Resumen de Quantum toroidal algebras and their representations

David Hernandez

• 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 group analogs of elliptic Cherednik algebras (elliptic double affine Hecke algebras) to which they are related via Schur–Weyl duality. In this review paper, we give a glimpse of some aspects of their very rich representation theory in the context of general quantum affinizations. We illustrate the theory with several examples. We also announce new results and explain possible further developments, in particular on finite-dimensional representations at roots of unity.