Affine highest weight categories and quantum affine Schur-Weyl duality of Dynkin quiver types
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Abstract:
For a Dynkin quiver $Q$ (of type $\mathrm {ADE}$), we consider a central completion of the convolution algebra of the equivariant $K$-group of a certain Steinberg type graded quiver variety. We observe that it is affine quasi-hereditary and prove that its category of finite-dimensional modules is identified with a block of Hernandez-Leclerc’s monoidal category $\mathcal {C}_{Q}$ of modules over the quantum loop algebra $U_{q}(L\mathfrak {g})$ via Nakajima’s homomorphism. As an application, we show that Kang-Kashiwara-Kim’s generalized quantum affine Schur-Weyl duality functor gives an equivalence between the category of finite-dimensional modules over the quiver Hecke algebra associated with $Q$ and Hernandez-Leclerc’s category $\mathcal {C}_{Q}$, assuming the simpleness of some poles of normalized $R$-matrices for type $\mathrm {E}$.References
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Additional Information
- Ryo Fujita
- Affiliation: Research Institute for Mathematical Sciences, Kyoto University, Oiwake-Kitashir- akawa, Sakyo, Kyoto 606-8502, Japan; and Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université de Paris, F-75013 Paris, France
- MR Author ID: 1243866
- ORCID: 0000-0003-4905-2402
- Email: rfujita@kurims.kyoto-u.ac.jp
- Received by editor(s): November 11, 2017
- Received by editor(s) in revised form: August 9, 2019
- Published electronically: March 18, 2022
- Additional Notes: The work of the author was supported in part by the Kyoto Top Global University program. It was also supported by Grant-in-Aid for JSPS Research Fellow (No. 18J10669) and by JSPS Overseas Research Fellowships during the revision.
- © Copyright 2022 American Mathematical Society
- Journal: Represent. Theory 26 (2022), 211-263
- MSC (2020): Primary 17B37; Secondary 17B67
- DOI: https://doi.org/10.1090/ert/601
- MathSciNet review: 4396615