36 research outputs found
On Shirshov bases of graded algebras
We prove that if the neutral component in a finitely-generated associative
algebra graded by a finite group has a Shirshov base, then so does the whole
algebra.Comment: 4 pages; v2: minor corrections in English; to appear in Israel J.
Mat
Group gradings on superinvolution simple superalgebras
AbstractIn this paper we describe all group gradings by an arbitrary finite group G on non-simple finite-dimensional superinvolution simple associative superalgebras over an algebraically closed field F of characteristic 0 or coprime to the order of G
Algebraic Geometry over Free Metabelian Lie Algebra I: U-Algebras and Universal Classes
This paper is the first in a series of three, the aim of which is to lay the
foundations of algebraic geometry over the free metabelian Lie algebra . In
the current paper we introduce the notion of a metabelian Lie -algebra and
establish connections between metabelian Lie -algebras and special matrix
Lie algebras. We define the -localisation of a metabelian Lie
-algebra and the direct module extension of the Fitting's radical of
and show that these algebras lie in the universal closure of .Comment: 34 page
Lyndon-Shirshov basis and anti-commutative algebras
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
definition of Lyndon-Shirshov basis, i.e., we find an anti-commutative
Gr\"{o}bner-Shirshov basis of a free Lie algebra such that is the
set of all non-associative Lyndon-Shirshov words, where is the set of
all monomials of , a basis of the free anti-commutative algebra on ,
not containing maximal monomials of polynomials from . Following from
Shirshov's anti-commutative Gr\"{o}bner-Shirshov bases theory \cite{S62a2}, the
set is a linear basis of a free Lie algebra