Weyl Modules for Classical and Quantum Affine Algebras

Abstract

The study of the irreducible finite–dimensional representations of quantum affine algebras has been the subject of a number of papers, [AK], [CP3], [CP5], [FR], [FM], [GV], [KS] to name a few. However, the structure of these representations is still unknown except in certain special cases. In this paper, we approach the problem by studying the classical (q → 1) limits of these representations. Standard results imply, for example, that if V is a finite–dimensional representation of Uq(ĝ), its q → 1 limit V has the same structure as a g–module as V has as a Uq(g)–module. We begin by studying an appropriate class of representations of the affine Lie algebra ĝ. The finite–dimensional irreducible representations of ĝ were classified in [C], [CP1], where it was shown that such representations are highest weight in a suitable sense, the highest weight being an n–tuple of polynomials π, where n is the rank of g. We therefore study the class of all highest weight finite–dimensional representations of ĝ. In fact, we prove that corresponding to each irreducible finite– dimensional representation V (π) there exists a unique (up to isomorphism) finite– dimensional highest weight module W (π), such that any finite dimensional highest weight module V with highest weight π is a quotient of W (π). We call these modules the Weyl modules because of an analogy with the modular representation theory of g, which we now explain. In [CP5], we showed that the irreducible representations of Uq(ĝ) are also highest weight and that their isomorphism classes are parametrized by a n–tuples of polynomials πq with coefficients in C(q). Under a natural condition on πq, the corresponding representation Vq(πq) of Uq(ĝ) specializes as q → 1 to a representation Vq(πq) of ĝ and is a quotient of W (π), where π is obtained from πq by setting q = 1. We conjecture that every W (π) is the classical limit of an irreducible Uq(ĝ)module. This is analogous to the fact that the Weyl modules for g in characteristic p are the mod p reductions of the irreducible modules in characteristic zero. We prove the conjecture in the case of g = sl2 in this paper. The conjecture is also true for the fundamental representations of Uq(ĝ), but the details of that will appear elsewhere. In Section 3, we prove a factorization property of Weyl modules analogous the one for the irreducible modules proved in [CP1]. In Section 5We obtain a necessary and sufficient condition for the Weyl modules to be irreducible: the interesting feature of this proof is that it uses the fact that the specialized irreducible modules for the quantum algebra are quotients of the Weyl module. Further, the condition for irreducibility of the Weyl modules is the same as a condition that appears first in the work of Drinfeld on the closely related Yangians, [Dr1].

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