Confluence laws and Hopf-Borel type theorem for operads

Abstract

In 2008, Loday shed light on the existence of Hopf-Borel theorems for operads. Using the vocabulary of category theory, Livernet, Mesablishvili and Wisbauer extended such theorems to monads. In both cases, the reasoning was to start from a mixed distributive law and then to prove that it induces an isomorphism of S-modules to finally get a rigidity theorem. Our reasoning goes here backward: we prove that from an isomorphism ofS-modules one can get what we called a confluence law, which generalises mixed distributive laws, and that it is enough to obtain a rigidity theorem. This enables us to show that for any operads P and Q having the same underlying S-module, there exists a confluence law α such that any conilpotent P coQ-bialgebra satisfying α is free and cofree over its primitive elements. Our reasoning permits us to generate many new examples.

Nonsymmetric operads in combinatorics

“…In [Lod08], Loday defined the notion of triples of operads, leading to the constructions of various kinds of bialgebras. This leads also to the discovery of analogs of the Poincaré-Birkhoff-Witt and Cartier-Milnor-Moore theorems and rigidity theorems (see as well [Cha02] and [BDO18]). Loday defined among others infinitesimal bialgebras, forming an example of bialgebras having an associative binary product and a coassociative binary coproduct satisfying a compatibility relation.…”