The algebra $U_q(\hat{sl}_\infty)$ and applications

TL;DR

This paper studies the algebra $U_q(\u221a{sl}_}_)$ and its integrable representations, aiming to inform character formulas for quantum toroidal algebras, and proves a positivity conjecture for specific modules.

Contribution

It introduces a positivity conjecture for representations of $U_q(}_)$ as quantum toroidal algebra modules and proves it for Kirillov-Reshetikhin modules.

Findings

Positivity conjecture for $U_q(}_)$ representations.

Proof of the conjecture for Kirillov-Reshetikhin modules.

Applications to character formula predictions for quantum toroidal algebras.

Abstract

In this note we consider the algebra $U_q(\hat{sl}_\infty)$ and we study the category O of its integrable representations. The main motivations are applications to quantum toroidal algebras, more precisely predictions of character formulae for representations of quantum toroidal algebras. In this context, we state a general positivity conjecture for representations of $U_q(\hat{sl}_\infty)$ viewed as representations of quantum toroidal algebras, that we prove for Kirillov-Reshetikhin modules.