# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1905.06277