Free rota-baxter algebras and rooted trees

Kurusch Ebrahimi, Li Guo

Research output: Contribution to journalArticlepeer-review

42 Scopus citations


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. Two constructions of free commutative Rota-Baxter algebras were obtained by Rota and Cartier in the 1970s and a third one by Keigher and one of the authors in the 1990s 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. This paper uses rooted trees and forests to give explicit constructions of free noncommutative Rota-Baxter algebras on modules and sets. This highlights the combinatorial nature of Rota-Baxter algebras and facilitates their further study. As an application, we obtain the unitarization of Rota-Baxter algebras.

Original languageEnglish (US)
Pages (from-to)167-194
Number of pages28
JournalJournal of Algebra and its Applications
Issue number2
StatePublished - 2008

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Applied Mathematics


  • Angular decoration
  • Free objects
  • Rooted trees
  • Rota-Baxter algebras
  • Unitarization


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