Summary: 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 \(\mathrm{Coh}^\alpha(X)\), \(\mathrm{Coh}^T(X)\) 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 \(\mathrm{Coh}^T(X)\) 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 \(\mathrm{Coh}^\alpha(X)\), \(\mathrm{Coh}^T(X)\) 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 \(\mathrm{Coh}^T(X)\).