## 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

Publisher Copyright:© 2021 The Japan Academy

## Keywords

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