Quantum extremal loop weight modules and monomial crystals

TL;DR

This paper introduces a new family of extremal loop weight modules for quantum toroidal algebras of type A, linking monomial crystal realizations with integrable representations and providing a conjectural construction method.

Contribution

It constructs extremal loop weight modules for quantum toroidal algebras using monomial crystal realizations and promotion operators, extending the understanding of these representations.

Findings

Constructed extremal loop weight modules for odd n

Linked monomial realizations with integrable representations

Provided a conjectural framework for module construction

Abstract

In this paper we construct a new family of representations for the quantum toroidal algebras of type $A_n$, which are $\ell$-extremal in the sense of Hernandez [24]. We construct extremal loop weight modules associated to level 0 fundamental weights $\varpi_\ell$ when $n=2r+1$ is odd and $\ell=1, r+1$ or $n$. To do it, we relate monomial realizations of level 0 extremal fundamental weight crystals with integrable representations of $\mathcal{U}_q(sl_{n+1}^{tor})$, and we introduce promotion operators for the level 0 extremal fundamental weight crystals. By specializing the quantum parameter, we get finite-dimensional modules of quantum toroidal algebras at roots of unity. In general, we give a conjectural process to construct extremal loop weight modules from monomial realizations of crystals.