Elliptic Central Characters and Blocks of Finite Dimensional Representations of Quantum Affine Algebras

Abstract

The category of finite dimensional (type 1) representations of a quantum affine algebra $U_q(\hat{{\mathfrak g}})$ is not semisimple. However, as any abelian category with finite-length objects, it admits a unique decomposition into a direct sum of indecomposable subcategorie (blocks). We define the elliptic central character of a finite dimensional (type 1) representation of $U_q(\hat{{\mathfrak g}})$. Then we show that the block decomposition of this category is paramentrized by these elliptic central characters.