15,458 research outputs found
Smallness problem for quantum affine algebras and quiver varieties
The geometric small property (Borho-MacPherson) of projective morphisms
implies a description of their singularities in terms of intersection homology.
In this paper we solve the smallness problem raised by Nakajima
(math.QA/0105173) for certain resolutions of quiver varieties (analogs of the
Springer resolution) : for Kirillov-Reshetikhin modules of simply-laced quantum
affine algebras, we characterize explicitly the Drinfeld polynomials
corresponding to the small resolutions. We use an elimination theorem for
monomials of Frenkel-Reshetikhin q-characters that we establish for non
necessarily simply-laced quantum affine algebras. We also refine results of
(math.QA/0501202) and extend the main result to general simply-laced quantum
affinizations, in particular to quantum toroidal algebras (double affine
quantum algebras).Comment: 33 pages; accepted for publication in Annales Scientifiques de
l'Ecole Normale Superieur
Advances in R-matrices and their applications (after Maulik-Okounkov, Kang-Kashiwara-Kim-Oh,...)
R-matrices are the solutions of the Yang-Baxter equation. At the origin of
the quantum group theory, they may be interpreted as intertwining operators.
Recent advances have been made independently in different directions.
Maulik-Okounkov have given a geometric approach to R-matrices with new tools in
symplectic geometry, the stable envelopes. Kang-Kashiwara-Kim-Oh proved a
conjecture on the categorification of cluster algebras by using R-matrices in a
crucial way. Eventually, a better understanding of the action of
transfer-matrices obtained from R-matrices led to the proof of several
conjectures about the corresponding quantum integrable systems.Comment: This is an English translation of the Bourbaki seminar 1129 (March
2017). The French version will appear in Ast\'erisqu
Quantum toroidal algebras and their representations
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.Comment: 24 pages. To appear in Selecta Mathematic
Algebraic Approach to q,t-Characters
Frenkel and Reshetikhin introduced q-characters to study finite dimensional
representations of quantum affine algebras. In the simply laced case Nakajima
defined deformations of q-characters called q,t-characters. The definition is
combinatorial but the proof of the existence uses the geometric theory of
quiver varieties which holds only in the simply laced case. In this article we
propose an algebraic general (non necessarily simply laced) new approach to
q,t-characters motivated by our deformed screening operators. The
t-deformations are naturally deduced from the structure of the quantum affine
algebra: the parameter t is analog to the central charge c. The q,t-characters
lead to the construction of a quantization of the Grothendieck ring and to
general analogues of Kazhdan-Lusztig polynomials in the same spirit as Nakajima
did for the simply laced case.Comment: 46 pages; accepted for publication in Advances in Mathematic
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