Jean-Louis Loday

"A duality between standard simplices and Stasheff polytopes"

The chain complexes of the standard simplices , resp. the cochain complexes of the Stasheff polytopes, assemble to give an operad. We show that these operads are Koszul dual of each other. They give rise to new types of algebras with 3 generating operations that we call associative trialgebras and dendriform trialgebras, respectively. The free dendriform trialgebra on one generator gives rise to an interesting arithmetic on the planar trees.