PBW theory for quantum affine algebras

Abstract

Let 𝘜'q(𝖌) be a quantum affine algebra of arbitrary type and let 𝒞𝖌⁰ be Hernandez-Leclerc’s category. We can associate the quantum affine Schur–Weyl duality functor 𝓕𝓓 to a duality datum 𝓓 in 𝒞𝖌⁰. In this paper, we introduce the notion of a strong (complete) duality datum 𝓓 and prove that, when 𝓓 is strong, the induced duality functor 𝓕𝓓 sends simple modules to simple modules and preserves the invariants Λ, Λ˜ and Λ∞ introduced by the authors. We next define the reflections 𝒮ₖ and 𝒮ₖ⁻¹ acting on strong duality data 𝓓. We prove that if 𝓓 is a strong (resp. complete) duality datum, then 𝒮ₖ(𝓓) and 𝒮ₖ⁻¹(𝓓) are also strong (resp. complete) duality data. This allows us to make new strong (resp. complete) duality data by applying the reflections 𝒮ₖ and 𝒮ₖ⁻¹ from known strong (resp. complete) duality data. We finally introduce the notion of affine cuspidal modules in 𝒞𝖌⁰ by using the duality functor 𝓕𝓓, and develop the cuspidal module theory for quantum affine algebras similar to the quiver Hecke algebra case. When 𝓓 is complete, we show that all simple modules in 𝒞𝖌⁰ can be constructed as the heads of ordered tensor products of affine cuspidal modules. We further prove that the ordered tensor products of affine cuspidal modules have the unitriangularity property. This generalizes the classical simple module construction using ordered tensor products of fundamental modules.