Journal Article

Simply laced root systems arising from quantum affine algebras

Oh, Se-jin, Kim, Myungho, Park, Euiyong, Kashiwara, Masaki

2022

Overview

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.

• Publisher: London Mathematical Society,Cambridge University Press
• Subject: Algebra / Decomposition / Group theory / Lattices (mathematics) / Modules