Rooted trees, Feynman graphs, and Hecke correspondences

We construct natural representations of the Connes-Kreimer Lie algebras on rooted trees/Feynman graphs arising from Hecke correspondences in the categories $\text{LRF},\text{LFG}$ constructed by K. Kremnizer and the author. We thus obtain the insertion/elimination representations constructed by Connes-Kreimer as well as an isomorphic pair we term top-insertion/top-elimination. We also construct graded finite-dimensional sub/quotient representations of these arising from “truncated” correspondences.