Geometry and Topology
Speaker: Francesco Sala (Western)
"Quantum toroidal algebras and K-theoretic Hall Algebra of the stack of torsion sheaves"
Time: 15:30
Room: MC 107
Starting from the works of Nakajima and Grojnowski, moduli spaces and stacks of sheaves on surfaces represent wonderful tools for the study of vertex and quantum algebras and their representations from a geometric point of view. For example, Schiffmann and Vasserot proved that the equivariant K-theory of the stack of zero-dimensional sheaves on $\mathbb C^2$ has an associative algebra structure and is isomorphic to the positive part of quantum toroidal algebra of type $\mathfrak{gl}(1)$; moreover, it acts on the equivariant K-theory of the Hilbert scheme of points on $\mathbb{C}^2$. Their result can be seen as a K-theoretic version of Nakajima-Grojnowski cohomological result for Hilbert schemes of points.
In the present talk, I would like to describe a new conjectural approach to the study of quantum toroidal algebras of type $\mathfrak{gl}(k)$ based on the study of algebra structures on the K-theory of the stacks of torsion sheaves over other noncompact surfaces (e.g. the stack of sheaves on the minimal resolution of the Du-Val singularity $\mathbb{C}^2/\mathbb{Z}_k$​, supported at an exceptional curve)​. (This is a work in progress with Olivier Schiffmann.)
Mathematics Calendar 
Week of October 18, 2015
