Communications in Mathematical Physics

, Volume 327, Issue 1, pp 151–179 | Cite as

General Dyson–Schwinger Equations and Systems

  • Loïc FoissyEmail author


We classify combinatorial Dyson–Schwinger equations giving a Hopf subalgebra of the Hopf algebra of Feynman graphs of the considered Quantum Field Theory. We first treat single equations with an arbitrary (eventually infinite) number of insertion operators. We distinguish two cases; in the first one, the Hopf subalgebra generated by the solution is isomorphic to the Faà di Bruno Hopf algebra or to the Hopf algebra of symmetric functions; in the second case, we obtain the dual of the enveloping algebra of a particular associative algebra (seen as a Lie algebra). We also treat systems with an arbitrary finite number of equations, with an arbitrary number of insertion operators, with at least one of degree 1 in each equation.


Hopf Algebra Rooted Tree Formal Series Feynman Graph Insertion Operator 
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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Laboratoire de MathématiquesUniversité de ReimsReims Cedex 2France

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