## Abstract

Let g_{0} be a simple Lie algebra of type ADE and let U^{'}_{q}(g) be the corresponding untwisted quantum affine algebra. We show that there exists an action of the braid group B(g_{0}) on the quantum Grothendieck ring K_{t}(g) of Hernandez-Leclerc’s category (Formula Presented). Focused on the case of type A_{N-1}, we construct a family of monoidal autofunctors (Formula Presented) on a localization T_{N} of the category of finite-dimensional graded modules over the quiver Hecke algebra of type A_{∞}. Under an isomorphism between the Grothendieck ring K(T_{N}) of T_{N} and the quantum Grothendieck ring (Formula Presented), the functors (Formula Presented) recover the action of the braid group B(A_{N-1}). We investigate further properties of these functors.

Original language | English |
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Pages (from-to) | 13-18 |

Number of pages | 6 |

Journal | Proceedings of the Japan Academy Series A: Mathematical Sciences |

Volume | 97 |

Issue number | 3 |

DOIs | |

State | Published - 2021 |

### Bibliographical note

Funding Information:(c) Let Q be a Q-data, and L := FQ(L(i)). Then the automorphism of Kt(g) induced by SL coincides with σi, i.e., the following diagram commutes: Acknowledgement. The research of Kashiwara was supported by Grant-in-Aid for Scientific Research (B) 15H03608, Japan Society for the Promotion of Science, the research of Kim was supported by the National Research Foundation of Korea(NRF) Grant funded by the Korea government(MSIP) (NRF-2017R1C1B2007824) and the research of Oh was supported by the Ministry of Education of the Republic of Korea and the NRF of Korea (NRF-2019R1A2C4069647).

Publisher Copyright:

© 2021 The Japan Academy

## Keywords

- Quantum affine algebra
- R-matrix.
- braid group action
- quantum Grothendieck ring
- quiver Hecke algebra