相关论文: Anti-commutative Groebner-Shirshov basis of a free…
In this paper, by using the Composition-Diamond lemma for non-associative algebras invented by A. I. Shirshov in 1962, we give Gr\"{o}bner-Shirshov bases for free Pre-Lie algebras and the universal enveloping non-associative algebra of an…
In this paper, by using Composition-Diamond lemma for Lie algebras, we give a Gr\"obner-Shirshov basis for free partially commutative Lie algebra over a commutative ring with unit. As an application, we obtain a normal form for such a Lie…
In this paper, we establish the Composition-Diamond lemma for free differential algebras. As applications, we give Groebner-Shirshov bases for free Lie-differential algebra and free commutative-differential algebra, respectively.
Chen, Fox, Lyndon 1958 \cite{CFL58} and Shirshov 1958 \cite{Sh58} introduced non-associative Lyndon-Shirshov words and proved that they form a linear basis of a free Lie algebra, independently. In this paper we give another approach to…
In this paper, we establish Composition-Diamond lemma for tensor product $k< X> \otimes k< Y>$ of two free algebras over a field. As an application, we construct a Groebner-Shirshov basis in $k< X> \otimes k< Y>$ by lifting a…
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…
Let $A$ be a brace algebra. This structure implies that $A$ is also a pre-Lie algebra. In this paper, we establish Composition-Diamond lemma for brace algebras. Using this Composition-Diamond lemma we prove that each pre-Lie algebra $L$ can…
In this paper, we study the concept of associative $n$-conformal algebra over a field of characteristic 0 and establish Composition-Diamond lemma for a free associative $n$-conformal algebra. As an application, we construct…
In this paper, we firstly establish Composition-Diamond lemma for $\Omega$-algebras. We give a Gr\"{o}bner-Shirshov basis of the free $L$-algebra as a quotient algebra of a free $\Omega$-algebra, and then the normal form of the free…
In this paper, we review Shirshov's method for free Lie algebras invented by him in 1962 which is now called the Groebner-Shirshov bases theory.
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, we construct a canonical linear basis for free commutative integro-differential algebras by applying the method of Gr\"obner-Shirshov bases. We establish the Composition-Diamond Lemma for free commutative differential…
In this paper we establish a Gr\"{o}bner-Shirshov bases theory for Lie algebras over commutative rings. As applications we give some new examples of special Lie algebras (those embeddable in associative algebras over the same ring) and…
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…
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…
We construct free modules over an associative conformal algebra. We establish Composition-Diamond lemma for associative conformal modules. As applications, Gr\"obner-Shirshov bases of the Virasoro conformal module and module over the…
In this paper we give some relationships among the Groebner-Shirshov bases in free associative algebras, free left modules and "double-free" left modules (free modules over a free algebra). We give the Chibrikov's Composition-Diamond lemma…
In this paper, we establish the Composition-Diamond lemma for $\lambda$-differential associative algebras over a field $K $ with multiple operators. As applications, we obtain Gr\"{o}bner-Shirshov bases of free $\lambda$-differential…
In this paper, we define the Gr\"obner-Shirshov basis for a dialgebra. The Composition-Diamond lemma for dialgebras is given then. As results, we give Gr\"obner-Shirshov bases for the universal enveloping algebra of a Leibniz algebra, the…
We establish the Composition-Diamond lemma for non-associative algebras over a free commutative algebra. As an application, we prove that every countably generated non-associative algebra over an arbitrary commutative algebra $K$ can be…