Shuffle algebras for quivers as quantum groups

TL;DR

This paper constructs a quantum loop group for any quiver, proves its isomorphism to a shuffle algebra and Hall algebra, and applies these results to describe the Hall algebra of smooth projective curves.

Contribution

It introduces a new quantum group associated to quivers, establishes its isomorphism to shuffle and Hall algebras, and extends results to non-generic parameters and cuspidal eigenforms.

Findings

Quantum loop group $ extbf{U}^+_Q$ is isomorphic to the shuffle algebra.

Provides a presentation of the spherical Hall algebra of a curve.

Describes the subalgebra generated by cuspidal eigenforms.

Abstract

We define a quantum loop group $\mathbf{U}^+_Q$ associated to an arbitrary quiver $Q=(I,E)$ and maximal set of deformation parameters, with generators indexed by $I \times \mathbb{Z}$ and some explicit quadratic and cubic relations. We prove that $\mathbf{U}^+_Q$ is isomorphic to the (generic, small) shuffle algebra associated to the quiver $Q$ and hence, by [Neg21a], to the localized K-theoretic Hall algebra of $Q$. For the quiver with one vertex and $g$ loops, this yields a presentation of the spherical Hall algebra of a (generic) smooth projective curve of genus $g$ (invoking the results of [SV12]). We extend the above results to the case of non-generic parameters satisfying a certain natural metric condition. As an application, we obtain a description by generators and relations of the subalgebra generated by absolutely cuspidal eigenforms of the Hall algebra of an arbitrary smooth projective curve (invoking the results of [KSV17]).