Related papers: Free $\Omega$-Rota-Baxter systems and Gr\"obner-Sh…
In this paper we apply the methods of rewriting systems and Gr\"obner-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…
We construct an explicit Gr\"obner--Shirshov basis for free associative Rota--Baxter algebras of weight zero with nilpotent operator $R^n=0$, where $n\ge 2$. First, we define a monomial order on the standard linear basis $RS(X)$ of the free…
In this paper, we give a linear basis of a free Rota-Baxter system on a set by using the Gr\"{o}bner-Shirshov bases method and then we obtain a left counital Hopf algebra structure on a free Rota-Baxter system.
In this paper, we establish the Composition-Diamond lemma for associative nonunitary Rota-Baxter algebras with weight $\lambda$. As applications, we obtain a linear basis of a free commutative Rota-Baxter algebra without unity and show that…
We establish the Gr\"obner-Shirshov bases theory for differential Lie $\Omega$-algebras. As an application, we give a linear basis of a free differential Lie Rota-Baxter algebra on a set.
In this paper, we generalize the Lyndon-Shirshov words to Lyndon-Shirshov $\Omega$-words on a set $X$ and prove that the set of all non-associative Lyndon-Shirshov $\Omega$-words forms a linear basis of the free Lie $\Omega$-algebra on the…
Differential operators and integral operators are linked together by the first fundamental theorem of calculus. Based on this principle, the notion of a differential Rota-Baxter algebra was proposed by Guo and Keigher from an algebraic…
In this paper, we construct free Lie Rota-Baxter superalgebra by using Gr\"{o}bner-Shirshov bases theory. We firstly construct free operated Lie superalgebras by the operated super-Lyndon-Shirshov monomials. Secondly, we establish…
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 Gr\"{o}bner-Shirshov bases.
A Rota-Baxter algebra, also known as a Baxter algebra, is an algebra with a linear operator satisfying a relation, called the Rota-Baxter relation, that generalizes the integration by parts formula. Most of the studies on Rota-Baxter…
We introduce a generalization of parametrized Rota-Baxter algebras, which includes family and matching Rota-Baxter algebras. We study the structure needed on the set $\Omega$ of parameters in order to obtain that free Rota-Baxter algebras…
The algebraic formulation of the derivation and integration related by the First Fundamental Theorem of Calculus (FFTC) gives rise to the notion of differential Rota-Baxter algebra. The notion has a remarkable list of categorical…
This paper introduces the structure of operated polygraphs as a categorical model for rewriting in operated algebras, generalizing Gr\"obner-Shirshov bases with non-monomial termination orders. We provide a combinatorial description of…
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…
In this paper, the Composition-Diamond lemma for commutative algebras with multiple operators is established. As applications, the Gr\"obner-Shirshov bases and linear bases of free commutative Rota-Baxter algebra, free commutative…
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…
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 $\Omega$-operated algebra is a an (associative) algebra equipped with a…
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, $\lambda$-differential algebra and…
M. Goncharov introduced and studied a Rota--Baxter operator on a cocommutative Hopf algebra. In the present paper we define relative Rota--Baxter operators on an arbitrary Hopf algebra. A particular case of this definition is Goncharov's…
It is known that if $A$ is a finite-dimensional unital algebra equipped with a Rota-Baxter operator $R$ of weight $\lambda$, then spectrum of $R$ is a subset of $\{0,-\lambda\}$. We are interested on finding all consequences of the…