The universe inside Hall algebras of coherent sheaves on toric resolutions

Abstract

Let $\mathfrak{g}\ne {\mathfrak{so}}_8$ be a simple Lie algebra of type $A,D,E$ with $\hat{\mathfrak{g}}$ the corresponding affine Kac-Moody algebra and ${\mathfrak{n}}_{-}\subset \hat{\mathfrak{g}}$ a nilpotent subalgebra. Given ${\mathfrak{n}}_{-}$ as above, we provide an infinite collection of cyclic finite abelian subgroups of $S{L}_3(\mathbb{C})$ with the following properties. Let $G$ be any group in the collection, $Y=G\operatorname{-}\mbox{Hilb}(\mathbb{C}^3)$ and $\Psi :D_G^b(Coh({\mathbb{C}}^3))\to D^b(Coh(Y))$ the derived equivalence of Bridgeland, King and Reid. We present an (explicitly described) subset of objects in $Co{h}_G({\mathbb{C}}^3)$, s.t. the Hall algebra generated by their images under $\Psi$ is isomorphic to $U({\mathfrak{n}}_{-})$. In case the field $\mathbb{k}$ (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 $v=\sqrt{|\mathbb{k}|}$.