# Finite dimensional modules over quantum toroidal algebras

## TL;DR

This paper proves that for most types of quantum toroidal algebras, there are no nontrivial finite-dimensional simple modules when the parameter q is generic, filling a gap in the representation theory of these algebras.

## Contribution

It establishes the nonexistence of finite-dimensional simple modules for quantum toroidal algebras of all types except A_1 at generic q, a previously unresolved question.

## Key findings

- No nontrivial finite-dimensional simple modules for non-A_1 types at generic q
- Completes the classification of finite-dimensional modules over quantum toroidal algebras
- Provides a foundation for future research on representations of 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 $U_q(\mathfrak{g}_{\rm tor})$ has no nontrivial finite dimensional simple module.