The universe inside Hall algebras of coherent sheaves on toric resolutions

Abstract

Let $\mathfrak{g}\neq \mathfrak{so}_8$ be a simple Lie algebra of type $A,D,E$ with $\widehat{\mathfrak{g}}$ the corresponding affine Kac-Moody algebra and $\mathfrak{n}_-\subset \widehat{\mathfrak{g}}$ a nilpotent subalgebra. Given $\mathfrak{n}_-$ as above, we provide an infinite collection of cyclic finite abelian subgroups of $SL_3(\mathbb{C})$ with the following properties. Let $G$ be any group in the collection, $Y=G\operatorname{-}\mbox{Hilb}(\mathbb{C}^3)$ and $\Psi: D^b_G(Coh(\mathbb{C}^3))\rightarrow D^b(Coh(Y))$ the derived equivalence of Bridgeland, King and Reid. We present an (explicitly described) subset of objects in $Coh_G(\mathbb{C}^3)$, s.t. the Hall algebra generated by their images under $\Psi$ is isomorphic to $U(\mathfrak{n}_-)$. In case the field $\Bbbk$ (in place of $\mathbb{C}$) is finite and $\mbox{char}(\Bbbk)$ is coprime with the order of $G$, we conjecture the isomorphisms of the corresponding 'counting' Ringel-Hall algebras and the specializations of quantized universal enveloping algebras $U_v(\mathfrak{n}_-)$ at $v=\sqrt{|\Bbbk|}$.