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Free Rota-Baxter algebras and rooted trees

机译：免费的Rota-Baxter代数和有根的树

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摘要

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.
机译：Rota-Baxter代数，也称为Baxter代数，是一种线性运算符，满足满足称为Rota-Baxter关系的关系的代数，该关系通过零件公式来概括积分。 Rota-Baxter代数的大多数研究都是针对可交换代数的。关于自由混合Rota-Baxter代数的两种构造是由Rota和Cartier在1970年代获得的，而第三种是由Keigher和一位作者在1990年代提出的，涉及可混合混洗。近来，非可交换的Rota Baxter代数在物理学中出现，涉及到Connes和Kreimer在扰动量子场理论中的重归一化的工作，以及在与Loday和Ronco在树状形代数和三阶代数的工作有关的数学中。本文使用有根的树木和森林在模块和集合上给出自由的非交换Rota-Baxter代数的显式构造。这突显了Rota-Baxter代数的组合性质，并促进了它们的进一步研究。作为应用，我们获得了Rota-Baxter代数的单位化。

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