Minimal affinizations of representations of quantum groups: the rank 2 case

If Uq (g) is a finite-dimensional complex simple Lie algebra, an affinization of a finite-dimensional irreducible representation V of Uq (g) is a finite-dimensional irreducible representation V of Uq (g) which contains V with multiplicity one, and is such that all other Uq (g) -types in V have highest weights strictly smaller than that of V. We define a natural partial ordering ^ on the set of affinizations of V. If g is of rank 2, we show that there is a unique minimal element with respect to this order and give its Uq (g) -module structure when g is of type AT. or €2Introduction If g is a finite-dimensional complex simple Lie algebra, the associated 'untwisted' affine Lie algebra § is a central extension, with one-dimensional centre, of the space of Laurent polynomial maps C —* g (on which a Lie bracket is defined using pointwise operations). Since the cocycle of the extension vanishes on the constant maps, we can regard g as a subalgebra of g. If V is any representation of g, it is easy to extend the action of g on V to an action of g on the same space. If a^C , evaluation at a gives a homomorphism eva'. Q~^ Q (under which the centre maps to zero) which is the identity on g, so pulling back V by eva gives the desired extension. It follows from the results of [2] that, if V is finite-dimensional and irreducible, these are, up to isomorphism, the only possible extensions. Quantum deformations Uq (g) and Uq (g) of the universal enveloping algebras of g and g were introduced in 1985 by V. G. Drinfel'd and M. Jimbo. These Communicated by T. Miwa, January 18, 1995. 1991 Mathematics Subject Classification (s): 17B. * Dept. of Math., Univ. of California, Riverside, CA92521, U.S.A. Partially supported by the NSF, DMS-9207701. 874 VYJAYANTHI CHARI algebras depend on a parameter #^C X ; we assume throughout this paper that q is transcendental. It is well-known (see [5] or [10] , for example) that, up to twisting by certain simple automorphisms, there is a natural one-to-one correspondence between the finite-dimensional representations of Uq (g) and those of g. Corresponding representations have the same character, and hence the same dimension. However, the structure of the finite-dimensional representations of Uq (g) is not well-understood. A parametrization of these representations in the spirit of Cartan's highest weight classification of the finite-dimensional irreducible representations of g is proved in the case Q = sl2 in [3], and in [6] in general. As in the classical situation, we may regard Uq (g) as a subalgebra of Uq (§) . If g is of type sln, the action of Uq (g) on any representation V extends to a representation of Uq (g) . However, if g is not of type sln, it is not usually possible to extend the action of Uq (g) on an irreducible finite-dimensional represention V to an action of Uq (g) on V. Thus, it is natural to ask how V can be 'enlarged' so as to obtain a representation of Uq (g) . To make this question precise, we define in this paper a natural partial ordering on the set of isomorphism classes of representations of Uq (g). By an affinization of a finitedimensional irreducible representation V of Uq (g) , we mean an irreducible representation V of Uq (g) which contains V as a Uq (g) -subrepresentation with multiplicity one, and such that all other irreducible Uq (g) -subrepresentations of V are strictly smaller than V. (There is a clear analogy with the classical Harish Chandra theory of (g, K) -modules here. ) We prove that any given representation V has only finitely many affinizations (at least one) up to Uq (g) -isomorphism, and one may ask if any of them is 'canonical'. A reasonable interpretation of this question is to look for the minimal affinization (s) of V, with respect to our partial order. If Q = sln, we show in [4] that every finite-dimensional irreducible representation of £/«(g) has, up to Uq(o) -isomorphism, a unique minimal affinization. In this paper, we prove that, if g is of type C2 or G^ there is again a unique minimal affinization, and we describe it precisely in terms of the highest weight classification of representations of Uq (g) mentioned above. In contrast to the sln case, the minimal affinization in these cases is not, in general, irreducible as a representation of Uq (g) . In fact, in the C% case we describe the structure of all minimal affinizations as representations of C/«(g); a consequence of this result is that the minimal affinization of V is irreducible under Uq (g) if and only if the value of its highest weight on the short simple root of g is 0 or 1. Subsequent papers will deal with the case when g has rank greater that 2. The problem of constructing aff inizations of representations of Uq (g) is important in several areas of mathematics and physics, as has been emphasized by I. B. Frenkel and N. Yu. Reshetikhin, among others (see Remark 4,2 in [9]). MINIMAL AFFINIZATIONS 875 As one example, recall that, to any finite-dimensional irreducible representation V of Uq(o) one can associate an R-matrix, i.e. an element R^End (V®V) which satisfies the 'quantum Yang-Baxter equation' (QYBE). There are many situations, however, in which it is important to have a solution of the 'QYBE with spectral parameters'. This is so, for example, in the theory of lattice models in statistical mechanics, for only when the R-matrix constructed from the Boltzmann weights of the model satisfies the QYBE with spectral parameters can one prove the existence of commuting transfer matrices and deduce the integrability of the model. (See [5], for example, for an introduction to these ideas.) Thus, it is natural to ask when R can be 'embedded' in a parameter-dependent R-matrix R(u) ^End (V® V). A sufficient condition for this is that the action of UQ(Q) on V extends to an action of UQ(Q) on V, for then V itself can be embedded in a 1-parameter family of representations of Uq(o) by twisting with a certain 1-parameter family of automorphisms of Uq(o) (which correspond, in the classical case, to 'rescaling' the C parameter in g). A second example concerns the affine Toda field theory associated to g. This admits Uq (g*) as a 'quantum symmetry group', where g* is the dual affine Lie algebra (whose generalized Cartan matrix is the transpose of that of g). It is well known that the classical solitons of this theory correspond essentially to the finite-dimensional irreducible representations of g. The solitons (or particle states) of the quantum theory should therefore correspond to the finitedimensional irreducible representations of Uq (g *). Since not all representations of Uq(o) are affinizable on the same space, the quantum solitons come in 'multiplets', and there are generally 'more' quantum solitons than classical ones. §1. Quantum Affine Algebras Let g be a finite-dimensional complex simple Lie algebra with Cartan subalgebra t) and Cartan matrix A = (an) ,-fj6j. Fix coprime positive integers (di) ,e/ such that (diatj) is symmetric. Let R be the set of roots and R a set of positive roots. The roots can be regarded as functions /— >Z; in particular, the simple roots at^R are given by Let 0= © 0 for all i^I\ 876 VYJAYANTHI CHARI be the set of dominant weights. Define a partial order > on P by iff XLet 6 be the unique highest root with respect to ^. Define a non-degenerate symmetric bilinear form ( , ) on t)* by (at, aj) =diaij , and denote by ( , ) also the induced form on t). Set d0 — y (0, 6), a0o — 2, and, for all i^I, 2(0, at) 2(0, a,-) 01 (0, 0) ' *° (a,-, a,-) Let /=/ II )0[ and A = (a//) ue/. Then, A is the generalized Cartan matrix of the untwisted affine Lie algebra g associated to g. From Section 5 onwards, we shall be interested in the case when g is of type C2. Then, /=|1,2| , d0 2 -I "-'. -* 2 the rows of A being numbered 0, 1, 2. Let q^C* be transcendental, and, for r, n&N, n>r, define