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

### Vyjayanthi Chari

University of California, Riverside, USA

## 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.