Braid group action on the module category of quantum affine algebras

Let ${\mathfrak{g}}_0$ be a simple Lie algebra of type ADE and let $U_q^{\mathrm{\prime }}(\mathfrak{g})$ be the corresponding untwisted quantum affine algebra. We show that there exists an action of the braid group $B({\mathfrak{g}}_0)$ on the quantum Grothendieck ring $K_t(\mathfrak{g})$ of Hernandez-Leclerc’s category $C_{\mathfrak{g}}^0$. Focused on the case of type $A_{N-1}$, we construct a family of monoidal autofunctors $\{{\mathcal{S}}_i{\}}_{i\in \mathbb{Z}}$ on a localization $T_N$ of the category of finite-dimensional graded modules over the quiver Hecke algebra of type $A_{\mathrm{\infty }}$. Under an isomorphism between the Grothendieck ring $K(T_N)$ of $T_N$ and the quantum Grothendieck ring $K_t(A_{N-1}^{(1)})$, the functors $\{{\mathcal{S}}_i{\}}_{1\le i\le N-1}$ recover the action of the braid group $B(A_{N-1})$. We investigate further properties of these functors.