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Classification of systems of Dyson-Schwinger equations in the Hopf algebra of decorated rooted trees. (English) Zbl 1234.16024

Summary: We consider systems of combinatorial Dyson-Schwinger equations (briefly, SDSE) \(X_1=B_1^+(F_1(X_1,\dots,X_N)),\dots,X_N=B_N^+(F_N(X_1,\dots,X_N))\) in the Connes-Kreimer Hopf algebra \(\mathcal H_I\) of rooted trees decorated by \(I=\{1,\dots,N\}\), where \(B_i^+\) is the operator of grafting on a root decorated by \(i\), and \(F_1,\dots,F_N\) are non-constant formal series. The unique solution \(X=(X_1,\dots,X_N)\) of this equation generates a graded subalgebra \(\mathcal H_{(S)}\) of \(\mathcal H_I\). We characterise here all the families of formal series \((F_1,\dots,F_N)\) such that \(\mathcal H_{(S)}\) is a Hopf subalgebra. More precisely, we define three operations on SDSE (change of variables, dilatation and extension) and give two families of SDSE (cyclic and fundamental systems), and prove that any SDSE \((S)\) such that \(\mathcal H_{(S)}\) is Hopf is the concatenation of several fundamental or cyclic systems after the application of a change of variables, a dilatation and iterated extensions.

MSC:

16T30 Connections of Hopf algebras with combinatorics
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
81T18 Feynman diagrams
05C05 Trees
17B81 Applications of Lie (super)algebras to physics, etc.
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