Rooted trees, Feynman graphs, and Hecke correspondences

TL;DR

This paper constructs natural representations of Connes-Kreimer Lie algebras using Hecke correspondences in categories related to rooted trees and Feynman graphs, unifying existing and new representation frameworks.

Contribution

It introduces a novel approach to representing Connes-Kreimer Lie algebras via Hecke correspondences, including new isomorphic and truncated representations.

Findings

Constructed natural representations from Hecke correspondences.

Unified insertion/elimination and top-insertion/top-elimination representations.

Developed finite-dimensional sub/quotient representations from truncated correspondences.

Abstract

We construct natural representations of the Connes-Kreimer Lie algebras on rooted trees/Feynman graphs arising from Hecke correspondences in the categories $\LRF, \LFG$ 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.