Daria Poliakova: Koszul self-duality phenomenon: from algebras to operads and polytopes

Abstract.

For two Koszul-dual algebras \(A\) and \(B\) their Poincare–Hilbert series satisfy the equation \(f_A(t) f_B(-t) = 1\). The same holds, when \(A\) and \(B\) have multiple objects; \(f_A\) and \(f_B\) are then interpreted as series-valued matrices. Koszul self-dual algebras are rare and of particular interest. A source of examples are polytopes: their incidence algebras are such, as shown by Patrick Polo (1995).

The theory was extended from algebras to operads by Ginzburg–Kapranov (1994); Poincare–Hilbert series of Koszul-dual operads are again (modulo signs) inverse to each other, but with respect to composition instead of product. In case of multiple objects, Poincare–Hilbert series are replaced by endomorphisms of \(T(V)\), where \(V\) is a vector space generated by colors of the operads. For a Koszul self-dual operad this gives an involution of \(T(V)\).

I explain how short directed polytopes should be a source of Koszul self-dual operads, with colors given by faces, and unary operations given by the incidence algebra. For simplices, self-duality is proved using operadic Groebner bases of Dotsenko–Khoroshkin (2010).

The motivation of this research is to extend Saneblidze–Umble (2002) diagonal for associahedra to a coherent structure, which I will explain, if the time permits.