$U_q(\hat{\frak{gl}}_{{}_N})$ action on $\hat{\frak{gl}}_{{}_N}$ -modules and quantum toroidal algebras

Abstract

We construct a vertex representation for the quantum toroidal algebra through the quantum general linear algebra. Using a new realization of the quantum general linear algebra we construct vertex operators for root vectors on the basic representation of the affine Lie algebra $gl_n$ and show that the simple generators give rise a realization of the quantum toroidal algebra with two parameters.