Braid group action on the module category of quantum affine algebras

Abstract

Let $\mathfrak{g}_{0}$ be a simple Lie algebra of type ADE and let $U'_{q}(\mathfrak{g})$ 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 $\mathcal{K}_{t}(\mathfrak{g})$ of Hernandez-Leclerc’s category $\mathcal{C}_{\mathfrak{g}}^{0}$. Focused on the case of type $A_{N-1}$, we construct a family of monoidal autofunctors $\{\mathcal{S}_{i}\}_{i\in \mathbf{Z}}$ on a localization $\mathcal{T}_{N}$ of the category of finite-dimensional graded modules over the quiver Hecke algebra of type $A_{\infty}$. Under an isomorphism between the Grothendieck ring $K(\mathcal{T}_{N})$ of $\mathcal{T}_{N}$ and the quantum Grothendieck ring $\mathcal{K}_{t}(A^{(1)}_{N-1})$, the functors $\{\mathcal{S}_{i}\}_{1\leq i\leq N-1}$ recover the action of the braid group $B(A_{N-1})$. We investigate further properties of these functors.