In mathematics, the noncommutative symmetric functions form a Hopf algebra NSymm analogous to the Hopf algebra of symmetric functions.
The Hopf algebra NSymm was introduced by Israel M. Gelfand, Daniel Krob, Alain Lascoux, Bernard Leclerc, Vladimir Retakh, and Jean-Yves Thibon.
The coproduct takes Zn to Σ Zi ⊗ Zn–i, where Z0 = 1 is the identity.
Michiel Hazewinkel showed[2] that a Hasse–Schmidt derivation on a ring A is equivalent to an action of NSymm on A: the part
The element Σ Zntn is a group-like element of the Hopf algebra of formal power series over NSymm, so over the rationals its logarithm is primitive.