The PBW basis of $U_{q,\bar{q}}(\ddot{\mathfrak{gl}}_n)$

TL;DR

This paper develops a new presentation of the quantum toroidal algebra of type A by analyzing its PBW basis and establishing generator commutation relations similar to known cases.

Contribution

It introduces a novel presentation of the quantum toroidal algebra of type A based on its PBW basis and commutation relations.

Findings

Established new generator commutation relations

Provided a new algebraic presentation for type A quantum toroidal algebra

Extended known relations from n=1 to general n

Abstract

We consider the PBW basis of the type A quantum toroidal algebra developed by the author, and prove commutation relations between its generators akin to the ones studied by Burban-Schiffmann for n=1. This gives rise to a new presentation of the quantum toroidal algebra of type A.