2020/08/25 by Jaiung Jun, Jun, Jaiung, Matt Szczesny +1 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2008.11302
We associate to a projective n-dimensional toric variety XΔ a pair of co-commutative (but generally non-commutative) Hopf algebras HαX, HTX. These arise as Hall algebras of certain categories \Cohα(X), \CohT(X) of coherent sheaves on XΔ viewed as a monoid scheme - i.e. a scheme obtained by gluing together spectra of commutative monoids rather than rings. When XΔ is smooth, the category \CohT(X) has an explicit combinatorial description as sheaves whose restriction to each \mathbbAn corresponding to a maximal cone σ∈ Δ is determined by an n-dimensional generalized skew shape. The (non-additive) categories \Cohα(X), \CohT(X) are treated via the formalism of proto-exact/proto-abelian categories developed by Dyckerhoff-Kapranov. The Hall algebras HαX, HTX are graded and connected, and so enveloping algebras HαX ≃ U(\nαX), HTX ≃ U(\nTX), where the Lie algebras \nαX, \nTX 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 \nTX to known Lie algebras. In particular, when X = ℙ1, \nTX is isomorphic to a non-standard Borel in \mathfrakgl2 [t,t-1]. When X is the second infinitesimal neighborhood of the origin inside \mathbbA2, \nTX is isomorphic to a subalgebra of \mathfrakgl2[t]. We also consider the case X=ℙ2, where we give a basis for \nTX by describing all indecomposable sheaves in \CohT(X).