Localizations for quiver Hecke algebras

Masaki Kashiwara (Kyoto University Institute for Advanced Study, Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan; and Korea Institute for Advanced Study, Seoul, South Korea)

Myungho Kim (Department of Mathematics, Kyung Hee University, Seoul, South Korea)

Se-Jin Oh (Department of Mathematics, Ewha Womans University, Seoul, South Korea)

Euiyong Park (Department of Mathematics, University of Seoul, South Korea)

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 $R$‑$\operatorname{gmod}$ the category of finite-dimensional graded $R$-modules. For an element $w$ of the Weyl group, $\mathscr{C}_w$ is the subcategory of $R$‑$\operatorname{gmod}$ which categorifies the quantum unipotent coordinate algebra $A_q (\mathfrak{n}(w))$. We construct the localization $\tilde{\mathscr{C}}_w$ of $\mathscr{C}_w$ by adding the inverses of simple modules $\mathsf{M} (w \Lambda_i , \Lambda_i)$ which correspond to the frozen variables in the quantum cluster algebra $A_q (\mathfrak{n}(w))$. The localization $\tilde{\mathscr{C}}_w$ is left rigid and it is conjectured that $\tilde{\mathscr{C}}_w$ is rigid.

Keywords

categorification, localization, monoidal category, quantum unipotent coordinate ring, quiver Hecke algebra

2010 Mathematics Subject Classification

Primary 16D90, 18D10. Secondary 81R10.

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In memory of Professor Bertram Kostant.

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).

Received 12 February 2020

Accepted 25 December 2020

Published 22 December 2021