Block decomposition for quantum affine algebras by the associated simply-laced root system

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-Leclerc. In this paper, we give the block decompositions of \(\mathscr{C}_\mathfrak{g}\) and \(\mathscr{C}_\mathfrak{g}^0\) for all untwisted and twisted quantum affine algebras by using the associated simply-laced finite type root system. We first define a certain abelian group \(\mathcal{W}\) (resp. \(\mathcal{W}_0\)) arising from simple modules of \( \mathscr{C}_\mathfrak{g}\) (resp. \(\mathscr{C}_\mathfrak{g}^0\)) by using the invariant \(\Lambda^\infty\) introduced in the previous work by the authors. The groups \(\mathcal{W}\) and \(\mathcal{W}_0\) have the 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 \(( \mathbb{R} \otimes_\mathbb{Z} \mathcal{W}_0, \Delta_0)\) is an irreducible simply-laced root system of finite type and the pair \(( \mathbb{R} \otimes_\mathbb{Z} \mathcal{W}, \Delta) \) is isomorphic to the direct sum of infinite copies of \(( \mathbb{R} \otimes_\mathbb{Z} \mathcal{W}_0, \Delta_0)\) as a root system. We next show that there exist direct decompositions of \(\mathscr{C}_\mathfrak{g}\) and \(\mathscr{C}_\mathfrak{g}^0\) parameterized by elements of \(\mathcal{W}\) and \(\mathcal{W}_0\) respectively, and prove that these decompositions are their block decompositions.