Lie idempotents and decompositions in the tensor algebra.

A classical problem in the study of words (or tensors) is to provide effective formulas implementing the Poincaré-Birkhoff-Witt isomorphism. The canonical solution is given by the works of Solomon and Garsia-Reutenauer (from the late 60's to the mid-80's), featuring Solomon's idempotent, but further progress was done later (eg as far as other Lie idempotents are concerned). We will survey some classical features of the problem and some recent developments on the subject (including possibly joint works with Reutenauer, Schocker -links with the theory of species and operads- and some applications to quantum field theory -joint works with Ebrahimi-Fard, Gracia-Bondia, Manchon).