JournalsjemsVol. 20, No. 5pp. 1161–1193

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
Affinizations and R-matrices for quiver Hecke algebras cover
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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\mathcal{D} and construct a tensor functor FD\clModgr(RD)Modgr(R)\mathfrak F^{\mathcal{D}}\cl \mathrm {Mod}_gr(R^{\mathcal D}) \to \mathrm {Mod}_gr(R) between graded module categories of quiver Hecke algebras RR and RDR^{\mathcal{D}} arising from D\mathcal{D}. The functor FD\mathfrak F^{\mathcal{D}} sends finite-dimensional modules to finite-dimensional modules, and is exact when RDR^{\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 RDR^{\mathcal{D}} is of finite type. We give several examples of the functors FD\mathfrak F^{\mathcal{D}} from the graded module category of the quiver Hecke algebra of type DD_\ell, CC_\ell, B1B_{\ell-1}, A1A_{\ell-1} to that of type AA_\ell, AA_\ell, BB_\ell, BB_\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