Let \({\mathcal P}=\bigoplus_{n\geq 0}{\mathbb Z}{\mathcal S}_n\) be the direct sum of the group rings of the symmetric groups, with the Hopf algebra structure introduced by Malvenuto and Reutenauer in 1995. The algebra \(\mathcal P\) has several subalgebras interesting both from the algebraic and combinatorial point of view. In the paper under review the author investigates the relation between the Lie convolution subalgebra \(\mathcal L\) (generated as an algebra by the Lie elements in \(\mathcal P\)) and the coplastic algebra \(\mathcal Q\) (defined combinatorially as the linear span of the sums of permutations with given Schensted \(Q\)-symbol, or, equivalently, of the sums of equivalence classes arising from the coplastic relations in \({\mathcal S}_n\) introduced by Knuth).
The main result of the paper is that the intersection of \(\mathcal L\) and \(\mathcal Q\) is equal to the direct sum \(\mathcal D\) of the Solomon descent algebras. The proof is essentially based on the fact that any Lie element in \(\mathcal P\) which is constant on coplastic classes is already contained in \(\mathcal D\), and this is combinatorially interesting for its own sake.