Extremal loop weight modules and tensor products for quantum toroidal algebras

TL;DR

This paper introduces a new class of extremal loop weight modules for quantum toroidal algebras of type A, constructed via tensor products and the Drinfeld coproduct, leading to new finite-dimensional representations at roots of unity.

Contribution

It defines integrable representations using tensor products, recovers known vector representations, and constructs new extremal loop weight modules and finite-dimensional representations.

Findings

Recovered vector representations as extremal loop weight modules

Constructed new extremal loop weight modules from tensor powers

Obtained finite-dimensional representations at roots of unity

Abstract

We define integrable representations of quantum toroidal algebras of type A by tensor product, using the Drinfeld "coproduct". This allow us to recover the vector representations recently introduced by Feigin-Jimbo-Miwa-Mukhin [6] and constructed by the author [21] as a subfamily of extremal loop weight modules. In addition we get new extremal loop weight modules as subquotients of tensor powers of vector representations. As an application we obtain finite-dimensional representations of quantum toroidal algebras by specializing the quantum parameter at roots of unity.