Infinitesimal or cocommutative dipterous bialgebras and good triples of operads

TL;DR

This paper explores new good triples of operads involving dipterous algebras, providing explicit constructions and connecting them to rooted trees, Schroeder numbers, and applications in quantum field theory.

Contribution

It introduces and proves the goodness of new triples of operads with dipterous structures, including semi-infinitesimal compatibility relations and explicit free algebra constructions.

Findings

The triple (As, Dipt, Grove) with semi-infinitesimal relations is good.

Explicit free dipterous and grove-algebra constructions are provided.

Connections to rooted trees, Schroeder numbers, and quantum field theory are established.

Abstract

The works of Poincare, Birkhoff, Witt and Cartier, Milnor, Moore on the connected cocommutative Hopf algebras translated in the language of operads means that the triple of operads (Com, As, Lie) endowed with the Hopf compatiblity relation is good. In this paper, we focus on left dipterous (resp. right dipterous) algebras which are associative algebras with an extra left (resp. right) module on themselves and look for good triples were $As$ is replaced by the dipterous operad Dipt. Since the work of Loday and Ronco, the triple of operads (As, Dipt, B_\infty) endowed with the semi-Hopf compatibility relations is known to be good. In this paper, we prove that the triple of operads (As, Dipt, Grove) endowed with the so-called (nonunital) semi-infinitesimal compatibility relations is good. For that, explicit constructions of the free dipterous algebra and the free grove-algebra over a K-vector space V are given. These constructions turn out to be related to rooted planar trees and the little an large Schroeder numbers. Many examples of dipterous algebras are given, notably the free L-dipterous algebras. As a corollary of our results, we also recover that the triple of operads (2As, Dipt, Vect) endowed both with the unital semi-Hopf and with the unital semi-infinitesimal compatibility relations is good, where 2As denotes the operad of 2-associative algebras. We also open this paper on a good triple, related to the Connes-Kreimer Hopf algebra in quantum field theory, (Com, Dipt, Prim_{Com} Dipt) endowed with the Hopf compatibility relations and also present a general theorem giving good triples from entangled dipterous like operads named associative molecules.