Quantum toroidal algebras and their representations

By D. Hernandez. 2008

55 citations recorded by the source corpus.

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 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b61d315ef5e6e6a328ab3c1b194998747a324df7

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. S. Shatashvili
2. Vasily Pestun
3. N. Nekrasov

In Geometry and Topology

1. N. Nekrasov
2. Vasily Pestun
3. S. Shatashvili
4. C. Vafa
5. Hee-Cheol Kim
6. Yuji Tachikawa
7. D. Gaiotto
8. A. Klemm

In Mathematical Physics

1. S. Shatashvili
2. Vasily Pestun
3. N. Nekrasov
4. C. Vafa
5. Marcos Mariño
6. Yuji Tachikawa
7. J. Teschner
8. A. Klemm

Citations in the corpus

• Quantum Geometry and Quiver Gauge Theories (2013, background): “If gΓ = ĝ is an affine Kac-Moody Lie algebra, then the gΓ-quantum affine algebra U aff q (gΓ) = U aff q (ĝ) = Uq( ̂̂g) is often called the g-quantum toroidal algebra [58, 113], because of the presence of two loops inside.”
• BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters (2015, background): “In [38] the qcharacters for the quantum affine algebras Uq(gγ ) for finite γ ’s and in [49] for affine γ ’s are constructed.”