Localizations for quiver Hecke algebras

We provide the localization procedure for monoidal categories by a real commuting family of braiders. For an element $w$ of the Weyl group, ${\mathcal{C}}_w$ is a subcategory of modules over quiver Hecke algebra which categorifies the quantum unipotent coordinate algebra $A_q[\mathfrak{n}(w)]$. We construct the localization $\widetilde{{\mathcal{C}}_w}$ of ${\mathcal{C}}_w$ by adding the inverses of simple modules which correspond to the frozen variables in the quantum cluster algebra $A_q[\mathfrak{n}(w)]$. The localization $\widetilde{{\mathcal{C}}_w}$ is left rigid and we expect that it is rigid.