New Hopf Structures on Binary Trees (Extended Abstract)

TL;DR

This paper explores new Hopf algebra structures on binary trees derived from multiplihedra, revealing novel combinatorial and algebraic relationships between well-known polytopes and algebraic objects.

Contribution

It introduces new Hopf structures on tree-like objects related to multiplihedra, connecting them with permutahedra and associahedra through module constructions.

Findings

Constructed a module over the Malvenuto-Reutenauer Hopf algebra

Developed a Hopf module over the Loday-Ronco algebra

Discovered new polytopes and a factorization of a projection from associahedra to hypercubes

Abstract

The multiplihedra {M_n} form a family of polytopes originating in the study of higher categories and homotopy theory. While the multiplihedra may be unfamiliar to the algebraic combinatorics community, it is nestled between two families of polytopes that certainly are not: the permutahedra {S_n} and associahedra {Y_n}. The maps between these families reveal several new Hopf structures on tree-like objects nestled between the Malvenuto-Reutenauer (MR) Hopf algebra of permutations and the Loday-Ronco (LR) Hopf algebra of planar binary trees. We begin their study here, constructing a module over MR and a Hopf module over LR from the multiplihedra. Rich structural information about this module is uncovered via a change of basis--using M\"obius inversion in posets built on the 1-skeleta of the {M_n}. Our analysis uses the notion of an interval retract, which should have independent interest in poset combinatorics. It also reveals new families of polytopes, and even a new factorization of a known projection from the associahedra to hypercubes.