Localizations for quiver hecke algebras

Masaki Kashiwara, Myungho Kim, Se Jin Oh, Euiyong Park

Research output: Contribution to journalArticlepeer-review

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Abstract

In this paper, we provide a generalization of the localization procedure for monoidal categories developed in [12] by Kang-Kashiwara-Kim by introducing the notions of braiders and a real commuting family of braiders. Let R be a quiver Hecke algebra of arbitrary symmetrizable type and R-gmod the category of finite-dimensional graded R-modules. For an element w of the Weyl group, Cw is the subcategory of R-gmod which categorifies the quantum unipotent coordinate algebra Aq (n(w)). We construct the localization C˜w of Cw by adding the inverses of simple modules M(wΛi, Λi ) which correspond to the frozen variables in the quantum cluster algebra Aq (n(w)). The localization C˜w is left rigid and it is conjectured that C˜w is rigid.

Original languageEnglish
Pages (from-to)1465-1548
Number of pages84
JournalPure and Applied Mathematics Quarterly
Volume17
Issue number4
DOIs
StatePublished - 2021

Bibliographical note

Funding Information:
Received February 12, 2020. 2010 Mathematics Subject Classification: Primary 18D10, 16D90; secondary 81R10. ∗The research of M. Ka. was supported by Grant-in-Aid for Scientific Research (B) 15H03608, Japan Society for the Promotion of Science. †The research of M. Kim was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) (NRF-2017R1C1B2007824). ‡The research of S.-j. Oh was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-2019R1A2C4069647). §The research of E. P. was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea Government (MSIP) (NRF-2017R1A1A1A05001058).

Funding Information:
The research of M. Ka. was supported by Grant-in-Aid for Scientific Research (B) 15H03608, Japan Society for the Promotion of Science. The research of M. Kim was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) (NRF-2017R1C1B2007824). The research of S.-j. Oh was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-2019R1A2C4069647). The research of E. P. was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea Government (MSIP) (NRF-2017R1A1A1A05001058). We thank Yoshiyuki Kimura for informing us his result with Hironori Oya and for fruitful communications, and the first author thank Christof Geiß and Bernard Leclerc for fruitful discussion. The second, third and fourth authors gratefully acknowledge for the hospitality of RIMS (Kyoto University) during their visit in 2018. The authors also would like to thank the anonymous referee for valuable comments.

Publisher Copyright:
© 2021, International Press, Inc.. All rights reserved.

Keywords

  • Categorification
  • Localization
  • Monoidal category
  • Quantum unipotent coordinate ring
  • Quiver Hecke algebra

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