FREE ROTA–BAXTER ALGEBRAS AND ROOTED TREES

By Kurusch Ebrahimi-Fard, Li Guo. 2005

71 citations recorded by the source corpus.

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, w...

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/50a26d0be663896adb2454ea79c6b903b05e2f25

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Citations in the corpus

Grobner-Shirshov Bases for Associative Algebras with Multiple Operators and Free Rota-Baxter Algebras ∗† (2008, background): “Now, we describe an element in S(X) by a labeled reduced planar rooted forest, see [27] and [ 30 ].”
Composition-Diamond lemma for associative $n$-conformal algebras (2004, background): “Rota-Baxter algebras (the latter is associative algebras with linear operation P(x) and the identity P(x)P(y) = P(P(x)y)+ P(xP(y))+ λP(xy), where λ is a ﬁx element of a ground ﬁeld, see, for example, [31]). Composition-Diamond Lemma for dialgebras [15] has an application to the PBW theorem for unive”
On Differential Rota-Baxter Algebras (2007, methodology): “Both constructions rely on the explicit construction of free Rota-Baxter algebras, in the commutative case in [18, 19] and in the noncommutative case in [2, 12, 13 ].”
Quillen homology for operads via Gröbner bases (2012, methodology): “We refer the reader to the paper of Ebrahimi–Fard and Guo [18] for an extensive discussion of applications and occurrences of those algebras in various areas of mathematics, and a combinatorial construction of the corresponding free algebras.”