Simply laced root systems arising from quantum affine algebras

Abstract

Let$U_q'({\mathfrak {g}})$be a quantum affine algebra with an indeterminate$q$, and let$\mathscr {C}_{\mathfrak {g}}$be the category of finite-dimensional integrable$U_q'({\mathfrak {g}})$-modules. We write$\mathscr {C}_{\mathfrak {g}}^0$for the monoidal subcategory of$\mathscr {C}_{\mathfrak {g}}$introduced by Hernandez and Leclerc. In this paper, we associate a simply laced finite-type root system to each quantum affine algebra$U_q'({\mathfrak {g}})$in a natural way and show that the block decompositions of$\mathscr {C}_{\mathfrak {g}}$and$\mathscr {C}_{\mathfrak {g}}^0$are parameterized by the lattices associated with the root system. We first define a certain abelian group$\mathcal {W}$(respectively$\mathcal {W} _0$) arising from simple modules of$\mathscr {C}_{\mathfrak {g}}$(respectively$\mathscr {C}_{\mathfrak {g}}^0$) by using the invariant$\Lambda ^\infty$introduced in previous work by the authors. The groups$\mathcal {W}$and$\mathcal {W} _0$have subsets$\Delta$and$\Delta _0$determined by the fundamental representations in$\mathscr {C}_{\mathfrak {g}}$and$\mathscr {C}_{\mathfrak {g}}^0$, respectively. We prove that the pair$( \mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} _0, \Delta _0)$is an irreducible simply laced root system of finite type and that the pair$( \mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} , \Delta )$is isomorphic to the direct sum of infinite copies of$( \mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} _0, \Delta _0)$as a root system.