Braid group action on the module category of quantum affine algebras

Abstract

Let g(0) be a simple Lie algebra of type ADE and let U-q' (g) be the corresponding untwisted quantum affine algebra. We show that there exists an action of the braid group B(g(0)) on the quantum Grothendieck ring K-t(g) of Hernandez-Leclerc's category C-g(0). Focused on the case of type AN 1, we construct a family of monoidal autofunctors {S-i}i is an element of Z on a localization T-N of the category of finite-dimensional graded modules over the quiver Hecke algebra of type A(infinity). Under an isomorphism between the Grothendieck ring K(T-N) of T-N and the quantum Grothendieck ring K-t(A(N-1)((1)))N, the functors {S-i}1 <= i <= N <= 1 recover the action of the braid group B(A(N-1)). We investigate further properties of these functors.