The Universe Inside Hall Algebras of Coherent Sheaves on Toric Resolutions

Let $\mathfrak {g}\neq \mathfrak {s}\mathfrak {o}_{8}$ be a simple Lie algebra of type $A,D,E$ with $\widehat {\mathfrak {g}}$ the corresponding affine Kac-Moody algebra and $\mathfrak {n}_{+}\subset \widehat {\mathfrak {g}}$ is the standard positive 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 establish isomorphisms of the corresponding "counting" Ringel-Hall algebras and the specializations of quantized universal enveloping algebras $U_{v}(\mathfrak {n}_{+})$ at $v=\sqrt {|\Bbbk |}$.