Hall algebras and shifted quantum affine algebras

TL;DR

This paper presents a geometric construction of a deformation of shifted quantum affine algebras using Hall algebras of quiver representations, linking quantum algebra with geometric and representation-theoretic methods.

Contribution

It introduces a new geometric realization of shifted quantum affine algebras via Hall algebras associated with a specific quiver, connecting different areas of algebra and geometry.

Findings

Constructs a deformation of shifted quantum affine algebras as Hall algebra of a quiver

Links the quiver to Rudakov's work on Lie algebra representations

Provides a new perspective on the structure of quantum affine algebras

Abstract

In \cite{FT19}, Finkelberg and Tsymbaliuk introduced the notion of shifted quantum affine algebras and described their role in the study of quantized Coulomb branches associated to certain 3D $N = 4$ quiver gauge theories. We describe a new geometric construction of a deformation of one of these shifted quantum affine algebras as the Hall algebra of the category of representations of a certain quiver $Q_{\textrm{Rud}}$ (modulo relations). This quiver first arose in the work of Rudakov in the study of the tame blocks of the category of restricted representations of the Lie algebra $\mathfrak{sl}_2(\mathbb{F}_q)$.