Quantum toroidal algebras and their representations

Details Quantum toroidal algebras and their representations Quantum toroidal algebras and their representations

2008-01-15

arxiv.org/abs/0801.2397v4

Selecta Math. 14 (2009), no. 3-4, 701-725

Mathematics

Quantum Algebra

24 pages. To appear in Selecta Mathematica

Scientific paper

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.

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