2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/72905Let NSymm be the Hopf algebra of noncommutative symmetric functions over the integers. In this paper a description is given of its Lie algebra of primitives over the integers, Prim(NSymm), in terms of recursion formulas. For each of the primitives of a basis of Prim(NSymm), indexed by Lyndon words, there is a recursively given divided power series over it. This gives another proof of the theorem that the algebra of quasi-symmetric functions is free over the integers.26 pages. The primitives are described over the integers (not over a characteristic zero field, which is much easier)Quantum AlgebraCombinatorics16W30, 05E05text