  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 $\Coh^T(X)$.
