When do the solutions of combinatorial Dyson-Schwinger equations generate a Hopf subalgebra?

Say I have a set of combinatorial Dyson-Schwinger equations of the form

\begin{align} X_1 &= \mathbb{1} + \alpha B_+^a (f_1(X_1,...X_N)) \\ & ... \tag{1} \\ X_N &= \mathbb{1} + \alpha B_+^n (f_1(X_1,...X_N)) \end{align}

that has an unique solution, solved by using the ansatz

$$X_j = \mathbb{1} + \sum_k^\infty c_{j,k} \alpha^k \tag{2}$$

How can I check if the elements $$c_{j,k}$$ form a Hopf subalgebra, by use of the coproduct? The coproduct is defined by

$$\Delta \circ B^i_+ = B^i_+ \otimes \mathbb{1} + (\text{id} \otimes B^i_+) \Delta \tag{3}$$

Thank you in advance.

• Is this from some reference? Looks like work of Connes & Kreimer. Which page? – Qmechanic Jun 18 at 22:51
• @Qmechanic Yes that's from Kreimer precisely. – Jxx Jun 19 at 18:58
• I think I found my answer, although I am not completely sure yet. It seems that a sufficient condition would be that the coproduct delivers terms only of the form $c_{i,k} \otimes c_{j,k}$ with $i,j = 1,...,N$ and $= 1,...,\infty$, or in other words that it is closed. It would be great if someone could confirm or contradict this assumption. – Jxx Jun 19 at 21:30