Pre-Lie algebras and systems of Dyson–Schwinger equations

Abstract

These lecture notes contain a review of the results of [15], [16], [17], and [19] about combinatorial Dyson–Schwinger equations and systems. Such an equation or system generates a subalgebra of a Connes–Kreimer Hopf algebra of decorated trees, and we shall say that the equation or the system is Hopf if the associated subalgebra is Hopf. We first give a classication of the Hopf combinatorial Dyson–Schwinger equations. The proof of the existence of the Hopf subalgebra uses pre-Lie structures and is different from the proof of [15] and [17]. We consider afterwards systems of Dyson-Schwinger equations. We give a description of Hopf systems, with the help of two families of special systems (quasi-cyclic and fundamental) and four operations on systems (change of variables, dilatation, extension, concatenation). We also give a few result on the dual Lie algebras. Again, the proof of the existence of these Hopf subalgebras uses pre-Lie structures and is different from the proof of [16].