Free weighted (modified) differential algebras, free (modified) Rota-Baxter algebras and Gröbner-Shirshov bases

Zhu, Zhicheng; Zhang, Huhu; Gao, Xing

doi:10.48550/arxiv.2108.03563

Cited by 2 publications

(2 citation statements)

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“…This led various authors to consider them independently. Motivated by such study, the authors in [34][35][36] considered the associative analogue of the modified CYBE, called the modified associative Yang-Baxter equation of weight κ ∈ k (in short modified AYBE of weight κ). A solution of the modified AYBE of weight κ is called a modified Rota-Baxter operator of weight κ.…”

Section: Introductionmentioning

confidence: 99%

A cohomological study of modified Rota-Baxter algebras

Das¹

2022

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A modified Rota-Baxter algebra is an algebra equipped with an operator that satisfies the modified Yang-Baxter equation. In this paper, we define the cohomology of a modified Rota-Baxter algebra with coefficients in a suitable bimodule. We relate our cohomology of a modified Rota-Baxter algebra with the known cohomology theory of a Rota-Baxter algebra. As applications of our cohomology, we study formal one-parameter deformations and abelian extensions of modified Rota-Baxter algebras. Contents 1. Introduction 1 2. Modified Rota-Baxter algebras 3 3. Cohomology of modified Rota-Baxter algebras 8 4. Deformations of modified Rota-Baxter algebras 11 5. Abelian extensions of modified Rota-Baxter algebras 13 References 15

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Section: Introductionmentioning

confidence: 99%

A cohomological study of modified Rota-Baxter algebras

Das¹

2022

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“…In order to deal with algebras endowed with operators, Guo et al introduced a GS basis theory in operated contexts in a series of papers [14,38,16,11] (see also [3]) with the goal to attack Rota's program [28] to classify "interesting" operators on algebras. For a global view about the state of art, we refer the reader to the survey paper [12] and for recent development, see [33,35,7,34,15,25,36,37].…”

Section: Introductionmentioning

confidence: 99%

Gröbner-Shirshov bases and linear bases for free differential type algebras over algebras

Zhou¹,

Qin²,

Zhou³

2021

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We study a question which can be roughly stated as follows: Given a (unital or nonunital) algebra A together with a Gröbner-Shirshov basis G, consider the free operated algebra B over A, such that the operator satisfies some polynomial identities Φ which are Gröbner-Shirshov in the sense of Guo et al., when does the union Φ ∪ G will be an operated Gröbner-Shirshov basis for B? We answer this question in the affirmative under a mild condition in our previous work with Wang. When this condition is satisfied, Φ ∪ G is an operated Gröbner-Shirshov basis for B and as a consequence, we also get a linear basis of B. However, the condition could not be applied directly to differential type algebras introduced by Guo, Sit and Zhang, including usual differential algebras.This paper solves completely this problem for differential type algebras. Some new monomial orders are introduced which, together with some known ones, permit the application of the previous result to most of differential type algebras, thus providing new operated GS bases and linear bases for these differential type algebras. Versions are presented both for unital and nonunital algebras. However, a class of examples are also presented, for which the natural expectation in the question is wrong and these examples are dealt with by direct inspection.

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Free weighted (modified) differential algebras, free (modified) Rota-Baxter algebras and Gröbner-Shirshov bases

Abstract: 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.

Cited by 2 publications

References 37 publications

A cohomological study of modified Rota-Baxter algebras

A cohomological study of modified Rota-Baxter algebras

Gröbner-Shirshov bases and linear bases for free differential type algebras over algebras

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