Higher Lie idempotents

Let T (X) be the tensor bialgebra over an alphabet X. It is a graded connected cocommutative bialgebra, canonically isomorphic to the envelopping bialgebra of the free Lie algebra over X, Lie(X). The subalgebra of its convolution algebra generated by the projections arising from the graduation is also an algebra for the composition of morphisms and is anti-isomorphic as such with the direct sum of the Solomon’s symmetric groups descent algebras [R2]. For example, the canonical projections arising from the isomorphism between T (X) and the envelopping bialgebra of Lie(X) belong to this convolution algebra and may be identified with certain idempotents of the symmetric groups algebras (see [GR] [R2]). More generally, over a field of caracteristic zero, any connected graded cocommutative bialgebra A is canonically isomorphic to the envelopping bialgebra of the Lie algebra of its primitive elements (Cartier-Milnor-Moore theorem, see [MM]). As for the tensor algebra, this isomorphism has a combinatorial description and the