Simply laced root systems arising from quantum affine algebras

Abstract

Let be a quantum affine algebra with an indeterminate, and let be the category of finite-dimensional integrable -modules. We write for the monoidal subcategory of introduced by Hernandez and Leclerc. In this paper, we associate a simply laced finite-type root system to each quantum affine algebra in a natural way and show that the block decompositions of and are parameterized by the lattices associated with the root system. We first define a certain abelian group (respectively) arising from simple modules of (respectively) by using the invariant introduced in previous work by the authors. The groups and have subsets and determined by the fundamental representations in and, respectively. We prove that the pair is an irreducible simply laced root system of finite type and that the pair is isomorphic to the direct sum of infinite copies of as a root system. ©