相关论文: Gr\"obner-Shirshov bases for Lie algebras over a c…
We show that if a countably generated Lie algebra $H$ does not contain isomorphic copies of certain finite-dimensional nilpotent Lie algebras $A$ and $B$ (satisfying some mild conditions), then $H$ embeds into a quotient of $A \ast B$ that…
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 Gr\"{o}bner-Shirshov bases theory for metabelian Lie algebras. As applications, we find the Gr\"{o}bner-Shirshov bases for partial commutative metabelian Lie algebras related to circuits, trees and some…
Let $\mathfrak{a},\mathfrak{b},\mathfrak{e}$ be algebras over a field $k$. Then $\mathfrak{e}$ is an extension of $\mathfrak{a}$ by $\mathfrak{b}$ if $\mathfrak{a}$ is an ideal of $\mathfrak{e}$ and $\mathfrak{b}$ is isomorphic to the…
In this survey, we formulate the Gr\"{o}bner-Shirshov bases theory for associative algebras and Lie algebras. Some new Composition-Diamond lemmas and applications are mentioned.
In this paper, linear bases for the partially commutative Lie algebras are found. The method of the Gr\"{o}bner--Shirshov bases is used. It easily follows from the structure that the equality problem is algorithmically solvable for the…
In this paper, by using Gr\"obner-Shirshov bases, we show that in the following classes, each (resp. countably generated) algebra can be embedded into a simple (resp. two-generated) algebra: associative differential algebras, associative…
In this paper, we elaborate Gr\"obner-Shirshov bases method for Leibniz (super)algebras. We show that there is a unique reduced Gr\"obner-Shirshov basis for every (graded) ideal of a free Leibniz (super)algebra. As applications, we…
We study the relation between Poisson algebras and representations of Lie conformal algebras. We establish a setting for the calculation of a Gr\"obner--Shirshov basis in a module over an associative conformal algebra and apply this…
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…
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.
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.
With this paper we present an extension of our recent ISSAC paper about computations of Groebner(-Shirshov) bases over free associative algebras Z<X>. We present all the needed proofs in details, add a part on the direct treatment of the…
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…
We show that a set of monic polynomials in the free Lie superalgebra is a Gr\"obner-Shirshov basis for a Lie superalgebra if and only if it is a Gr\"obner-Shirshov basis for its universal enveloping algebra. We investigate the structure of…
One of the natural ways to prove that the Hall words (Philip Hall, 1933) consist of a basis of a free Lie algebra is a direct construction: to start with a linear space spanned by Hall words, to define the Lie product of Hall words, and…
The aim of this paper is to introduce and study Lie algebras and Lie groups over noncommutative rings. For any Lie algebra $\gg$ sitting inside an associative algebra $A$ and any associative algebra $\FF$ we introduce and study the algebra…
We give Gr\"obner-Shirshov bases for Drinfeld-Kohno Lie algebra $\textbf{L}_{n}$ in \cite{[Et]} and Kukin Lie algebra $A_P$ in \cite{Kukin}, where $P$ is a semigroup. As applications, we show that as $\mathbb{Z}$-module $\textbf{L}_{n}$ is…
Let $R$ be a finite commutative ring with identity. In this paper, we give a necessary condition for the existence of an orthogonal decomposition of the special linear Lie algebra over $R$. Additionally, we study orthogonal decompositions…
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…