Twenty-five years of representation theory of quantum groups (11w5096)
Organizers
Nicolas Guay (University of Alberta)
Pavel Etingof (Massachusetts Institute of Technology)
Victor Ginzburg (University of Chicago)
Alistair Savage (University of Ottawa)
David Hernandez (Centre National de la Recherche Scientifique and Ecole Polytechnique)
Objectives
After an initial period of gestation in the first half of the 1980's, quantum groups rose to prominence following the address of V. Drinfeld at the International Congress of Mathematicians in 1986. The initial motivation to study quantum groups came from mathematical physics, but it was soon discovered that they are important in other branches of mathematics and that they possess a very rich representation theory.
Several different approaches to the study of representations of quantum groups have been fruitful. An algebro-combinatorial approach has led to the discovery of crystal bases and of more general crystal structures, which have had a profound influence in Lie theory. Geometric realizations of representations have been obtained via equivariant K-theory of appropriate varieties, e.g. Steinberg varieties and quiver varieties. Of course, Lie theorists have been very much interested in highest weight modules for quantum groups, their characters and also in some specific families of finite dimensional modules, e.g. Weyl modules, Kirillov-Reshetikhin modules.
Among the recent notable results achieved in connection with popular topics are the categorification of quantum Kac-Moody groups, the discovery of cluster structures in the representation theory of quantized enveloping algebras of affine type, and the geometric realizations of representations of double affine quantum algebras. There is no doubt that quantum groups will continue to be a source of mathematical discoveries for a long time.
The objective of this workshop is to bring together experienced mathematicians, many of whom have contributed to some of the main advances in the theory of quantum groups, and younger researchers with a special interest in this topic. This workshop will give these mathematicians an opportunity to survey the important open conjectures from the past which are still relevant today and pave the way for future research on quantum groups in light of the recent advances on this subject.
