PBW theory for quantum affine algebras

  • Masaki Kashiwara

    Kyoto University, Japan; Korea Institute for Advanced Study, Seoul, Korea
  • Myungho Kim

    Kyung Hee University, Seoul, Korea
  • Se-jin Oh

    Ewha Womans University, Seoul, Republic of Korea
  • Euiyong Park

    University of Seoul, Korea
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Let be a quantum affine algebra of arbitrary type and let be Hernandez-Leclerc’s category. We can associate the quantum affine Schur–Weyl duality functor to a duality datum in . In this paper, we introduce the notion of a strong (complete) duality datum and prove that, when is strong, the induced duality functor sends simple modules to simple modules and preserves the invariants , and introduced by the authors. We next define the reflections and acting on strong duality data . We prove that if is a strong (resp.\ complete) duality datum, then and are also strong (resp. complete) duality data. This allows us to make new strong (resp. complete) duality data by applying the reflections and from known strong (resp. complete) duality data. We finally introduce the notion of affine cuspidal modules in by using the duality functor , and develop the cuspidal module theory for quantum affine algebras similar to the quiver Hecke algebra case. When is complete, we show that all simple modules in 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.

Cite this article

Masaki Kashiwara, Myungho Kim, Se-jin Oh, Euiyong Park, PBW theory for quantum affine algebras. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1323