Integrable and Weyl Modules for Quantum Affine sl₂

Abstract

Let t be an arbitrary symmetrizable Kac-Moody Lie algebra and U q (t) the corresponding quantized enveloping algebra of t defined over C(q). If µ is a dominant integral weight of t then one can associate to it in a natural way an irreducible integrable U q (t)-module L(µ). These modules have many nice properties and are well understood,More generally, given any integral weight λ, Kashiwara [K] defined an integrable U q (t)-module V max (λ) generated by an extremal vector v λ . If w is any element of the Weyl group W of t, then one has, where w 0 ∈ W is such that w 0 λ is dominant integral. In the case when λ is not in the Tits cone, the module V max (λ) is not irreducible and very little is known about it, although it is known that it admits a crystal basis, [K].In the case when t is an affine Lie algebra, an integral weight λ is not in the Tits cone if and only if λ has level zero. Choose w 0 ∈ W so that w 0 λ is dominant with respect to the underlying finite-dimensional simple Lie algebra of t. In as yet unpublished work, Kashiwara proves that V max (λ) ∼ = W q (w 0 λ), where W q (w 0 λ) is an integrable U q (t)-module defined by generators and relations analogous to the definition of L(µ).In [CP4], we studied the modules W q (λ) further. In particular, we showed that they have a family W q (π) of non-isomorphic finite-dimensional quotients which are maximal, in the sense that any another finite-dimensional quotient is a proper quotient of some W q (π). In this paper, we show that, if t is the affine Lie algebra associated to sl 2 and λ = m ∈ Z + , the modules W q (π) all have the same dimension 2 m . This is done by showing that the modules W q (π), under suitable conditions, have a q = 1 limit, which allows us to reduce to the study of the corresponding problem in the classical case carried out in [CP4]. The modules W q (π) have a unique irreducible quotient V q (π), and we show that these are all the irreducible finitedimensional U q (t)-modules. In [CP1], [CP2], a similar classification was obtained by regarding q as a complex number and U q (t) as an algebra over C; in the present situation, we have to allow modules defined over finite extensions of C(q).We are then able to realize the modules W q (m) as being the space of invariants of the action of the Hecke algebra H m on the tensor product (V ⊗ C(q)[t, t −1 ]) ⊗m , where V is a two-dimensional vector space over C(q). Again, this is done by reducing to the case of q = 1.In the last section, we indicate how to extend some of the results of this paper to the general case. We conjecture that the dimension of the modules W q (π) depends only on λ, and we give a formula for this dimension.

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“…A convenient procedure for introducing the latter is to take Drinfeld's double (see [45,46] and B.1) of the algebra H generated by the screenings.…”

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“…(5.14). In other words, the twisted Leibniz rule emerges from the application of the standard one to Φ 1 ⋆ Φ 2 and taking into account the transformation of the star-product 15) i.e., in transforming a star-product of operators we have consider the star-product as a differential operator with its own transformation properties. Notice that the transformation of the star-product depends on the representation of the fields we multiply.…”

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