Minimal affinizations of representations of quantum groups: the irregular case

Abstract

Let\(\mathfrak{g}\) be a finite-dimensional complex simple Lie algebra and Uq(\(\mathfrak{g}\)) the associated quantum group (q is a nonzero complex number which we assume is transcendental). IfV is a finitedimensional irreducible representation of Uq(\(\mathfrak{g}\)), an affinization ofV is an irreducible representationVV of the quantum affine algebra Uq(\(\hat {\mathfrak{g}}\)) which containsV with multiplicity one and is such that all other irreducible Uq(\(\mathfrak{g}\))-components ofV have highest weight strictly smaller than the highest weight λ ofV. There is a natural partial order on the set of Uq(\(\mathfrak{g}\)) classes of affinizations, and we look for the minimal one(s). In earlier papers, we showed that (i) if\(\mathfrak{g}\) is of typeA, B, C, F orG, the minimal affinization is unique up to Uq(\(\mathfrak{g}\))-isomorphism; (ii) if\(\mathfrak{g}\) is of typeD orE and λ is not orthogonal to the triple node of the Dynkin diagram of\(\mathfrak{g}\), there are either one or three minimal affinizations (depending on λ). In this paper, we show, in contrast to the regular case, that if Uq(\(\mathfrak{g}\)) is of typeD4 and λ is orthogonal to the triple node, the number of minimal affinizations has no upper bound independent of λ.