## Abstract

We introduce the notion of affinizations and R-matrices for arbitrary quiver Hecke algebras. It is shown that they enjoy similar properties to those for symmetric quiver Hecke algebras. We next define a duality datum D and construct a tensor functor F^{D}: Mod_{gr}(R^{D}) ? Mod_{gr}(R) between graded module categories of quiver Hecke algebras R and R^{D} arising from D. The functor F^{D} sends finite-dimensional modules to finite-dimensional modules, and is exact when R^{D} is of finite type. It is proved that affinizations of real simple modules and their R-matrices give a duality datum. Moreover, the corresponding duality functor sends every simple module to a simple module or zero when R^{D} is of finite type. We give several examples of the functors F^{D} from the graded module category of the quiver Hecke algebra of type D, C, B_{?1}, A_{?1} to that of type A, A, B, B, respectively.

Original language | English |
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Pages (from-to) | 1161-1193 |

Number of pages | 33 |

Journal | Journal of the European Mathematical Society |

Volume | 20 |

Issue number | 5 |

DOIs | |

State | Published - 2018 |

## Keywords

- Affinization
- Duality functor
- Quiver Hecke algebra
- R-matrix