Feynman graphs, rooted trees, and Ringel-Hall algebras

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.

References in corpus (5)

• Hopf Algebras, Renormalization and Noncommutative Geometry
• Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem
• Stability conditions and Stokes factors
• Combinatorics of Rooted Trees and Hopf Algebras
• Growth estimates for Dyson-Schwinger equations

Cited by in corpus (7)

• Renormalization and computation I: motivation and background
• Representations of Quivers over F1
• Algebra for quantum fields
• Pre-Lie algebras and Incidence Categories of Colored Rooted Trees
• Rooted trees, Feynman graphs, and Hecke correspondences
• Colored trees and noncommutative symmetric functions
• The Hopf algebra of skew shapes, torsion sheaves on A^n/F_1, and ideals in Hall algebras of monoid representations