Resumen de Hall Lie algebras of toric monoid schemes
We associate to a projective n-dimensional toric variety X a pair of co-commutative (but generally non-commutative) Hopf algebras Hα X , HT X . These arise as Hall algebras of certain categories Cohα(X),CohT (X) of coherent sheaves on X viewed as a monoid scheme—i.e. a scheme obtained by gluing together spectra of commutative monoids rather than rings. When X is smooth, the category CohT (X) has an explicit combinatorial description as sheaves whose restriction to each An corresponding to a maximal cone σ ∈ is determined by an n-dimensional generalized skew shape. The (non-additive) categories Cohα(X),CohT (X) are treated via the formalism of protoexact/proto-abelian categories developed by Dyckerhoff–Kapranov. The Hall algebras Hα X , HT X are graded and connected, and so enveloping algebras Hα X U(nα X ), HT X U(nT X ), where the Lie algebras nα X , nT 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 nT X to known Lie algebras. In particular, when X = P1, nT X is isomorphic to a non-standard Borel in gl2[t, t−1].When X is the second infinitesimal neighborhood of the origin inside A2, nT X is isomorphic to a subalgebra of gl2[t]. We also consider the case X = P2, where we give a basis for nT X by describing all indecomposable sheaves in CohT (X).
