Beyond Kirillov-Reshetikhin modules

TL;DR

This survey reviews the structure and classification of finite-dimensional representations of untwisted quantum affine algebras, focusing on recent advances in understanding specific families like Kirillov-Reshetikhin modules.

Contribution

It provides a comprehensive overview of foundational results and recent developments in the structure of key irreducible modules for quantum affine algebras.

Findings

Classification of simple modules and q-characters

Understanding of Kirillov-Reshetikhin modules and minimal affinizations

Recent structural insights into specific irreducible representations

Abstract

In this survey, we shall be concerned with the category of finite-dimensional representations of the untwisted quantum affine algebras when the quantum parameter q is not a root of unity. We review the foundational results of the subject, including the Drinfeld presentation, the classification of simple modules and q-characters. We then concentrate on particular families of irreducible representations whose structure has recently been understood: Kirillov-Reshetikhin modules, minimal affinizations and beyond.