The universe inside Hall algebras of coherent sheaves on toric resolutions

TL;DR

This paper constructs explicit objects in the derived category of G-equivariant coherent sheaves on C^3 whose Hall algebras are isomorphic to universal enveloping algebras of certain nilpotent Lie subalgebras, revealing deep connections between geometry and Lie theory.

Contribution

It provides an explicit description of objects in G-equivariant coherent sheaves on C^3 that generate Hall algebras isomorphic to universal enveloping algebras of nilpotent Lie subalgebras, linking geometric and algebraic structures.

Findings

Hall algebras generated by specific objects are isomorphic to $U( _-)$

Explicit objects in $Coh_G(C^3)$ are constructed

Conjecture relating counting Ringel-Hall algebras to quantum groups at roots of unity

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 $v=\sqrt{|\Bbbk|}$.