. R A ] 1 3 O ct 2 00 5 ON FREE ROTA – BAXTER ALGEBRAS

Most of the studies on Rota–Baxter algebras (also known as Baxter algebras) have been for commutative algebras. Free commutative Rota–Baxter algebras were constructed by Rota and Cartier in the 1970s. A later construction was obtained by Keigher and one of the authors in terms of mixable shuffles. Recently, noncommutative Rota–Baxter algebras have appeared both in physics in connection with the work of Connes and Kreimer on renormalization theory in perturbative quantum field theory, and in mathematics in connection with the work of Loday and Ronco on dendriform dialgebras and trialgebras. We give two explicit constructions of free noncommutative Rota–Baxter algebras. One construction is in terms of words for which we also give a recursive definition. The other one is in terms of angularly decorated planar rooted trees. This makes it more transparent the relation between Rota–Baxter algebras and dendriform algebras.