Recent advances in stochastic PDEs, Hopf algebras of typed trees and integral equations have inspired the study of algebraic structures with replicating operations. To understand their algebraic and combinatorial nature, we first use rooted forests with multiple decoration sets to construct free Hopf algebras with multiple Hochschild 1-cocycle conditions. Applying the universal property of the underlying operated algebras and the method of Gröbner-Shirshov bases, we then construct free objects in the category of matching Rota-Baxter algebras which is a generalization of Rota-Baxter algebras to allow multiple Rota-Baxter operators. Finally the free matching Rota-Baxter algebras are equipped with a cocycle Hopf algebra structure.
In this paper, we study the averaging operator by assigning a rewriting system to it. We obtain some basic results on the kind of rewriting system we used. In particular, we obtain a sufficient and necessary condition for the confluence. We supply the relationship between rewriting systems and Grobner-Shirshov bases based on bracketed polynomials. As an application, we give a basis of the free unitary averaging algebra on a non-empty set.
In this paper we apply the methods of rewriting systems and Gröbner-Shirshov bases to give a unified approach to a class of linear operators on associative algebras. These operators resemble the classic Rota-Baxter operator, and they are called {\it Rota-Baxter type operators}. We characterize a Rota-Baxter type operator by the convergency of a rewriting system associated to the operator. By associating such an operator to a Gröbner-Shirshov basis, we obtain a canonical basis for the free algebras in the category of associative algebras with that operator. This construction include as special …
Many years ago, Rota proposed a program on determining algebraic identities that can be satisfied by linear operators. After an extended period of dormant, progress on this program picked up speed in recent years, thanks to perspectives from operated algebras and Gröbner-Shirshov bases. These advances were achieved in a series of papers from special cases to more general situations. These perspectives also indicate that Rota's insight can be manifested very broadly, for other algebraic structures such as Lie algebras, and further in the context of operads. This paper gives a survey on the moti…
Motivated by the pivotal role played by linear operators, many years ago Rota proposed to determine algebraic operator identities satisfied by linear operators on associative algebras, later called Rota's program on algebraic operators. Recent progresses on this program have been achieved in the contexts of operated algebra, rewriting systems and Groebner-Shirshov bases. These developments also suggest that Rota's insight can be applied to determine operator identities on Lie algebras, and thus to put the various linear operators on Lie algebras in a uniform perspective. This paper carries out…
In this paper, we generalize the Lyndon-Shirshov words to Lyndon-Shirshov $Ω$-words on a set $X$ and prove that the set of all non-associative Lyndon-Shirshov $Ω$-words forms a linear basis of the free Lie $Ω$-algebra on the set $X$. From this, we establish Gröbner-Shirshov bases theory for Lie $Ω$-algebras. As applications, we give Gröbner-Shirshov bases for free $λ$-Rota-Baxter Lie algebras, free modified $λ$-Rota-Baxter Lie algebras and free Nijenhuis Lie algebras and then linear bases of such three free algebras are obtained.
In this paper, we construct free Lie Rota-Baxter superalgebra by using Gröbner-Shirshov bases theory. We firstly construct free operated Lie superalgebras by the operated super-Lyndon-Shirshov monomials. Secondly, we establish Gröbner-Shirshov bases theory for operated Lie superalgebras. Thirdly, we find a Gröbner-Shirshov basis of a free Lie Rota-Baxter superalgebra on a $\mathbb{Z}_2$-graded set. Consequently, we can obtain a linear basis of a free Lie Rota-Baxter superalgebra by the composition-diamond lemma for operated Lie superalgebras.
We establish the Gröbner-Shirshov bases theory for differential Lie $Ω$-algebras. As an application, we give a linear basis of a free differential Lie Rota-Baxter algebra on a set.
In this paper, we obtain respectively some new linear bases of free unitary (modified) weighted differential algebras and free nonunitary (modified) Rota-Baxter algebras, in terms of the method of Gröbner-Shirshov bases.
In this paper we revisit Rota's Classification Problem on classifying algebraic identities for linear operator. We reformulate Rota's Classification Problem in the contexts of rewriting systems and Gröbner-Shirshov bases, through which Rota's Classification Problem amounts to the classification of operators, given by their defining operator identities, that give convergent rewriting systems or Gröbner-Shirshov bases. Relationship is established between the reformulations in terms of rewriting systems and that of Gröbner-Shirshov bases. We provide an effective condition that gives Gröbner-Shirs…
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.
In this paper, we establish the Composition-Diamond lemma for associative nonunitary Rota-Baxter algebras with weight $λ$. As applications, we obtain a linear basis of a free commutative Rota-Baxter algebra without unity and show that every countably generated Rota-Baxter algebra with weight 0 can be embedded into a two-generated Rota-Baxter algebra.
In this paper, we establish the Composition-Diamond lemma for associative algebras with multiple linear operators. As applications, we obtain Groebner-Shirshov bases of free Rota-Baxter algebra, $λ$-differential algebra and $λ$-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 those obtained by Ebrahimi-Fard and Guo, and Guo and Keigher recently by using other methods.
In this paper, we propose the concept of an $Ω$-Rota-Baxter system, which is a generalization of a Rota-Baxter system and an $Ω$-Rota-Baxter algebra of weight zero. In the framework of operated algebras, we obtain a linear basis of a free $Ω$-Rota-Baxter system for an extended diassociative semigroup $Ω$, in terms of bracketed words and the method of Gröbner-Shirshov bases. As applications, we introduce the concepts of Rota-Baxter system family algebras and matching Rota-Baxter systems as special cases of $Ω$-Rota-Baxter systems, and construct their free objects. Meanwhile, free $Ω$-Rota-Baxte…
Quite recently, Bremner et al. introduced a new approach to Rota's Classification Problem and classified some (new) operated polynomial identities. In this paper, we prove that all operated polynomial identities classified by Bremner et al. are Gröbner-Shirshov.
Quite much recent studies has been attracted to the operated algebra since it unifies various notions such as the differential algebra and the Rota-Baxter algebra. An $Ω$-operated algebra is a an (associative) algebra equipped with a set $Ω$ of linear operators which might satisfy certain operator identities such as the Leibniz rule. A free $Ω$-operated algebra $B$ can be generated on an algebra $A$ similar to a free algebra generated on a set. If $A$ has a Gröbner-Shirshov basis $G$ and if the linear operators $Ω$ satisfy a set $Φ$ of operator identities, it is natural to ask when the union $…
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…
In this paper, we construct a canonical linear basis for free commutative integro-differential algebras by applying the method of Gröbner-Shirshov bases. We establish the Composition-Diamond Lemma for free commutative differential Rota-Baxter algebras of order $n$. We also obtain a weakly monomial order on these algebras, allowing us to obtain Gröbner-Shirshov bases for free commutative integro-differential algebras on a set. We finally generalize the concept of functional derivations to free differential algebras with arbitrary weight and generating sets from which to construct a canonical li…
A long standing problem of Gian-Carlo Rota for associative algebras is the classification of all linear operators that can be defined on them. In the 1970s, there were only a few known operators, for example, the derivative operator, the difference operator, the average operator, and the Rota-Baxter operator. A few more appeared after Rota posed his problem. However, little progress was made to solve this problem in general. In part, this is because the precise meaning of the problem is not so well understood. In this paper, we propose a formulation of the problem using the framework of operat…
This paper investigates algebraic objects equipped with an operator, such as operated monoids, operated algebras etc. Various free object functors in these operated contexts are explicitly constructed. For operated algebras whose operator satisfies a set $Φ$ of relations (usually called operated polynomial identities (aka. OPIs)), Guo defined free objects, called free $Φ$-algebras, via universal algebra. Free $Φ$-algebras over algebras are studied in details. A mild sufficient condition is found such that $Φ$ together with a Gröbner-Shirshov basis of an algebra $A$ form a Gröbner-Shirshov basi…
In this paper, by using Gröbner-Shirshov bases for Rota-Baxter algebras, we prove that every dendriform dialgebra over a field of characteristic 0 can be embedded into its universal enveloping Rota-Baxter algebra of weight 0.
In this paper, we give a linear basis of a free Rota-Baxter system on a set by using the Gröbner-Shirshov bases method and then we obtain a left counital Hopf algebra structure on a free Rota-Baxter system.
Rota-Baxter associative algebras and Rota-Baxter Lie algebras are both important in mathematics and mathematical physics, with the former a basic structure in quantum field renormalization and the latter a operator form of the classical Yang-Baxter equation. An outstanding problem posed by Gubarev is to determine whether there is a Poincaré-Birkhoff-Witt theorem for the universal enveloping Rota-Baxter associative algebra of a Rota-Baxter Lie algebra. This paper resolves this problem positively, working with operated algebras and applying the method of Gröbner-Shirshov bases.
In this paper, we establish the Composition-Diamond lemma for $λ$-differential associative algebras over a field $K $ with multiple operators. As applications, we obtain Gröbner-Shirshov bases of free $λ$-differential Rota-Baxter algebras. In particular, linear bases of free $λ$-differential Rota-Baxter algebras are obtained and consequently, the free $λ$-differential Rota-Baxter algebras are constructed by words.
With the help of Rota-Baxter operators and the Groebner-Shirshov bases, we prove that any pre-Lie algebra injectively embeds into its universal enveloping preassociative algebra.
In this survey, we outline two recent constructions of free commutative integro-differential algebras. They are based on the construction of free commutative Rota-Baxter algebras by mixable shuffles. The first is by evaluations. The second is by the method of Gröbner-Shirshov bases.
