Toric Hall Algebras and infinite-dimensional Lie algebras
Appears in collection : Algebraic Structures in Perturbative Quantum Field Theory: a conference in honour of Dirk Kreimer's 60th birthday
The process of counting extensions in categories yields an associative (and sometimes Hopf) algebra called a Hall algebra. Applied to the category of Feynman graphs, this process recovers the Connes-Kreimer Hopf algebra. Other examples abound, yielding various combinatorial Hopf algebras. I will discuss joint work with J. Jun which attaches a Hopf algebra to a projective toric variety X. This Hopf algebra arises as the Hall algebra of a category of coherent sheaves on X locally modeled on n-dimensional skew partitions.