The primitives of the Hopf algebra of noncommutative symmetric functions. (Q2390129)

The non-commutative Hopf algebra NSymm of symmetric functions or the Leibniz Hopf algebra is a free associative ring with countable many free generators \(Z_1,Z_2,\dots\). Comultiplication in NSymm is defined by the rule: \(\Delta(Z_n)=\sum_{i=0}^nZ_i\otimes Z_{n-i}\) where \(Z_0=1\). The aim of the paper is to find the Lie ring P(Nsymm) of primitive elements in NSymm. The Lie ring P(Nsymm) contains a free Lie subring \(FL(P)\) freely generated by Newton primitives \[ P_n(Z)=\sum_{r_1+\cdots+r_k=n,\;r_i>0}(-1)^{k+1}Z_{r_1}\cdots Z_{r_k}. \] There is found the index of the homogeneous part of \(FL(P)_n\) in \(\text{NSymm}_n\) for \(n=1,\dots,6\). Generalizing Newton primitives the author introduces homogeneous primitives \(P_\alpha\) parameterized by Lyndon words \(\alpha\). It is shown that primitives \(P_\alpha\) form a base of P(Nsymm) as a free additive Abelian group. -- It is also shown that the graded dual of NSymm, the ring QSymm of quasisymmetric functions, is a free commutative ring.