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

Rota–Baxter algebras came historically from several sources, including enumerative combinatorics problems (Gian-Carlo Rota), $q$-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.

References

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

K. Ebrahimi-Fard, L. 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 $sl(2,C)$ 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.

Edmundo Castillo, Rafael Diaz, Rota-Baxter categories, arxiv/0706.1068

Among motivatring 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