Quantum loop groups for symmetric Cartan matrices

TL;DR

This paper introduces a new quantum loop group for symmetric Cartan matrices, establishing a perfect Hopf pairing and applying it to describe the localized K-theoretic Hall algebra of loopless quivers with a specific $C^*$ action.

Contribution

It constructs a quantum loop group for symmetric Cartan matrices with minimal relations ensuring a perfect Hopf pairing, and applies this to quiver Hall algebras.

Findings

Defined a quantum loop group with minimal relations for symmetric Cartan matrices.

Established a perfect Hopf pairing between positive and negative parts.

Described the localized K-theoretic Hall algebra for loopless quivers with a $C^*$ action.

Abstract

We introduce a quantum loop group associated to a general symmetric Cartan matrix, by imposing just enough relations between the usual generators $\{e_{i,k}, f_{i,k}\}_{i \in I, k \in \mathbb{Z}}$ in order for the natural Hopf pairing between the positive and negative halves of the quantum loop group to be perfect. As an application, we describe the localized K-theoretic Hall algebra of any quiver without loops, endowed with a particularly important $\mathbb{C}^*$ action.