2020/03/06 by Kashiwara, Masaki, Kim, Myungho, Oh, Se-jin +1
#FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2003.03265
Let Uq'(\mathfrakg) be a quantum affine algebra with an indeterminate q and let \mathscrC_\mathfrakg be the category of finite-dimensional integrable Uq'(\mathfrakg)-modules. We write \mathscrC_\mathfrakg0 for the monoidal subcategory of \mathscrC_\mathfrakg introduced by Hernandez-Leclerc. In this paper, we associate a simply-laced finite type root system to each quantum affine algebra Uq'(\mathfrakg) in a natural way, and show that the block decompositions of \mathscrC_\mathfrakg and \mathscrC_\mathfrakg0 are parameterized by the lattices associated with the root system. We first define a certain abelian group W (resp. W0) arising from simple modules of \mathscrC_\mathfrakg (resp. \mathscrC_\mathfrakg0) by using the invariant Λ^∞ introduced in the previous work by the authors. The groups W and W0 have the subsets Δ and Δ0 determined by the fundamental representations in \mathscrC_\mathfrakg and \mathscrC_\mathfrakg0 respectively. We prove that the pair ( ℝ ⊗_ℤ W0, Δ0) is an irreducible simply-laced root system of finite type and the pair ( ℝ ⊗_ℤ W, Δ) is isomorphic to the direct sum of infinite copies of ( ℝ ⊗_ℤ W0, Δ0) as a root system.