Insertion and Elimination: the doubly infinite Lie algebra of Feynman graphs

TL;DR

This paper explores the Lie algebra structure formed by insertion and elimination operations on Feynman graphs, revealing a complex algebraic framework that extends the understanding of graph manipulations in quantum field theory.

Contribution

It characterizes the larger Lie algebra generated by insertion and elimination derivations on Feynman graphs, providing a detailed algebraic structure of these operations.

Findings

Insertion and elimination derivations form a non-commutative Lie algebra.

The algebraic structure extends the known symmetries of Feynman graphs.

Provides a foundation for further algebraic analysis of quantum field theory graphs.

Abstract

The Lie algebra of Feynman graphs gives rise to two natural representations, acting as derivations on the commutative Hopf algebra of Feynman graphs, by creating or eliminating subgraphs. Insertions and eliminations do not commute, but rather establish a larger Lie algebra of derivations which we here determine.