Trialgebras and families of polytopes

TL;DR

This paper explores duality and Koszul properties of operads related to simplices, Stasheff polytopes, and cubes, introducing new algebraic structures called trialgebras with potential applications in algebraic topology.

Contribution

It establishes duality and Koszulity of operads associated with polytopes and introduces novel trialgebra structures with specific operations and relations.

Findings

Standard simplices and Stasheff polytopes form dual Koszul operads.

Introduction of associative and dendriform trialgebras with specific relations.

Cubes give rise to a self-dual operad under Koszul duality.

Abstract

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. As a consequence the generating series of the standard simplices and the generating series of the Stasheff polytopes are inverse to each other. The two operads give rise to new types of algebras with 3 generating operations, 11 relations, respectively 7 relations, that we call {\it associative trialgebras} and {\it dendriform trialgebras} respectively. The free dendriform trialgebra, which is based on planar trees, has an interesting Hopf algebra structure, which will be dealt with in another paper. Similarly the family of cubes gives rise to an operad which happens to be self-dual for Koszul duality.