Foldings of KLR algebras

TL;DR

This paper proves that the categorification of folded quantum groups via KLR algebras occurs over the original base ring, confirming McNamara's conjecture and clarifying the algebraic structure involved.

Contribution

The paper demonstrates that the extension ring used in categorification coincides with the original base ring, resolving a conjecture posed by McNamara.

Findings

Confirmed that the extension ring equals the original ring ${f A}$.

Provided a positive answer to McNamara's question.

Clarified the algebraic structure of categorification for folded quantum groups.

Abstract

Let ${\mathbf U}^-_q$ be the negative half of the quantum group associated to a Kac-Moody algebra ${\mathfrak g}$, and $\underline{\mathbf U}^-_q$ the quantum group obtained by a folding of ${\mathfrak g}$. Let ${\mathbf A} = {\mathbf Z}[q,q^{-1}]$. McNamara showed that $\underline{\mathbf U}^-_q$ is categorified over a certain extenion ring $\widetilde{\mathbf A}$ of ${\mathbf A}$, by uing the folding theory of KLR algebras. He posed a question whether $\widetilde{\mathbf A}$ coincides with ${\mathbf A}$ or not. In this paper, we give an affirmative answer for this problem.