[Paper Review] Trialgebras and families of polytopes

Jean-Louis Loday, Marı́a Ronco|arXiv (Cornell University)|May 6, 2002

Advanced Topics in Algebra20 citations

TL;DR

This paper establishes a duality between the operads of standard simplices and Stasheff polytopes (associahedra) via Koszul duality, proving both are Koszul operads. It introduces associative and dendriform trialgebras—algebras with three binary operations satisfying 11 and 7 relations, respectively—whose free algebras are generated by cells of simplices and associahedra. The key result is that the generating series of the two operads are formal inverses, with the generating series of the Stasheff polytopes expressed as a root of a quadratic equation in terms of the simplex series.

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.

Motivation & Objective

• To establish a Koszul duality between the operads of standard simplices and Stasheff polytopes.
• To define and study associative and dendriform trialgebras as algebras with three binary operations satisfying specific associativity-like relations.
• To prove that the operads associated with these trialgebras are Koszul, implying the generating series of their respective polytope families are formal inverses.
• To extend the framework to cubical trialgebras, showing the associated operad is self-dual and also Koszul.

Proposed method

• Constructing the free associative trialgebra on one generator as the chain complex of standard simplices, with operations defined via face and degeneracy maps.
• Defining dendriform trialgebras via three operations (≺, ≻, ·) satisfying 7 relations, and showing their free algebra is isomorphic to the algebra of planar trees.
• Using the duality between the operads of simplices and associahedra to prove that the associated Koszul complex is acyclic, implying Koszulness.
• Proving acyclicity by constructing explicit deformation retractions of simplicial complexes associated to the free algebras.
• Extending the framework to cubical trialgebras by introducing three operations (≺, ·, ≻) satisfying 9 relations, and showing the resulting operad is self-dual.
• Verifying the Koszul property of the cubical trialgebra operad by showing its homology is concentrated in degree 1, using the chain complex of hypercubes.

Experimental results

Research questions

• RQ1 Are the operads of standard simplices and Stasheff polytopes Koszul dual to each other?
• RQ2 Can the free dendriform trialgebra be realized as an algebra over planar trees, and does this give rise to a Hopf algebra structure?
• RQ3 What is the relationship between the generating series of the operads of simplices and associahedra, and how does Koszul duality manifest in this context?
• RQ4 Is there a self-dual operad associated with the family of hypercubes, and if so, what are its algebraic and homological properties?
• RQ5 Can the Koszul complex of the cubical trialgebra operad be shown to be acyclic, and what does this imply about the operad's Koszulity?

Key findings

• The operads of standard simplices and Stasheff polytopes are Koszul dual to each other, as established by the acyclicity of the Koszul complex associated with the dendriform trialgebra.
• The generating series of the Stasheff polytopes, $ f^{ ilde{ ext{K}}}_t(x) $, is the compositional inverse of the generating series of the standard simplices, $ f^{ ilde{ ext{Δ}}}_t(x) $, satisfying $ f^{ ilde{ ext{Δ}}}_t(f^{ ilde{ ext{K}}}_t(x)) = x $.
• The generating series of the Stasheff polytopes is explicitly given by $ f^{ ilde{ ext{K}}}_t(x) = \frac{-(1+(2+t)x) + \sqrt{1+2(2+t)x + t^2x^2}}{2(1+t)x} $, derived from the Poincaré polynomial of the associahedra.
• The free associative trialgebra on one generator is linearly generated by the cells of the standard simplices, with the chain complex isomorphic to the normalized chain complex of $ \Delta^{n-1} $, which is acyclic.
• The cubical trialgebra operad is self-dual, and its generating series $ f^I_t(x) = \frac{-x}{1 + (2+t)x} $ is its own compositional inverse, confirming its Koszul property.
• The homology of the free cubical trialgebra on a vector space $ V $ is $ H_n = V $ if $ n=1 $, and 0 otherwise, proving the Koszul property of the cubical operad via acyclicity of the associated chain complex.