nLab Rota-Baxter algebra

RotaBaxter algebras

Rota–Baxter algebras


Given a commutative unital ring kk, an associative unital kk-algebra AA is a Rota–Baxter algebra of weight λ\lambda if it is equipped with a Rota–Baxter operator P:AAP\colon A \to A, which is a kk-linear endomorphism such that

P(x)P(y)=P(P(x)y)+P(xP(y))+λP(xy) P(x) P(y) = P(P(x) y) + P(x P(y)) + \lambda P(x y)

Historical sources and motivation

Rota–Baxter algebras came historically from several sources, including enumerative combinatorics problems (Gian-Carlo Rota), qq-calculus (Jackson integral) and integrable systems (Baxter). More recently the application to the combinatorics of renormalization and Birkhoff decomposition have been found, showing that the essence of renormalization is not that crucially existing only in the presence of complex analysis.


A large bibliography on Rota–Baxter algebras can be found at Li Guo’s Rota–Baxter algebra page.

  • K. Ebrahimi-Fard, L. Guo, Rota-Baxter algebras and dendriform dialgebras, Jour. Pure Appl. Algebra 212 (2008), 320-339, arXiv:math/0503647

  • Kurusch Ebrahimi-Fard, Li Guo?, Rota–Baxter algebras in renormalization of perturbative quantum field theory, In Universality and Renormalization, I. Binder and D. Kreimer, editors, Fields Institute Communicatins, v. 50, AMS 2007, 47-105, arXiv:hep-th/0604116.

  • K. Ebrahimi-Fard, L. Guo, D. Manchon, Birkhoff type decompositions and the Baker–Campbell–Hausdorff recursion, Comm. Math. Physics 267 (2006) 821-845, arXiv:math-ph/0602004

  • Jun Pei, Chengming Bai, Li Guo, Rota-Baxter operators on sl(2,C)sl(2,C) and solutions of the classical Yang-Baxter equation, arxiv/1311.0612

A generalization of Rota-Baxter operators to algebraic operads is found in connection to certain splitting phenomenon for operads:

  • Jun Pei, Chengming Bai, Li Guo, Splitting of operads and Rota-Baxter operators on operads, arxiv/1306.3046

There is also a categorification of the concept of Rota–Baxter algebra, namely the Rota–Baxter category, cf.

Among motivating results, there are also Spitzer identities from

  • F. Spitzer, A combinatorial lemma and its application to probability theory, Trans. Amer. Math. Soc. 82 (1956), 323–339, MR0079851, doi
category: algebra

Last revised on May 26, 2023 at 08:52:51. See the history of this page for a list of all contributions to it.