# Affinizations and R-matrices for quiver Hecke algebras

### Masaki Kashiwara

Kyoto University, Japan and Korea Institute for Advanced Study, Seoul, Korea### Euiyong Park

University of Seoul, Republic of Korea

## 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 $\mathcal{D}$ and construct a tensor functor $\mathfrak F^{\mathcal{D}}\cl \mathrm {Mod}_gr(R^{\mathcal D}) \to \mathrm {Mod}_gr(R)$ between graded module categories of quiver Hecke algebras $R$ and $R^{\mathcal{D}}$ arising from $\mathcal{D}$. The functor $\mathfrak F^{\mathcal{D}}$ sends finite-dimensional modules to finite-dimensional modules, and is exact when $R^{\mathcal{D}}$ 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 a simple module to a simple module or zero when $R^{\mathcal{D}}$ is of finite type. We give several examples of the functors $\mathfrak F^{\mathcal{D}}$ from the graded module category of the quiver Hecke algebra of type $D_\ell$, $C_\ell$, $B_{\ell-1}$, $A_{\ell-1}$ to that of type $A_\ell$, $A_\ell$, $B_\ell$, $B_\ell$, respectively.

## Cite this article

Masaki Kashiwara, Euiyong Park, Affinizations and R-matrices for quiver Hecke algebras. J. Eur. Math. Soc. 20 (2018), no. 5, pp. 1161–1193

DOI 10.4171/JEMS/785