  For any $n \geq 3$ we obtain the decomposition in simple factors of the Lie
subalgebra of the group algebra of the symmetric group on $n$ letters generated
by the transpositions. This enables us to determine the algebraic hull of the
braid group $\B_n$ and of several of its subgroups inside the representations
of the Iwahori-Hecke algebra of type A.
