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

Abstract. 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 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 cases several previous ones for free objects in similar categories, such as those of Rota-Baxter algebras and Nijenhuis algebras.

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“…These identities are then characterized by means of their rewriting systems [4] and associated Gröbner-Shirshov bases [10,12,13].…”

Section: P(x)p(y) = P(xp(y) + P(x)y − P(xy)) Leroux's Td Operator P(mentioning

confidence: 99%

Rota-Baxter type operators, rewriting systems and Gröbner-Shirshov bases

Guo

Sit

Gao

et al. 2015

ACM Commun. Comput. Algebra

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“…It is often important to be precise about the location of a subword. This is the case, for example, in the study of rewriting systems and Gröbner-Shirshov bases [2,7,13,17]. For this purpose, we need additional information for the location of the subword u.…”

Section: Relative Locations Of Subwordsmentioning

confidence: 99%

“…For this purpose, we need additional information for the location of the subword u. Such information is provided by the concept of ⋆-words [7]. Definition 2.2.…”

Section: Relative Locations Of Subwordsmentioning

confidence: 99%

“…The classification of relative locations can be quite subtle in some structures, especially when a non-identity operator is present, such as in differential algebras, Rota-Baxter algebras [12] and, more generally, operated algebras [5,7,13,17]. Further by [11] free operated semigroups have natural combinatorial presentation as rooted trees which serve as the baby models for Feynman graphs [8,14].…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2 we deal with the first subtle point which is already present in studying relative locations of two subwords in comparison with two subsets, namely that one subword can appear at multiple locations in a given word. The concept of a ⋆-word [7] allows us to give a unique label to each appearance of a subword, called a placement. We then show that each placement corresponds uniquely to a substring of the string corresponding to the given word, converting the problem of studying subwords to that of subsets which can be solved easily as mentioned above.…”

Section: Introductionmentioning

confidence: 99%

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Relative locations of subwords in free operated semigroups and Motzkin words

Zheng

Guo

2014

Front. Math. China

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Abstract. Bracketed words are basic structures both in mathematics (such as Rota-Baxter algebras) and mathematical physics (such as rooted trees) where the locations of the substructures are important. In this paper we give the classification of the relative locations of two bracketed subwords of a bracketed word in an operated semigroup into the separated, nested and intersecting cases. We achieve this by establishing a correspondence between relative locations of bracketed words and those of words by applying the concept of Motzkin words which are the algebraic forms of Motzkin paths.

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Constructions of Free Commutative Integro-Differential Algebras

Gao

Guo

2014

Algebraic and Algorithmic Aspects of Differential and Integral Operators

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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. 1 2 XING GAO AND LI GUO The study of differential algebra began with Ritt's classic work [35,36]. After the fundamental work of Kolchin [33], differential algebra has evolved into a vast area of mathematics that is important in both theory [15,44] and applications: for instance, in mechanic theorem proving by W.-T. Wu [45,46]. Free (commutative) differential algebras, in the form of differential polynomial algebras (Theorem 2.3), are essential for studying differential equations, as polynomial algebras are for commutative algebras.The algebraic study of integrals came much later. In fact the development did not start from an algebraic abstraction of integrals, but from the effort of G. Baxter [5] in 1960 to understand a formula in probability theory. As a result, the concept is not called an integral algebra, but called a (Rota-)Baxter algebra (Eq. (3)) which is the integral counterpart of the derivation, the difference operator, and divided differences (see Eq. (1)). Soon afterwards Rota noticed its importance in combinatorics and promoted its study through research and survey articles (see e.g. [40,41]). Independently, Rota-Baxter operators on Lie algebras were found to be closely related to the classical Yang-Baxter equation [42]. Since the turn of this century, the theory of Rota-Baxter algebra has experienced rapid development with broad applications in mathematics and physics [4,21,29,40,41,42], especially noteworthy in the Hopf algebra approach of Connes-Kreimer to renormalization of quantum field theory [16,19,29]. Here again a fundamental role is played by free (commutative) Rota-Baxter algebras that were first constructed by Rota [40] and Cartier [14], and then by in terms of mixable shuffles (Theorem 2.4).The fusion of differential and Rota-Baxter algebras, motivated by algebraic study of calculus as a whole, appeared about five years ago. It is amazing that two structures for this purpose were introduced at about the same time. One is a relatively simple coupling of differential algebra and Rota-Baxter algebra through section axiom (Eq. (4)) that reflects the First Fundamental Theorem of Calculus. It is called differential Rota-Baxter algebra [26]. The other one is a more faithful abstraction of the integration-by-parts formula (see Eq. (7)), giving rise to the concept of an integro-differential algebra [37] which has generated much interest [1,2,38,39]. As suggested in previous cases, free objects for these algebraic structures are important in their studies. Because of the relative independence of the differential and integral (Rota-Baxter) structures in a differential Rota-Baxter algebra, the free object was constructed by a clear combination of the free objects on the differential and Rota-Baxter sides and were obtai...

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

Cited by 78 publications

References 52 publications

Rota-Baxter type operators, rewriting systems and Gröbner-Shirshov bases

Rota-Baxter type operators, rewriting systems and Gröbner-Shirshov bases

Relative locations of subwords in free operated semigroups and Motzkin words

Constructions of Free Commutative Integro-Differential Algebras

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