Graftings and cograftings for species

Abstract

We introduce graftings and cograftings for vector species, modelled on the elementary operations of adjoining and deleting a root in a rooted forest. We relate these maps to operads, up and down operators, and obtain nontrivial balanced pairs, admitting pair of creation/annihilation operators with commuting relations. We establish universal properties for the species of rooted forests. Using Fock functors, we recover the classical Connes--Kreimer Hopf algebra, its noncommutative planar analogue, and Foissy's $q$-infinitesimal Hopf algebra.

arXiv comment45 pages, 13 figures