A functor for constructing $R$-matrices in the category $\mathcal{O}$ of Borel quantum loop algebras

TL;DR

This paper introduces a new functor to construct $R$-matrices in the category $ ext{O}$ of Borel quantum loop algebras, linking different subcategories and providing new algebraic relations and interpretations.

Contribution

It defines an exact functor $_q$ that connects categories $ ext{O}$ for $U_q(g)$ and $U_{q^{-1}}(g)$, enabling the construction of $R$-matrices and new algebraic insights.

Findings

Constructed $R$-matrices for $ ext{O}^+$ using functor $_q$

Derived new relations in the Grothendieck ring $K_0( ext{O})$

Provided a functorial interpretation of the isomorphism $K_0( ext{O}^+) o K_0( ext{O}^-)$

Abstract

We tackle the problem of constructing $R$-matrices for the category $\mathcal{O}$ associated to the Borel subalgebra of an arbitrary untwisted quantum loop algebra $U_q(\mathfrak{g})$. For this, we define an exact functor $\mathcal{F}_q$ from the category $\mathcal{O}$ linked to $U_{q^{-1}}(\mathfrak{g})$ to the one linked to $U_q(\mathfrak{g})$. This functor $\mathcal{F}_q$ is compatible with tensor products, preserves irreducibility and interchanges the subcategories $\mathcal{O}^+$ and $\mathcal{O}^-$ of (D. Hernandez, B. Leclerc, Algebra Number Theory, 2016). We construct $R$-matrices for $\mathcal{O}^+$ by applying $\mathcal{F}_q$ on the braidings already found for $\mathcal{O}^-$ in (D. Hernandez, Rep. Theory, 2022). We also use the factorization of the latter intertwiners in terms of stable maps to deduce an analogous factorization for our new braidings. We finally obtain as byproducts new relations for the Grothendieck ring $K_0(\mathcal{O})$ as well as a functorial interpretation of a remarkable ring isomorphism $K_0(\mathcal{O}^+)\simeq K_0(\mathcal{O}^-)$ of Hernandez--Leclerc.