Jianjun Qiu, Yuqun Chen
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.
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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 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.
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.
Gröbner-Shirshov Bases for Commutative Algebras with Multiple Operators and Free Commutative Rota-Baxter Algebras
本文建立了含重算子的交换代数的合成钻石引理。作为应用，分别给出了自由交换Rota-Baxter代数、自由交换$λ$ -微分代数和自由交换$λ$ -微分Rota-Baxter代数的Gröbner-Shirshov基和线性基.因此，这三个自由代数直接由交换$Ω$ -字构造。
Gröbner-Shirshov bases for Rota-Baxter algebras
本文建立了带权$λ$的结合非酉Rota-Baxter代数的合成钻石引理.作为应用，我们得到了一个不含单位元的自由交换Rota-Baxter代数的线性基，并证明了每个权为0的可数生成的Rota-Baxter代数都可以嵌入到一个二元生成的Rota-Baxter代数中.
Groebner-Shirshov Bases for Associative Algebras with Multiple Operators and Free Rota-Baxter Algebras
本文建立了含多个线性算子的结合代数的合成-钻石引理。作为应用，我们分别得到了自由Rota-Baxter代数、$λ$ -微分代数和$λ$ -微分Rota-Baxter代数的Groebner-Shirshov基.特别地，分别得到了这三个自由代数的线性基，它们与Ebrahimi-Fard和Guo以及Guo和Keigher最近用其他方法得到的线性基基本相同或相似.
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 $…
Construction of free quasi-idempotent differential Rota-Baxter algebras by Gröbner-Shirshov bases
本文在拟幂等算子背景下，为自由交换拟幂等微分代数、Rota-Baxter代数及微分Rota-Baxter代数建立了Gröbner-Shirshov基，并借助Composition-Diamond引理为这三个范畴的自由对象提供了线性基。
Some new operated Lie polynomial identities and Gröbner-Shirshov bases
本文证明Bremner-Elgendy分类中新的带算子结合多项式恒等式所对应的李多项式恒等式均为Gröbner-Shirshov，肯定回答了Guo等人提出的问题，推进了李代数背景下Rota纲领的发展。
Gröbner-Shirshov bases of Rota-Baxter algebra of weight $λ$ with spectrum lying in $\{0,-λ\}$
本文研究谱包含在{0,-λ}中的Rota-Baxter代数，在一般情况下求得了由Rota-Baxter关系与R^k(R+λid)^l=0生成的理想的Gröbner-Shirshov基，推广了已有特例结果。
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 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…
A Gröbner--Shirshov Basis for Nilpotent Rota--Baxter Algebras of Weight Zero
本文构建了自由结合Rota-Baxter代数的Gröbner-Shirshov基，针对nilpotent运算R^n=0(n≥2)的情况，通过定义单子顺序和基本恒等式，给出了有限的基和正常形式，解决nilpotent Rota-Baxter代数的字问题。
Differential Type Operators and Gröbner-Shirshov Bases
Gian-Carlo Rota关于结合代数的一个长期存在的问题是所有可定义在其上的线性算子的分类。在20世纪70年代，只有少数已知的算子，例如，导数算子，差分算子，平均算子和Rota-Baxter算子。在罗塔提出他的问题后，又出现了几个。然而，在解决这一问题方面总体上进展甚微。部分原因是人们对这个问题的确切含义还没有很好地理解。在本文中，我们提出了一个制定的问题，使用的框架操作代数和查看一个结合代数与线性算子作为一个满足一定的操作多项式单位。这个框架还允许我们应用重写系统和Gröbner-Shirshov基的理论。为了缩小我们的重点更多的运营商，罗塔感兴趣的，我们进一步考虑两个特殊类别的运营商，即那些推广微分或Rota-Baxter运营商。事实证明，这两类算子对应于那些在两种不同的单项序下拥有Gröbner-Shirshov基的算子。在这个框架下工作，并借助计算机代数，我们能够得出这两类运算符的列表，并提供一些证据证明这些列表可能是完整的。我们的搜索已经发现了相当多的新的运营商的这些类型，其性质预计类似的微分算子和Rota-Baxter运营商分别。最近，在相关领域出现了一个更统一的方法，如差代数和微分代数，Rota-Baxter代数和Nijenhuis代数。这些理论中的相似之处可以通过罗塔问题的进展来更有效地探索。
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…
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…
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…
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…
Para-differential Rota-Baxter algebras and their free objects by Gröbner-Shirshov bases
本文研究了 para-微分 Rota-Baxter 代数，建立了其 Gröbner-Shirshov 基础理论，并给出了自由代数的显式构造。
Embedding dendriform dialgebra into its universal enveloping Rota-Baxter algebra
利用Rota-Baxter代数的Gröbner-Shirshov基，证明了特征为0的域上的每一个树状双代数都可以嵌入到其权为0的泛包络Rota-Baxter代数中.
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.
Composition-Diamond lemma for $λ$-differential associative algebras with multiple operators
本文建立了域$K $上含重算子的$λ$ -微分结合代数的合成钻石引理。作为应用，我们得到了自由$λ$ -微分Rota-Baxter代数的Gröbner-Shirshov基.特别地，得到了自由$λ$ -微分Rota-Baxter代数的线性基，从而用字构造了自由$λ$ -微分Rota-Baxter代数.
Construction of free commutative integro-differential algebras by the method of Gröbner-Shirshov bases
本文利用Gröbner-Shirshov基的方法构造了自由交换积分-微分代数的一个标准线性基.建立了$n$阶自由交换微分Rota-Baxter代数的合成钻石引理.我们还得到了这些代数上的弱单项式序，使我们能够获得自由交换积分-微分代数在一个集合上的Gröbner-Shirshov基。最后，我们将泛函导子的概念推广到具有任意权和生成集的自由微分代数，并由此构造了自由交换积分-微分代数的典范线性基。
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.
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.
Constructions of free commutative integro-differential algebras
本文概述了自由交换积分-微分代数的两个最新构造。它们是基于自由交换Rota-Baxter代数的混合洗牌的建设。第一是通过评价。第二种是Gröbner-Shirshov基方法。
Polygraphic resolutions for operated algebras
本文引入运算多图结构作为运算代数重写的范畴模型，利用多自动机描述其关键分支，展示自由运算代数多图分解的构造方法，并应用于Rota-Baxter代数等多个家族，推广了Gröbner-Shirshov基。
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