups: the non-simply-laced case, preprint (1994). V. G. Drinfel’d, Quantum groups, Proceedings of the International Congress of Mathematicians, Berkeley, 1986, American Mathematical Society, 1987, pp. 798–820. , A new realization of Yangians and quantized affine algebras, Soviet Math. Dokl. 36 (1988), 212–216. M. Jimbo, A q-analog of U (gl(N + 1)), Hecke algebra and the Yang–Baxter equation, Lett. Math. Phys. 11 (1986), 247–252. G. Lusztig, Introduction to Quantum Groups, Birkhäuser, Boston, 1993. Department of Mathematics, University of California, Riverside, CA 92521. E-mail address: [email protected] Department of Mathematics, King’s College, London, WC2R 2LS. E-mail address: [email protected]