Quantum toroidal algebras and their representations

TL;DR

This paper reviews the rich representation theory of quantum toroidal algebras, exploring their connections to elliptic Cherednik algebras and discussing new results on finite-dimensional representations at roots of unity.

Contribution

It provides an overview of quantum toroidal algebras' representation theory, introduces new findings, and discusses potential future research directions.

Findings

Illustrated with several examples of representations.

Announced new results on finite-dimensional representations at roots of unity.

Explored connections to elliptic Cherednik algebras.

Abstract

Quantum toroidal algebras (or double affine quantum algebras) are defined from quantum affine Kac-Moody algebras by using the Drinfeld quantum affinization process. They are quantum groups analogs of elliptic Cherednik algebras (elliptic double affine Hecke algebras) to whom they are related via Schur-Weyl duality. In this review paper, we give a glimpse on some aspects of their very rich representation theory in the context of general quantum affinizations. We illustrate with several examples. We also announce new results and explain possible further developments, in particular on finite dimensional representations at roots of unity.