Commutative Rota-Baxter Algebras Research Articles

Commutative matching Rota-Baxter operators, shuffle products with decorations and matching Zinbiel algebras

The Rota-Baxter algebra and the shuffle product are both algebraic structures arising from integral operators and integral equations. Free commutative Rota-Baxter algebras provide an algebraic framework for integral equations with the simple Riemann integral operator. The Zinbiel algebras form a category in which the shuffle product algebra is the free object. Motivated by algebraic structures underlying integral equations involving multiple integral operators and kernels, we study commutative matching Rota-Baxter algebras and construct the free objects making use of the shuffle product with multiple decorations. We also construct free commutative matching Rota-Baxter algebras in a relative context, to emulate the action of the integral operators on the coefficient functions in an integral equation. We finally show that free commutative matching Rota-Baxter algebras give the free matching Zinbiel algebra, generalizing the characterization of the shuffle product algebra as the free Zinbiel algebra obtained by Loday.

Braided Rota–Baxter algebras, quantum quasi-shuffle algebras and braided dendriform algebras

Rota–Baxter algebras and the closely related dendriform algebras have important physics applications, especially to renormalization of quantum field theory. Braided structures provide effective ways of quantization such as for quantum groups. Continuing recent study relating the two structures, this paper considers Rota–Baxter algebras and dendriform algebras in the braided contexts. Applying the quantum shuffle and quantum quasi-shuffle products, we construct free objects in the categories of braided Rota–Baxter algebras and braided dendriform algebras, under the commutativity condition. We further generalize the notion of dendriform Hopf algebra to the braided context and show that quantum shuffle algebra gives a braided dendriform Hopf algebra. Enveloping braided commutative Rota–Baxter algebras of braided commutative dendriform algebras are obtained.

Left counital Hopf algebra structures on free commutative Nijenhuis algebras

Motivated by the Hopf algebra structures established on free commutative Rota-Baxter algebras, we explore Hopf algebra related structures on free commutative Nijenhuis algebras. Applying a cocycle condition, we first prove that a free commutative Nijenhuis algebra on a left counital bialgebra (in the sense that the right-sided counicity needs not hold) can be enriched to a left counital bialgebra. We then establish a general result that a connected graded left counital bialgebra is a left counital right antipode Hopf algebra in the sense that the antipode is also only right-sided. We finally apply this result to show that the left counital bialgebra on a free commutative Nijenhuis algebra on a connected left counital bialgebra is connected and graded, hence is a left counital right antipode Hopf algebra.

Universal Enveloping Commutative Rota–Baxter Algebras of Pre- and Post-Commutative Algebras

Universal enveloping commutative Rota-Baxter algebras of pre- and postcommutative algebras are constructed. We prove that the pair of varieties (commutative Rota-Baxter algebras of nonzero weight,postcommutative algebras) is a PBW-pair and the pair (commutative Rota-Baxter algebras of zero weight,precommutative algebras) is not.

Generating functions from the viewpoint of Rota–Baxter algebras

We study generating functions in the context of Rota–Baxter algebras. We show that exponential generating functions can be naturally viewed as a very special case of complete free commutative Rota–Baxter algebras. This viewpoint allows us to apply free Rota–Baxter algebras to give a large class of algebraic structures in which generalizations of generating functions can be studied. We generalize the product formula and composition formula for exponential power series. We also give generating functions both for known number families such as Stirling numbers of the second kind and partition numbers, and for new number families such as those from not necessarily disjoint partitions and partitions of multisets.

Gröbner–Shirshov bases for commutative algebras with multiple operators and free commutative Rota–Baxter algebras

In this paper, the Composition-Diamond lemma for commutative algebras with multiple operators is established. As applications, the Gröbner–Shirshov bases and linear bases of free commutative Rota–Baxter algebra, free commutative λ-differential algebra and free commutative λ-differential Rota–Baxter algebra are given, respectively. Consequently, these three free algebras are constructed directly by commutative Ω-words.

Localization of Rota–Baxter algebras

A commutative Rota–Baxter algebra can be regarded as a commutative algebra that carries an abstraction of the integral operator. With the motivation of generalizing the study of algebraic geometry to Rota–Baxter algebras, we extend the central concept of localization for commutative algebras to commutative Rota–Baxter algebras. The existence of such a localization is proved and, under mild conditions, its explicit construction is obtained. The existence of tensor products of commutative Rota–Baxter algebras is also proved and the compatibility of localization and the tensor product of Rota–Baxter algebras is established. We further study Rota–Baxter coverings and show that they form a Grothendieck topology.

A noncommutative Bohnenblust–Spitzer identity for Rota–Baxter algebras solves Bogoliubov’s recursion

The Bogoliubov recursion is a particular procedure appearing in the process of renormalization in perturbative quantum field theory. It provides convergent expressions for otherwise divergent integrals. We develop here a theory of functional identities for noncommutative Rota–Baxter algebras which is shown to encode, among others, this process in the context of Connes–Kreimer's Hopf algebra of renormalization. Our results generalize the seminal Cartier–Rota theory of classical Spitzer-type identities for commutative Rota–Baxter algebras. In the classical, commutative, case these identities can be understood as deriving from the theory of symmetric functions. Here we show that an analogous property holds for noncommutative Rota–Baxter algebras. That is, we show that functional identities in the noncommutative setting can be derived from the theory of noncommutative symmetric functions. Lie idempotents, and particularly the Dynkin idempotent, play a crucial role in the process. Their action on the pro-unipotent groups such as those of perturbative renormalization is described in detail along the way.

FREE ROTA–BAXTER ALGEBRAS AND ROOTED TREES

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.