The diagonal of the Stasheff polytope

TL;DR

This paper develops a new A-infinity algebra structure on tensor products using the simplicial decomposition of the Stasheff polytope, introducing a novel operad with a coassociative diagonal.

Contribution

It constructs an operad based on the simplicial Stasheff polytope that admits a coassociative diagonal and relates it to existing cubical decomposition methods.

Findings

Constructed an operad AA-infinity with a coassociative diagonal.

Established a deformation retract of A-infinity from AA-infinity.

Compared simplicial and cubical decomposition approaches.

Abstract

We construct an A-infinity structure on the tensor product of two A-infinity algebras by using the simplicial decomposition of the Stasheff polytope. The key point is the construction of an operad AA-infinity based on the simplicial Stasheff polytope. The operad AA-infinity admits a coassociative diagonal and the operad A-infinity is a retract by deformation of it. We compare these constructions with analogous constructions due to Saneblidze-Umble and Markl-Shnider based on the Boardman-Vogt cubical decomposition of the Stasheff polytope.