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OK, The argument goes as follows : you do two or three commutative diagrams showing that morphism of bi-algebras commute with convolution (I can send you them) and end by

log(I\otimes

\begin{eqnarray} &&log(I\otimes I)=log((I\otimes e)*(e\otimes I))= log(I\otimes I))=\cr &&log(I\otimes e)+log(e\otimes I)=log(I)\otimes e +e\otimes log(I) \end{eqnarray}

This proves that log(I) maps INTO the set of primitives and to end the proof just remark that log(I)(f)=f when f is primitive because the series ends at the first term.

1

OK, The argument goes as follows : you do two or three commutative diagrams showing that morphism of bi-algebras commute with convolution (I can send you them) and end by

log(I\otimes I)=log((I\otimes e)*(e\otimes I))= log(I\otimes e)+log(e\otimes I)=log(I)\otimes e +e\otimes log(I)

This proves that log(I) maps INTO the set of primitives and to end the proof just remark that log(I)(f)=f when f is primitive because the series ends at the first term.