Categorical relations between Langlands dual quantum affine algebras: Doubly laced types

TL;DR

This paper establishes categorical isomorphisms between Grothendieck rings of certain quantum affine algebra representations, confirming aspects of Langlands duality conjectures for finite-dimensional modules.

Contribution

It proves isomorphisms between Grothendieck rings of categories over quantum affine algebras and their Langlands duals for specific types, advancing understanding of Langlands duality in quantum affine settings.

Findings

Grothendieck rings are isomorphic for categories over dual quantum affine algebras.

Results confirm parts of Frenkel-Hernandez conjectures on Langlands duality.

Provides categorical equivalences for types A_{2n-1} and D_{n+1}.

Abstract

We prove that the Grothendieck rings of category $\mathcal{C}^{(t)}_Q$ over quantum affine algebras $U_q'(\g^{(t)})$ $(t=1,2)$ associated to each Dynkin quiver $Q$ of finite type $A_{2n-1}$ (resp. $D_{n+1}$) is isomorphic to one of category $\mathcal{C}_{\mQ}$ over the Langlands dual $U_q'({^L}\g^{(2)})$ of $U_q'(\g^{(2)})$ associated to any twisted adapted class $[\mQ]$ of $A_{2n-1}$ (resp. $D_{n+1}$). This results provide partial answers of conjectures of Frenkel-Hernandez on Langlands duality for finite-dimensional representation of quantum affine algebras.