From polytopes to operads and back

TL;DR

This paper constructs colored operads from directed polytopes, linking their cellular complexes to algebraic structures, and verifies conjectures about their Koszul and self-dual properties for specific polytope classes.

Contribution

It introduces a novel method to associate operads with directed polytopes and proves their Koszul and self-dual properties for simplices, polygons, and their products.

Findings

Operads encode operations on polytope cellular complexes.

Conjecture that certain polytope-derived operads are Koszul and self-dual.

Verification of the conjecture for simplices, polygons, and products.

Abstract

For a directed polytope, we construct a colored operad whose Poincare-Hilbert series encodes certain operations on the cellular complex of the polytope. We conjecture that for a class of short polytopes the constructed operads are Koszul and self-dual. We verify the conjecture for simplices, polygons, and products thereof.