Construction of Nijenhuis operators and dendriform trialgebras

We construct Nijenhuis operators from particular bialgebras called dendriform‐Nijenhuis bialgebras. It turns out that such Nijenhuis operators commute with TD‐operators, a kind of Baxter‐Rota operators, and are therefore closely related to dendriform trialgebras. This allows the construction of associative algebras, called dendriform‐Nijenhuis algebras, made out of nine operations and presenting an exotic combinatorial property. We also show that the augmented free dendriform‐Nijenhuis algebra and its commutative version have a structure of connected Hopf algebras. Examples are given.