227 research outputs found
Gr\"obner-Shirshov bases for Lie algebras over a commutative algebra
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 non-special Lie algebras (following a suggestion of P.M. Cohn (1963)
\cite{Conh}). In particular, Cohn's Lie algebras over the characteristic
are non-special when . We present an algorithm that one can check
for any , whether Cohn's Lie algebras is non-special. Also we prove that any
finitely or countably generated Lie algebra is embeddable in a two-generated
Lie algebra
Gr\"{o}bner-Shirshov bases for metabelian Lie algebras
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 cubes.Comment: 20 page
Groebner-Shirshov basis for HNN extensions of groups and for the alternating group
In this paper, we generalize the Shirshov's Composition Lemma by replacing
the monomial order for others. By using Groebner-Shirshov bases, the normal
forms of HNN extension of a group and the alternating group are obtained
Groebner-Shirshov basis for the braid group in the Artin-Garside generators
In this paper, we give a Groebner-Shirshov basis of the braid group
in the Artin--Garside generators. As results, we obtain a new algorithm for
getting the Garside normal form, and a new proof that the braid semigroup
is the subsemigroup in
Gr\"obner-Shirshov bases for -algebras
In this paper, we firstly establish Composition-Diamond lemma for
-algebras. We give a Gr\"{o}bner-Shirshov basis of the free -algebra
as a quotient algebra of a free -algebra, and then the normal form of
the free -algebra is obtained. We secondly establish Composition-Diamond
lemma for -algebras. As applications, we give Gr\"{o}bner-Shirshov bases of
the free dialgebra and the free product of two -algebras, and then we show
four embedding theorems of -algebras: 1) Every countably generated
-algebra can be embedded into a two-generated -algebra. 2) Every
-algebra can be embedded into a simple -algebra. 3) Every countably
generated -algebra over a countable field can be embedded into a simple
two-generated -algebra. 4) Three arbitrary -algebras , , over a
field can be embedded into a simple -algebra generated by and if
and , where is the free product of
and .Comment: 22 page
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