ar X iv : m at h / 05 10 26 6 v 3 [ m at h . R A ] 2 1 Fe b 20 06 ON FREE ROTA – BAXTER ALGEBRAS

A Rota–Baxter algebra, also known as a Baxter algebra, is an algebra with a linear operator satisfying a relation, called the Rota–Baxter relation, that generalizes the integration by parts formula. Most of the studies on Rota–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 in perturbative quantum field theory, and in mathematics related to the work of Loday and Ronco on dendriform dialgebras and trialgebras. We give explicit constructions of free noncommutative Rota–Baxter algebras in various contexts. Our strategy is to obtain these free objects from Rota–Baxter algebra structures on classes of trees and forests. Elements of free Rota–Baxter algebras are expressed in terms of angularly decorated planar rooted forests. This furthers our understanding of free Rota–Baxter algebras and facilitates their further study. Such a forest interpretation then translates naturally into one in terms of bracketed words, which has already appeared in special cases in our recent study of dendriform algebras, thereby making the relation between Rota–Baxter algebras and dendriform algebras more transparent.