An identity of metabelian algebras

1988 ◽  
Vol 28 (3) ◽  
pp. 357-358
Author(s):  
A. Sh. Abakarov
Keyword(s):  
Metabelian Algebras

scholarly journals Topological rank of (-1, 1) metabelian algebras

2021 ◽  
Vol 40 ◽  
pp. 1-6
Author(s):  
Karamsi Jayalakshmi ◽  
Kommaddi Hari Babu
Keyword(s):  
Index 2 ◽  
Torsion Free ◽  
Metabelian Algebras

In 1981, Pchelintsev developed the idea for arranging non-nilpotent subvarieties in a given variety by using topological rank for spechtian varieties of algebra as a fixed tool. In this paper we show that for a given topological rank over a field of 2, 3 ? torsion free of (-1; 1) metabelian algebra solvable of index 2 that are Lie-nilpotent of step not more than p is equal to P.

scholarly journals On almost nilpotent varieties in theclass of commutative metabelian algebras

2017 ◽  
Vol 21 (3) ◽  
pp. 21-28
Author(s):  
S.P. Mishchenko ◽  
O.V. Shulezhko
Keyword(s):  
Positive Integer ◽  
Linear Algebra ◽  
Associative Algebra ◽  
Main Field ◽  
Nilpotent Algebra ◽  
Nilpotent Variety ◽  
Zero Characteristic ◽  
Fixed Set ◽  
Engel Identity ◽  
Metabelian Algebras

A well founded way of researching the linear algebra is the study of it using the identities, consequences of which is the identity of nilpotent. We know the Nagata-Higman’s theorem that says that associative algebra with nil condition of limited index over a field of zero characteristic is nilpotent. It is well known the result of E.I.Zel’manov about nilpotent algebra with Engel identity. A set of linear algebras where a fixed set of identities takes place, following A.I. Maltsev, is called a variety. The variety is called almost nilpotent if it is not nilpotent, but each its own subvariety is nilpotent. Here in the case of the main field with zero characteristic, we proved that for any positive integer m there exist commutative metabelian almost nilpotent variety of exponent is equal to m.

PRIMITIVE ELEMENTS OF FREE METABELIAN ALGEBRAS OF RANK TWO

2003 ◽  
Vol 13 (01) ◽  
pp. 17-33 ◽  
Author(s):  
VESSELIN DRENSKY ◽  
JIE-TAI YU
Keyword(s):  
Lie Algebras ◽  
Free Algebra ◽  
Primitive Element ◽  
Variety Of Algebras ◽  
Primitive Elements ◽  
Associative Algebras ◽  
Polynomial Algebras ◽  
Rank 2 ◽  
Metabelian Algebras ◽  
Injective Endomorphism

Let F(x,y) be a relatively free algebra of rank 2 in some variety of algebras over a field K of characteristic 0. In this paper we consider the problem whether p(x,y) ∈ F(x,y) is a primitive element (i.e. an automorphic image of x): (i) If F(x,y)/(p(x,y)) ≅ F(z), the relatively free algebra of rank 1 (ii) If p(f,g) is primitive for some injective endomorphism (f,g) of F(x,y) (iii) If p(x,y) is primitive in a relatively free algebra of larger rank. These problems have positive solutions for polynomial algebras in two variables. We give the complete answer for the free metabelian associative and Lie algebras and some partial results for free associative algebras.

On the Varieties of Commutative Metabelian Algebras

2018 ◽  
Vol 233 (5) ◽  
pp. 713-723
Author(s):  
S. P. Mishchenko ◽  
N. P. Panov ◽  
Yu. Yu. Frolova ◽  
Trang Nguyen
Keyword(s):  
Metabelian Algebras
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