Schur duality in the toroidal setting

Varagnolo, Michela; Vasserot, Éric

doi:10.1007/bf02517898

Cited by 75 publications

(73 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A reversed functor is also defined in [VV1]. Recently analogous results were studied in [G1, G2] in the trigonometric and rational cases, where affine Yangians and new algebras (deformed double current algebras) are used in the duality and have the PBW-property.…”

Section: Schur-weyl Duality and Cherednik Algebrasmentioning

confidence: 92%

Quantum toroidal algebras and their representations

Hernandez

2009

Sel. math., New ser.

View full text Add to dashboard Cite

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. (2000). Primary 17B37; Secondary 81R50, 82B23. Mathematics Subject Classification

show abstract

Section: Schur-weyl Duality and Cherednik Algebrasmentioning

confidence: 92%

Quantum toroidal algebras and their representations

Hernandez

2009

Sel. math., New ser.

View full text Add to dashboard Cite

show abstract

“…In particular, the knot superpolynomials of [49][50][51], constructed with the help of double-affine Hecke algebras (DAHA) [52], still lack a clear R-matrix realization within the Reshetikhin-Turaev (RT) formalism, either original [53][54][55] or modern [56][57][58]. On the other hand, the DIM algebra is naturally related with DAHA by a kind of Schur duality (see [59] for a degenerate version of this correspondence). There is another way to naturally associate these two algebras: the DIM algebra is the limit of spherical DAHA for large number of strands (see [60][61][62] for a degenerate version of this correspondence).…”

Section: Jhep10(2016)047mentioning

confidence: 99%

Toric Calabi-Yau threefolds as quantum integrable systems. ℛ $$ \mathrm{\mathcal{R}} $$ -matrix and ℛ T T $$ \mathrm{\mathcal{R}}\mathcal{T}\mathcal{T} $$ relations

Awata

Kanno

Миронов

et al. 2016

J. High Energ. Phys.

View full text Add to dashboard Cite

R-matrix is explicitly constructed for simplest representations of the DingIohara-Miki algebra. Calculation is straightforward and significantly simpler than the one through the universal R-matrix used for a similar calculation in the Yangian case by A. Smirnov but less general. We investigate the interplay between the R-matrix structure and the structure of DIM algebra intertwiners, i.e. of refined topological vertices and show that the R-matrix is diagonalized by the action of the spectral duality belonging to the SL(2, Z) group of DIM algebra automorphisms. We also construct the T -operators satisfying the RT T relations with the R-matrix from refined amplitudes on resolved conifold. We thus show that topological string theories on the toric Calabi-Yau threefolds can be naturally interpreted as lattice integrable models. Integrals of motion for these systems are related to q-deformation of the reflection matrices of the Liouville/Toda theories.

show abstract

“…More general quantum toroidal Kac-Moody algebras were constructed in [14] using Drinfeld presentation and vertex representations where the quantum Serre relations were found to be closely connected with some nontrivial relations of Hall-Littlewood symmetric functions. The quantum toroidal algebras have been studied in various contexts: toroidal Schur-Weyl duality [24], its general representation theory [18], vertex representations [21], McKay correspondence [6], toroidal actions on level one representations [22], higher level analogs for quantum affine algebras [23], fusion products [11] and an excellent survey can be found in [12]. Recently quantum toroidal algebras have found more interesting rich structures and applications in [1,2,3].…”

Section: Introductionmentioning

confidence: 99%