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 ( g ^ ) U_q(\widehat {{\mathfrak g}}) is not semisimple. However, as any abelian category with finite-length objects, it admits a unique decomposition in a direct sum of indecomposable subcategories (blocks). We define the elliptic central character of a finite dimensional (type 1) representation of U q ( g ^ ) U_q(\widehat {{\mathfrak g}}) and show that the block decomposition of this category is parametrized by these elliptic central characters.