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

Abstract

If $U_q(\mathfrak{g})$ is a finite-dimensional complex simple Lie algebra, an affinization of a finite-dimensional irreducible representation $V$ of $U_q(\mathfrak{g})$ is a finite-dimensional irreducible representation $\hat{V}$ of $U_q(\mathfrak{g})$ which contains $\hat{V}$ with multiplicity one, and is such that all other $U_q(\mathfrak{g})$-types in $\hat{V}$ have highest weights strictly smaller than that of $V$. We define a natural partial ordering $\preccurlyeq$ on the set of affinizations of $V$. If $\mathfrak{g}$ is of rank $2$, we show that there is a unique minimal element with respect to this order and give its $U_q(\mathfrak{g})$-module structure when $\mathfrak{g}$ is of type $A_2$ or $C_2$.