Braid group action on the module category of quantum affine algebras

TL;DR

This paper demonstrates a braid group action on the quantum Grothendieck ring of certain quantum affine algebras, constructing monoidal autofunctors that recover braid group actions in type A cases, revealing new symmetries in module categories.

Contribution

It introduces a braid group action on the quantum Grothendieck ring for quantum affine algebras and constructs monoidal autofunctors that realize this action in type A.

Findings

Braid group acts on the quantum Grothendieck ring of category C_g^0.

Constructs monoidal autofunctors on a localization of quiver Hecke algebra modules.

Recovers classical braid group actions in type A cases.

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 $K_t(\mathfrak{g})$ 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 $\{\mathscr{S}_i\}_{i\in \mathbb{Z}}$ on a localization $T_N$ of the category of finite-dimensional graded modules over the quiver Hecke algebra of type $A_{\infty}$. Under an isomorphism between the Grothendieck ring $K(T_N)$ of $T_N$ and the quantum Grothendieck ring $K_t({A^{(1)}_{N-1}})$, the functors $\{\mathscr{S}_i\}_{1\le i\le N-1}$ recover the action of the braid group $B(A_{N-1})$. We investigate further properties of these functors.