Related papers: Groebner-Shirshov bases for extensions of algebras
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 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…
In this paper, by using the Groebner-Shirshov bases, we give characterizations of the Schreier extensions of groups when the group is presented by generators and relations. An algorithm to find the conditions of a group to be a Schreier…
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…
In this paper, we compute the Gr\"obner-Shirshov bases for certain regular double extension algebras by means of an algorithm implemented in Matlab, which facilitates the underlying algebraic computations. Moreover, we establish that these…
In this note we show how to apply the Gr\"obner--Shirshov bases (GSB) method for modules over an associative algebra to the study of vertex algebras defined by generators and relations. We compute GSBs for a series of vertex algebras and…
As a fundamental notion, the free differential algebra on a set is concretely constructed as the polynomial algebra on the differential variables. Such a construction is not known for the more general notion of the free differential algebra…
We review some applications of Gr\"obner-Shirshov bases, including PBW theorems, linear bases of free universal algebras, normal forms for groups and semigroups, extensions of groups and algebras, embedding of algebras.
We establish Gr\"obner--Shirshov bases theory for commutative dialgebras. We show that for any ideal $I$ of $Di[X]$, $I$ has a unique reduced Gr\"obner--Shirshov basis, where $Di[X]$ is the free commutative dialgebra generated by a set $X$,…
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 review Shirshov's method for free Lie algebras invented by him in 1962 which is now called the Groebner-Shirshov bases theory.
We develop the theory of Gr\"obner bases for ideals in a polynomial ring with countably infinite variables over a field. As an application we reconstruct some of the one-one correspondences among various sets of partitions by using division…
Two fundamental questions in the theory of Groebner bases are decision ("Is a basis G of a polynomial ideal a Groebner basis?") and transformation ("If it is not, how do we transform it into a Groebner basis?") This paper considers the…
A finite Grobner-Shirshov basis is constructed for the plactic algebra of rank 3 over a field K. It is also shown that plactic algebras of rank exceeding 3 do not have finite Grobner-Shirshov bases associated to the natural…
We establish Gr\"{o}bner-Shirshov bases theory for Gelfand-Dorfman-Novikov algebras over a field of characteristic $0$. As applications, a PBW type theorem in Shirshov form is given and we provide an algorithm for solving the word problem…
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 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…
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…
We explicitly describe the structure of HNN extensions of Lie superalgebras. We specify their bases. Moreover, we prove that the HNN extension is a direct sum of two subalgebras: original Lie superalgebra, and the free Lie superalgebra,…
This extended abstract gives a construction for lifting a Gr\"obner basis algorithm for an ideal in a polynomial ring over a commutative ring R under the condition that R also admits a Gr\"obner basis for every ideal in R.