Monoidal categories of modules over quantum affine algebras of type A and B

TL;DR

This paper constructs a functor linking categories of modules over quantum affine algebras of types A and B, establishing isomorphisms between their Grothendieck rings and connecting their simple modules.

Contribution

It introduces a new tensor functor from modules over quiver Hecke algebras of type A to modules over quantum affine algebras of type B, revealing a deep connection between these categories.

Findings

Establishes a ring isomorphism between Grothendieck rings of related module categories.

Provides a bijection between classes of simple modules across different quantum affine algebra types.

Connects categories of modules over types A and B quantum affine algebras through functorial constructions.

Abstract

We construct an exact tensor functor from the category $\mathcal{A}$ of finite-dimensional graded modules over the quiver Hecke algebra of type $A_\infty$ to the category $\mathscr C_{B^{(1)}_n}$ of finite-dimensional integrable modules over the quantum affine algebra of type $B^{(1)}_n$. It factors through the category $\mathcal T_{2n}$, which is a localization of $\mathcal{A}$. As a result, this functor induces a ring isomorphism from the Grothendieck ring of $\mathcal T_{2n}$ (ignoring the gradings) to the Grothendieck ring of a subcategory $\mathscr C^{0}_{B^{(1)}_n}$ of $\mathscr C_{B^{(1)}_n}$. Moreover, it induces a bijection between the classes of simple objects. Because the category $\mathcal T_{2n}$ is related to categories $\mathscr C^{0}_{A^{(t)}_{2n-1}}$ $(t=1,2)$ of the quantum affine algebras of type $A^{(t)}_{2n-1}$, we obtain an interesting connection between those categories of modules over quantum affine algebras of type $A$ and type $B$. Namely, for each $t =1,2$, there exists an isomorphism between the Grothendieck ring of $\mathscr C^{0}_{A^{(t)}_{2n-1}}$ and the Grothendieck ring of $\mathscr C^{0}_{B^{(1)}_n}$, which induces a bijection between the classes of simple modules.