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세미나

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AUTHOR Kashiwara, Masaki
TITLE Simply laced root systems arising from quantum affine algebras
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JOURNAL COMPOSITIO MATHEMATICA, 2022
ABSTRACT Let U-q(') (g) be a quantum affine algebra with an indeterminate q, and let C-g be the category of finite-dimensional integrable U-q' (g)-modules. We write C-g(0) for the monoidal subcategory of C-g introduced by Hernandez and Leclerc. In this paper, we associate a simply laced finite-type root system to each quantum affine algebra U-g'(') (g) in a natural way and show that the block decompositions of C-g and C-g(0) are parameterized by the lattices associated with the root system. We first define a certain abelian group W (respectively W-0) arising from simple modules of C-g (respectively C-g(0)) by using the invariant Lambda(infinity) introduced in previous work by the authors. The groups W and W-0 have subsets Delta and Delta(0) determined by the fundamental representations in C-g and C-g(0), respectively. We prove that the pair (R circle times(Z) W-0, Delta(0)) is an irreducible simply laced root system of finite type and that the pair (R circle times(Z) W, Delta) is isomorphic to the direct sum of infinite copies of (R circle times(Z) W-0, Delta(0)) as a root system.
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