Hall Lie algebras of toric monoid schemes

TL;DR

This paper constructs and analyzes Hall algebras associated with categories of coherent sheaves on toric monoid schemes, revealing connections to known Lie algebras and providing explicit examples.

Contribution

It introduces a novel framework for Hall algebras of monoid scheme coherent sheaves and relates these to classical Lie algebras in specific cases.

Findings

Hall algebras are graded and connected, isomorphic to enveloping algebras of certain Lie algebras.

Explicit descriptions of Lie algebras for specific toric varieties like P^1 and neighborhoods in A^2.

Identification of the Lie algebra for P^1 with a non-standard Borel in gl_2[t,t^{-1}].

Abstract

We associate to a projective $n$-dimensional toric variety $X_{\Delta}$ a pair of co-commutative (but generally non-commutative) Hopf algebras $H^{\alpha}_X, H^{T}_X$. These arise as Hall algebras of certain categories $\Coh^{\alpha}(X), \Coh^T(X)$ of coherent sheaves on $X_{\Delta}$ viewed as a monoid scheme - i.e. a scheme obtained by gluing together spectra of commutative monoids rather than rings. When $X_{\Delta}$ is smooth, the category $\Coh^T(X)$ has an explicit combinatorial description as sheaves whose restriction to each $\mathbb{A}^n$ corresponding to a maximal cone $\sigma \in \Delta$ is determined by an $n$-dimensional generalized skew shape. The (non-additive) categories $\Coh^{\alpha}(X), \Coh^T(X)$ are treated via the formalism of proto-exact/proto-abelian categories developed by Dyckerhoff-Kapranov. The Hall algebras $H^{\alpha}_X, H^{T}_X$ are graded and connected, and so enveloping algebras $H^{\alpha}_X \simeq U(\n^{\alpha}_X)$, $H^{T}_X \simeq U(\n^{T}_X)$, where the Lie algebras $\n^{\alpha}_X, \n^{T}_X$ are spanned by the indecomposable coherent sheaves in their respective categories. We explicitly work out several examples, and in some cases are able to relate $\n^T_X$ to known Lie algebras. In particular, when $X = \mathbb{P}^1$, $\n^T_X$ is isomorphic to a non-standard Borel in $\mathfrak{gl}_2 [t,t^{-1}]$. When $X$ is the second infinitesimal neighborhood of the origin inside $\mathbb{A}^2$, $\n^T_X$ is isomorphic to a subalgebra of $\mathfrak{gl}_2[t]$. We also consider the case $X=\mathbb{P}^2$, where we give a basis for $\n^T_X$ by describing all indecomposable sheaves in $\Coh^T(X)$.