Hall algebras and shifted quantum affine algebras

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 $\mathfrak{sl}_2(\mathbb{F}_q)$.