The primitives of the Hopf algebra of noncommutative symmetric functions

Abstract

Let NSymm be the Hopf algebra of noncommutative symmetricfunctions over the integers. In this paper a description is givenof its Lie algebra of primitives over the integers, Prim(NSymm), interms of recursion formulas. For each of the primitives of a basis ofPrim(NSymm), indexed by Lyndon words, there is a recursively givendivided power series over it. This gives another proof of the theoremthat the algebra of quasi-symmetric functions is free over the integers.