Quantum loop groups and critical convolution algebras

TL;DR

This paper introduces a geometric realization of simple modules for quantum loop groups using new algebraic structures derived from quivers with potentials, expanding the framework of convolution algebras.

Contribution

It develops a novel family of algebras based on quivers with potentials, utilizing critical K-theory and homology, generalizing Nakajima's convolution algebras.

Findings

Realization of simple modules including Kirillov-Reshetikhin representations

Introduction of algebras attached to quivers with potentials

Generalization of convolution algebras via critical theories

Abstract

We realize geometrically a family of simple modules of (shifted) quantum loop groups including Kirillov-Reshetikhin and prefundamental representations. To do this, we introduce a new family of algebras attached to quivers with potentials, using critical K-theory and critical Borel-Moore homology, which generalizes the convolution algebras attached to quivers defined by Nakajima.