Primitive elements in a free dendriform algebra (Q2708918)

Let \(K\) be a field of characteristic zero. For \(n\geq 1\) and \(X\) a finite set, let \(Y_{n,X}\) be the set of planar binary trees with \(n+1\) leaves, whose vertices are colored with the elements of \(X\). The graded space \(\overline{K[Y_{\infty,X}]}=\sum_{n\geq 1}\oplus K[Y_{n,X}]\) has a natural structure of a Hopf algebra, in general neither commutative nor cocommutative. \textit{J.-L. Loday} showed that \(\overline{K[Y_{\infty,X}]}\) is the free dendriform algebra on \(X\), now denoted \(\text{Dend}(X)\) [Dialgebras, Prépublication de l'Inst. de Recherche Math. Avancée (Strasbourg) 14 (1999)]. A dendriform algebra is an associative algebra whose multiplication is the sum of two binary operations with certain properties. The paper under review describes the primitive elements \(P(X)\) of \(\text{Dend}(X)\), which turn out to be the span of certain \(n\)-ary linear operations on \(\text{Dend}(X)\), for all \(n\geq 1\). The main theorem of the article is that \(\text{Dend}(X)\) and the cotensor algebra \(\overline T(P(X))\) are isomorphic as coalgebras.NEWLINENEWLINEFor the entire collection see [Zbl 0955.00038].