International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 49, Pages 2595-2615

Construction of Nijenhuis operators and dendriform trialgebras

Philippe Leroux

Institut de Recherche Mathématique de Rennes, Université de Rennes I and CNRS UMR 6625, Campus de Beaulieu, Rennes Cedex 35042, France

Received 16 February 2004

Copyright © 2004 Philippe Leroux. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.