Minimal Affinizations of Representations of Quantum Groups: the Rank 2 Case

Abstract

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 _A_2 or _C_2.