Feynman graphs, rooted trees, and Ringel-Hall algebras

TL;DR

This paper constructs categories of rooted forests and Feynman graphs, linking them to Ringel-Hall algebras and providing a new algebraic interpretation of Connes-Kreimer Lie algebras.

Contribution

It introduces symmetric monoidal categories of rooted forests and Feynman graphs, connecting them to Ringel-Hall algebras and offering a novel perspective on Connes-Kreimer Lie algebras.

Findings

Ringel-Hall Hopf algebras are dual to Connes-Kreimer Hopf algebras.

Connes-Kreimer Lie algebras are interpreted as Ringel-Hall Lie algebras.

Categories resemble finitary abelian categories, enabling algebraic constructions.

Abstract

We construct symmetric monoidal categories $\LRF, \FD$ of rooted forests and Feynman graphs. These categories closely resemble finitary abelian categories, and in particular, the notion of Ringel-Hall algebra applies. The Ringel-Hall Hopf algebras of $\LRF, \FD$, $\HH_{\LRF}, \HH_{\FD}$ are dual to the corresponding Connes-Kreimer Hopf algebras on rooted trees and Feynman graphs. We thus obtain an interpretation of the Connes-Kreimer Lie algebras on rooted trees and Feynman graphs as Ringel-Hall Lie algebras.