Finite dimensional modules over quantum toroidal algebras

Abstract

The representations of the quantum toroidal algebras have been widely studied by many authors. However, no one has constructed some finite dimensional modules for them while $q$ is generic. In this paper, for all $\mathfrak{g}$-generic $q$, if $\mathfrak{g}$ is not of type $A_1$, we prove that the quantum toroidal algebra $U_q({\mathfrak{g}}_{\mathrm{tor}})$ has no nontrivial finite dimensional simple module.

7 pages

References in corpus (2)

• Schur duality in the toroidal setting
• Minimal Affinizations of Representations of Quantum Groups: the simply--laced case