PBW theory for quantum affine algebras

Abstract

Let Uq́.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 ƒ, ƒz and ƒ1 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.