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 \(H^{\alpha }_X, H^{T}_X\). These arise as Hall algebras of certain categories \({\text {Coh}}^{\alpha }(X), {\text {Coh}}^T(X)\) 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 \({\text {Coh}}^T(X)\) 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 \({\text {Coh}}^{\alpha }(X), {\text {Coh}}^T(X)\) 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 \(H^{\alpha }_X \simeq U(\mathfrak {n}^{\alpha }_X)\), \(H^{T}_X \simeq U(\mathfrak {n}^{T}_X)\), where the Lie algebras \(\mathfrak {n}^{\alpha }_X, \mathfrak {n}^{T}_X\) 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 \(\mathfrak {n}^T_X\) to known Lie algebras. In particular, when \(X = \mathbb {P}^1\), \(\mathfrak {n}^T_X\) 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 \(\mathbb {A}^2\), \(\mathfrak {n}^T_X\) 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 \(\mathfrak {n}^T_X\) by describing all indecomposable sheaves in \({\text {Coh}}^T(X)\).