Combinatorial Hopf algebras from PROs

TL;DR

This paper introduces a general method to construct Hopf algebras from stiff PROs, extending classical operad-based constructions and providing new algebraic structures with applications in combinatorics.

Contribution

It presents a novel construction that generalizes operad-to-Hopf algebra methods to stiff PROs, broadening the scope of algebraic structures derivable from PROs.

Findings

Constructs Hopf algebras from stiff PROs with precise generator-relation descriptions.

Generalizes classical operad-based Hopf algebra constructions.

Provides examples including Hopf algebras on heaps of pieces and deformed Faà di Bruno algebras.

Abstract

We introduce a general construction that takes as input a so-called stiff PRO and that outputs a Hopf algebra. Stiff PROs are particular PROs that can be described by generators and relations with precise conditions. Our construction generalizes the classical construction from operads to Hopf algebras of van der Laan. We study some of its properties and review some examples of application. We get in particular Hopf algebras on heaps of pieces and retrieve some deformed versions of the noncommutative Fa\`a di Bruno algebra introduced by Foissy.