Hall Lie algebras of toric monoid schemes

Abstract

We associate to a projective $n$-dimensional toric variety $X_{\Delta }$ a pair of co-commutative (but generally non-commutative) Hopf algebras $H_X^{\alpha },H_X^T$. These arise as Hall algebras of certain categories $\Coh^α(X), \Coh^T(X)$ 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 $\Coh^α(X), \Coh^T(X)$ are treated via the formalism of proto-exact/proto-abelian categories developed by Dyckerhoff-Kapranov. The Hall algebras $H_X^{\alpha },H_X^T$ are graded and connected, and so enveloping algebras $H^α_X \simeq U(\n^α_X)$, $H^{T}_X \simeq U(\n^{T}_X)$, where the Lie algebras $\n^α_X, \n^{T}_X$ 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 $\Coh^T(X)$.

arXiv commentTitle changed from earlier version "Toric Hall algebras and infinite-dimensional Lie algebras". Significant changes in approach to T-sheaves (their definition etc.)