Beyond Kirillov-Reshetikhin modules

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 representations 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.