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

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