How to sheafify an elliptic quantum group

TL;DR

This paper surveys the construction of a sheafified elliptic quantum group linked to symmetric Kac-Moody Lie algebras, using equivariant elliptic cohomology of moduli spaces, and explores its actions and relations to affine Grassmannians.

Contribution

It introduces a sheafified elliptic quantum group via elliptic cohomology, connecting it to Nakajima quiver varieties and the global affine Grassmannian.

Findings

Sheafification obtained through equivariant elliptic cohomology.

Elliptic quantum group acts on equivariant elliptic cohomology of quiver varieties.

Relation established between the sheafified elliptic quantum group and the affine Grassmannian.

Abstract

We give an introductory survey of the results in arXiv: 1708.01418. We discuss a sheafified elliptic quantum group associated to any symmetric Kac-Moody Lie algebra. The sheafification is obtained by applying the equivariant elliptic cohomological theory to the moduli space of representations of a preprojective algebra. By construction, the elliptic quantum group naturally acts on the equivariant elliptic cohomology of Nakajima quiver varieties. As an application, we obtain a relation between the sheafified elliptic quantum group and the global affine Grassmannian over an elliptic curve.