Trialgebras and families of polytopes

We show that the family of standard simplices and the family of Stasheff polytopes are dual to each other in the following sense. The chain modules of the standard simplices, resp. the Stasheff polytopes, assemble to give an operad. We show that these operads are dual of each other in the operadic sense. The main result of this paper is to show that they are both Koszul operads. This theorem appears as a refinement of the functional equation which relates the generating series of the standard simplices to the generating series of the Stasheff polytopes. The two operads give rise to new types of algebras with 3 generating operations, 11 relations, respectively 7 relations, that we call associative trialgebras and dendriform trialgebras respectively. Similarly, the family of cubes gives rise to an operad which happens to be self-dual for Koszul duality. Introduction. We introduce a new type of associative algebras characterized by the fact that the associative product ∗ is the sum of three binary operations : x ∗ y := x≺ y + x y + x · y , MSC 2000: 05C05, 17A50, 17D99,18D50, 18G35, 18G60.