Recent developments in the theory of Hall
algebras
From 20th to 24th November 2006 at the CIRM
Introduction
The theory of Hall algebras is closely related to the theory of
representations
and algebraic geometry. In a broad sense, a Hall algebra provides a
tool allowing one
to code a category. The object of this conference is to present current
developments of
the theory whose history we shall describe briefly here.
- The term is due to Ringel, who made reference to the works of
Hall on symmetric
functions where he gave a correspondence between an associative algebra
and the category
of nilpotent representations of the quiver consisting of a single
vertex and a single cycle.
In the early 90's, Ringel gave a new start to the theory. In the
setting of hereditary
categories, he defined a quantum version of Hall algebras, giving a
realisation
of the positive part of quantum Kac-Moody algebras. Note that it is
thanks to this
link established between the geometry of quiver representations and
quantum groups
that Lusztig developed his theory of canonical basis.
- In studying a Hall algebra associated to the category of
coherent sheaves on the projective line, Kapranov opened a new
direction in the theory. One can
extend his study to the case of elliptic curves and of surfaces - see
the works of
Schiffmann and of Kapranov-Vasserot.
- Another recent development of the theory consists of associating
a Hall type
algebra to triangulated categories, or differential graded categories
(Xiao, Toën, Caldero-Keller). One can thus realise Kac-Moody type
Lie algebras, and also cluster algebras from certain triangulated
categories.
The aim of this conference is to bring together some of the experts in
the field, and
also to provide an introduction of the theory to PhD students and young
researchers.
The following speakers have confirmed their participation :
P. Baumann, T. Bridgeland, I. Burban,
W. Crawley-Boevey, A. Hubery,
M. Kapranov, B. Keller, B. Leclerc, C. M. Ringel,
O. Schiffmann, B. Toën, E. Vasserot, J. Xiao.
This conference is organised by the
GDR
2432 and receives support from the European Research Training
Network
Liegrits.
Back
to the website of the GDR
http://www.math.jussieu.fr/gdralgebre/hall2006/
24 octobre 2005.