Chapter

Higher Structures in Geometry and Physics

Volume 287 of the series Progress in Mathematics pp 269-292

Date:

The Diagonal of the Stasheff Polytope

  • Jean-Louis LodayAffiliated withInstitut de Recherche Mathématique Avancée, CNRS et Université de Strasbourg Email author 

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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.

Key words

Stasheff polytope Associahedron Operad Bar–cobar construction Cobar construction A-infinity algebra AA-infinity algebra Diagonal