Toric Hall Algebras and infinite-dimensional Lie algebras
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Citation (published version)Maciej Szczesny, Jaiung Jun. 2020. "Toric Hall Algebras and infinite-dimensional Lie algebras." https://arxiv.org/abs/2008.11302
We associate to a projective n-dimensional toric variety X_𝛥 a pair of cocommutative (but generally non-commutative) Hopf algebras H^𝛂_X, H^T_X. These arise as Hall algebras of certain categories Coh^𝛂 (X), Coh^T (X) of coherent sheaves on X_𝛥 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 Coh^T (X) has an explicit combinatorial description as sheaves whose restriction to each A^n corresponding to a maximal cone is determined by an n-dimensional generalized skew shape. The (non-additive) categories Coh^𝛂 (X), Coh^T (X) are treated via the formalism of protoexact/proto-abelian categories developed by Dyckerhoff-Kapranov. The Hall algebras H^𝛂_X, H^T_X are graded and connected, and so enveloping algebras H^𝛂_X ≃ U(n^𝛂_X), H^T_X ≃ U(n^T_X), where the Lie algebras n^𝛂_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 = P^1, n^T_X is isomorphic to a non-standard Borel in gl_2[t, t^-1]. When X is the second infinitesimal neighborhood of the origin inside A^2, n^T_X is isomorphic to a subalgebra of gl_2[t]. We also consider the case X = P^2, where we give a basis for n^T_X by describing all indecomposable sheaves in Coh^T (X).