Free Rota-Baxter Algebras Research Articles

Free Rota-Baxter Family Algebras and Free (tri)dendriform Family Algebras

In this paper, we first construct the free Rota-Baxter family algebra generated by some set X in terms of typed angularly X-decorated planar rooted trees. As an application, we obtain a new construction of the free Rota-Baxter algebra only in terms of angularly decorated planar rooted trees (not forests), which is quite different from the known construction via angularly decorated planar rooted forests by K. Ebrahimi-Fard and L. Guo. We then embed the free dendriform (resp. tridendriform) family algebra into the free Rota-Baxter family algebra of weight zero (resp. one). Finally, we prove that the free Rota-Baxter family algebra is the universal enveloping algebra of the free (tri)dendriform family algebra.

Hopf algebra structure on free Rota–Baxter algebras by angularly decorated rooted trees

By means of a new notion of subforests of an angularly decorated rooted forest, we give a combinatorial construction of a coproduct on the free Rota–Baxter algebra on angularly decorated rooted forests. We show that this coproduct equips the Rota–Baxter algebra with a bialgebra structure and further a Hopf algebra structure.

Hopf algebras of rooted forests, cocyles, and free Rota-Baxter algebras

The Hopf algebra and the Rota-Baxter algebra are the two algebraic structures underlying the algebraic approach of Connes and Kreimer to renormalization of perturbative quantum field theory. In particular, the Hopf algebra of rooted trees serves as the “baby model” of Feynman graphs in their approach and can be characterized by certain universal properties involving a Hochschild 1-cocycle. Decorated rooted trees have also been applied to study Feynman graphs. We will continue the study of universal properties of various spaces of decorated rooted trees with such a 1-cocycle, leading to the concept of a cocycle Hopf algebra. We further apply the universal properties to equip a free Rota-Baxter algebra with the structure of a cocycle Hopf algebra.

The Shuffle Quasimonad and Modules with Differentiation and Integration

Differential linear logic and the corresponding categorical structure, differential categories, introduced the idea of differential structure associated to a (co)monad. Typically in settings such as algebraic geometry, one expresses differential structure for an algebra by having a module with a derivation, i.e. a map satisfying the Leibniz rule. In the monadic approach, we are able to continue to work with algebras and derivations, but the additional structure allows us to define other rules of the differential calculus for such modules; in particular one can define a monadic version of the chain rule as well as other basic identities.In attempting to develop a similar theory of integral linear logic, we were led to consider the shuffle multiplication. This was shown by Guo and Keigher to be fundamental in the construction of the free Rota-Baxter algebra, the Rota-Baxter equation being the integral analogue of the Leibniz rule. This shuffle multiplication induces a quasimonad on the category of vector spaces. The notion of quasimonad, called r-unital monad by Wisbauer, is slightly weaker than that of monad, but is still sufficient to define a sensible notion of module with differentiation and integration.In this paper, we demonstrate this quasimonad structure, show that its free modules have both differential and integral operators satisfying the Leibniz and Rota-Baxter rules and satisfy the fundamental theorems of calculus.

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.

Free integro-differential algebras and Gröbner–Shirshov bases

The notion of commutative integro-differential algebra was introduced for the algebraic study of boundary problems for linear ordinary differential equations. Its noncommutative analog achieves a similar purpose for linear systems of such equations. In both cases, free objects are crucial for analyzing the underlying algebraic structures, e.g. of the (matrix) functions.In this paper we apply the method of Gröbner–Shirshov bases to construct the free (noncommutative) integro-differential algebra on a set. The construction is from the free Rota–Baxter algebra on the free differential algebra on the set modulo the differential Rota–Baxter ideal generated by the noncommutative integration by parts formula. In order to obtain a canonical basis for this quotient, we first reduce to the case when the set is finite. Then in order to obtain the monomial order needed for the Composition-Diamond lemma, we consider the free Rota–Baxter algebra on the truncated free differential algebra. A Composition-Diamond lemma is proved in this context, and a Gröbner–Shirshov basis is found for the corresponding differential Rota–Baxter ideal.

Natural endomorphisms of quasi-shuffle Hopf algebras

The Hopf algebra of word-quasi-symmetric functions ($\WQSym$), a noncommutative generalization of the Hopf algebra of quasi-symmetric functions, can be endowed with an internal product that has several compatibility properties with the other operations on $\WQSym$. This extends constructions familiar and central in the theory of free Lie algebras, noncommutative symmetric functions and their various applications fields, and allows to interpret $\WQSym$ as a convolution algebra of linear endomorphisms of quasi-shuffle algebras. We then use this interpretation to study the fine structure of quasi-shuffle algebras (MZVs, free Rota-Baxter algebras...). In particular, we compute their Adams operations and prove the existence of generalized Eulerian idempotents, that is, of a canonical left-inverse to the natural surjection map to their indecomposables, allowing for the combinatorial construction of free polynomial generators for these algebras.

Enumeration and Generating Functions of Rota–Baxter Words

In this paper, we prove results on enumerations of sets of Rota–Baxter words ( $${{{\tt RBWs}}}$$ ) in a single generator and one unary operator. Examples of operators are integral operators, their generalization to Rota–Baxter operators, and Rota–Baxter type operators. $${{{\tt RBWs}}}$$ are certain words formed by concatenating generators and images of words under the operators. Under suitable conditions, they form canonical bases of free Rota–Baxter type algebras which are studied recently in relation to renormalization in quantum field theory, combinatorics, number theory, and operads. Enumeration of a basis is often a first step to choosing a data representation in implementation. We settle the case of one generator and one operator, starting with the idempotent case (more precisely, the exponent 1 case). Some integer sequences related to these sets of $${{{\tt RBWs}}}$$ are known and connected to other sequences from combinatorics, such as the Catalan numbers, and others are new. The recurrences satisfied by the generating series of these sequences prompt us to discover an efficient algorithm to enumerate the canonical basis of certain free Rota–Baxter algebras.

Enumeration and Generating Functions of Differential Rota–Baxter Words

In this paper, the methods and results in enumeration and generation of Rota–Baxter words in Guo and Sit (Algebraic and Algorithmic Aspects of Differential and Integral Operators (AADIOS), Math. Comp. Sci., vol. 4, Sp. Issue (2,3), 2011) are generalized and applied to a free, non-commutative, non-unitary, ordinary differential Rota–Baxter algebra with one generator. A differential Rota–Baxter algebra is an associative algebra with two operators modeled after the differential and integral operators, which are related by the First Fundamental Theorem of Calculus. Differential Rota–Baxter words are words formed by concatenating differential monomials in the generator with images of words under the Rota–Baxter operator. Their totality is a canonical basis of a free, non-commutative, non-unitary, ordinary differential Rota–Baxter algebra. A free differential Rota–Baxter algebra can be constructed from a free Rota–Baxter algebra on a countably infinite set of generators. The order of the derivation gives another dimension of grading on differential Rota–Baxter words, enabling us to generalize and refine results from Guo and Sit to enumerate the set of differential Rota–Baxter words and outline an algorithm for their generation according to a multi-graded structure. Enumeration of a basis is often a first step to choosing a data representation in implementation of algorithms involving free algebras, and in particular, free differential Rota–Baxter algebras and several related algebraic structures on forests and trees. The generating functions obtained can be used to provide links to other combinatorial structures.

Generalized Shuffles Related to Nijenhuis and TD-Algebras

Shuffle type products are well known in mathematics and physics. They are intimately related to Loday's dendriform algebras and were extensively used to give explicit constructions of free Rota–Baxter algebras. In the literature there exist at least two other Rota–Baxter type algebras, namely, the Nijenhuis algebra and the so-called TD-algebra. The explicit construction of the free unital commutative Nijenhuis algebra uses a modified quasi-shuffle product, called the right-shift shuffle. We show that another modification of the quasi-shuffle, the so-called left-shift shuffle, can be used to give an explicit construction of the free unital commutative TD-algebra. We explore some basic properties of TD-operators. Our construction is related to Loday's unital commutative tridendriform algebra, including the involutive case. The concept of Rota–Baxter, Nijenhuis and TD-bialgebras is introduced at the end, and we show that any commutative bialgebra provides such objects.

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

In this paper, we establish the Composition-Diamond lemma for associative algebras with multiple linear operators. As applications, we obtain Gröbner–Shirshov bases of free Rota–Baxter algebra, free λ -differential algebra and free λ -differential Rota–Baxter algebra, respectively. In particular, linear bases of these three free algebras are respectively obtained, which are essentially the same or similar to the recent results obtained by K. Ebrahimi-Fard–L. Guo, and L. Guo–W. Keigher by using other methods.

Operated semigroups, Motzkin paths and rooted trees

Combinatorial objects such as rooted trees that carry a recursive structure have found important applications recently in both mathematics and physics. We put such structures in an algebraic framework of operated semigroups. This framework provides the concept of operated semigroups with intuitive and convenient combinatorial descriptions, and at the same time endows the familiar combinatorial objects with a precise algebraic interpretation. As an application, we obtain constructions of free Rota-Baxter algebras in terms of Motzkin paths and rooted trees.

Rota–Baxter algebras and dendriform algebras

In this paper we study the adjoint functors between the category of Rota–Baxter algebras and the categories of dendriform dialgebras and trialgebras. In analogy to the well-known theory of the adjoint functor between the category of associative algebras and Lie algebras, we first give an explicit construction of free Rota–Baxter algebras and then apply it to obtain universal enveloping Rota–Baxter algebras of dendriform dialgebras and trialgebras. We further show that free dendriform dialgebras and trialgebras, as represented by binary planar trees and planar trees, are canonical subalgebras of free Rota–Baxter algebras.