Systems of Dyson-Schwinger equations
Abstract
We consider systems of combinatorial Dyson-Schwinger equations (briefly, SDSE) X1=B+1(F1(X1,...,XN))...XN=B+N(FN(X1,...,XN)) in the Connes-Kreimer Hopf algebra HI of rooted trees decorated by I=1,...,N,where B+i is the operator of grafting on a root decorated by i, and F1...,FN are non-constant formal series.The unique solution X=(X1,...,XN) of this equation generates a graded subalgebra HS of HI. We characterize here all the families of formal series (F1,...,FN) such that HS 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 HS is Hopf is the concatenation of several fundamental or cyclic systems after the application of a change of variables, a dilatation and iterated extensions. We also describe the Hopf algebra HS as the dual of the enveloping algebra of a Lie algebra gS of one of the following types: 1. gS is a Lie algebra of paths associated to a certain oriented graph. 2. gS is an iterated extension of the Fa\`a di Bruno Lie algebra. 3. gS is an iterated extension of an abelian Lie algebra.