# 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.

## Key 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.