Toric Hall algebras and infinite-dimensional Lie algebras

Abstract

We associate to a projective n-dimensional toric variety X∆ 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∆ 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, 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, whenWe 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).

Algebraic K-theory and Grothendieck-Witt theory of monoid schemes

“…In the other direction, monoid schemes can be seen as a direct generalization of toric geometry and Kato fans of logarithmic schemes; see [26], [8], [5], [2], [7] among others. The central position of monoid schemes within F 1 -geometry is confirmed by the multiple links to other disciplines, such as Weyl groups as algebraic groups over F 1 [28], computational methods for toric geometry [6], [7], [13], a framework for tropical scheme theory [14], applications to representation theory [41], [19] and, last but not least, stable homotopy theory as K-theory over F 1 [9], [2], the theme on which we dwell in this paper.…”

Proto-exact categories of modules over semirings and hyperrings

“…• in perspective of algebraic geometry over "the field with one element" along with other interesting examples. Also, the first and the second authors further explored certain categories of coherent sheaves arising from algebraic geometry over monoids in [JS20b].…”

“…1 Several interesting "combinatorial" categories are equipped with a proto-exact structure, for instance, the category of matroids [EJS20], the category of representations over a quiver (and more generally any monoid) over "the field with one element" [Szc12], [JS20a], [JS21]. Categories with more algebro-geometric flavors, which are not additive, have been explored in [Szc18], [JS20b], [ELY20].…”

Coefficient Quivers, $\mathbb{F}_1$-Representations, and Euler Characteristics of Quiver Grassmannians

“…We call them coefficient quivers due to their resemblance to the notion of coefficient quivers for quiver representations introduced by Ringel [Rin98]. 4 Szczesny explored several aspects of representation theory over F 1 in [Szc12,Szc14,Szc18,JS20b]. In particular, he introduced a notion of quiver representations over F 1 based on an idea that vector spaces over F 1 are finite pointed sets.…”

“…In fact, Szczesny's main observation was that the Hall algebra H Q,F 1 of Rep(Q, F 1 ) behaves in some ways like the specialization at q = 1 of the Hall algebra H Q,F q of Rep(Q, F q ). 5 This line of ideas was further pursued with the first named author in [JS20b] to compute the Hall algebra of coherent sheaves on P 2 by using its degenerate combinatorial model of monoid schemes.…”