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Insertion and elimination: The doubly infinite Lie algebra of Feynman graphs. (English) Zbl 1033.81061
The aim is to algebraically describe the operations of elimination and insertion of subgraphs in the context of Feynman graphs. This is exhibited by identitfying such operations with representations of certain Hopf algebras and Lie algebra associated to Feynman graphs. The authors use their previous groundbreaking work regarding the Hopf algebras $${\mathcal H}_{\text{cm}}$$ and $${\mathcal H}_{\text{rt}}$$ introduced in [Commun. Math. Phys. 199, 203-242 (1998; Zbl 0932.16038)]. The insertions and eliminations do not commute, so the authors construct a larger Lie algebra which is studied in detail.
This work is important because the algebraic structures provided cover all operations in the perturbative expansion of a QFT.

##### MSC:
 81T18 Feynman diagrams 81T15 Perturbative methods of renormalization applied to problems in quantum field theory 81T05 Axiomatic quantum field theory; operator algebras 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 17B81 Applications of Lie (super)algebras to physics, etc. 81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
##### Keywords:
quantum field theory; Hopf algebras; Feynman graphs
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