Primitive, almost primitive, test, and \(\Delta\)-primitive elements of free algebras with the Nielsen-Schreier property
type: Item
instance of: scholarly article
title: Primitive, almost primitive, test, and \(\Delta\)-primitive elements of free algebras with the Nielsen-Schreier property
zbMATH Open document ID: 0956.17001
author: A. A. Mikhalev
author: Jie-Tai Yu
published in: Journal of Algebra
publication date: 2001-03-14T00:00:00Z
review text: Let \(k\) be a field of characteristic different from 2 and \(F\) a free algebra in a Schreier variety \(V\) of nonassociative \(k\)-algebras defined by a set of homogeneous identities. An element \(v\in F\) is primitive if \(v\) can be included into some set of generators of \(F\). Let \(F\) have two free generators and \(u\in F\setminus 0\). Suppose that \(\phi\) is an endomorphism of \(F\) such that the automorphic orbit of \(u\) is stable under the action of \(\phi\). Then \(\phi\) is an automorphism of \(F\). A non-primitive element \(u\in F\) is almost primitive if it is primitive in any proper subalgebra of \(F\) containing \(u\). It is shown that \(x_1x_2+x_3x_4+\cdots +x_{2m-1}x_{2m}\) is almost primitive in \(F\) provided \(\{x_1,x_2,\ldots, x_{2m}\}\) is the set of free generators in \(F\). Similarly \(x_1x_2+x_3x_4+\cdots +x_{2m-1}x_{2m}+x_{2m}x_{2m+1}\) is almost primitive in the free nonassociative algebra \(F\) provided \(\{x_1,x_2,\ldots, x_{2m+1}\}\) is the set of free generators in \(F\). There are given various examples of almost primitive elements in a free Lie algebra, free Lie \(p\)-algebra, free Lie superalgebra, free Lie \(p\)-superalgebra. An element \(a\neq 0\) in an algebra \(A\) is a test element if for any endomorphism \(\phi\) of \(A\) the equality \(\phi(a)=a\) means that \(\phi\) is an automorphism. An almost primitive element \(a\in F\setminus 0\) is a test one if its constituent of degree 1 vanishes. It means that all examples of almost primitive elements listed above are in fact test elements. Let \(F\) be a free Lie algebra with a free set of generators \(X=\{x_1,\ldots,x_n\}\). Then the universal envelope \(U\) is a free associative algebra on \(X\). If \(u\in U\), then \(u=\alpha+x_1a_1+\cdots +x_na_n\), where \(a_i\in U,\alpha\in k\). Put \(a_i=\frac{\partial u}{\partial x_i}\). An element \(a\in F\) is \(\Delta\)-primitive if the elements \(\frac{\partial a}{\partial x_1},\ldots \frac{\partial u}{\partial x_n}\) generate the l…
Mathematics Subject Classification ID: 17A50
Mathematics Subject Classification ID: 17B01
zbMATH DE Number: 1471831
zbMATH Keywords: Lie superalgebras
zbMATH Keywords: primitive elements
zbMATH Keywords: almost primitive
zbMATH Keywords: test elements
zbMATH Keywords: almost elements
zbMATH Keywords: free algebras
reviewed by: Vyacheslav A. Artamonov
MaRDI profile type: Publication
full work available at URL: jabr.2000.8288
OpenAlex ID: W1967515256
cites work: PROJECTIVE METABELIAN GROUPS AND LIE ALGEBRAS
cites work: Infinite dimensional Lie superalgebras
cites work: Supports of derivations, free factorizations, and ranks of fixed subgroups in free groups
cites work: Q4300256
cites work: On Almost Primitive Elements of Free Groups With an Application to Fuchsian Groups
cites work: Two dimensional coordinate polynomials and dominant maps
cites work: What does a basis of F(a,b) look like?
cites work: Markoff forms and primitive words
cites work: Generic elements of free groups
cites work: Retracts of polynomial rings
cites work: A Commutator Test for Two Elements to Generate the Free Algebra of Rank Two
cites work: Q3722604
cites work: Q3994461
cites work: Nullhomologous words in free groups which are not nullhomologous in any proper subgroup
cites work: Test polynomials for automorphisms of polynomial and free associative algebras
cites work: Orbits in free algebras of rank two
cites work: The Mal'tsev-Nielsen equation in a free metabelian group of rank two
cites work: Some combinatorial questions about polynomial mappings
cites work: New classes of test polynomials of polynomial algebras
cites work: Test words, generic elements and almost primitivity
cites work: Q4244058
cites work: Nielsen's commutator test for two-generator groups
cites work: Commutators, generators and conjugacy equations in groups
cites work: On endomorphisms of free groups that preserve primitivity
cites work: Q5622399
cites work: A solution of the problem of van den Essen and Shpilrain
cites work: Automorphisms of the fuchsian groups op type (0;2,2,2,q;0)
cites work: Q5801396
cites work: On Schreier Varieties of Linear Algebras
cites work: Q5513066
cites work: Q3697196
cites work: Q3748446
cites work: Q3789741
cites work: Q4332787
cites work: Q4939384
cites work: Symmetrization in the geometric theory of functions of a complex variable
cites work: Q4332803
cites work: Test elements and retracts of free Lie algebras‡
cites work: Test Elements, Retracts and Automorphic Orbits of Free Algebras
cites work: Q4851269
cites work: RANK AND PRIMITIVITY OF ELEMENTS OF FREE COLOUR LIE (p-)SUPERALGEBRAS
cites work: Automorphisms and primitive elements of free lie superalgebras
cites work: Test elements for monomorphisms of free lie algebras and lie superalgebras†
cites work: Q4888904
cites work: Primitives in the free group on two generators
cites work: On components of polynomial automorphisms in two variables
cites work: Q3134850
cites work: Commutator equations in free groups
cites work: Alternierende Produkte in freien Gruppen
cites work: Q3216798
cites work: Q4729967
cites work: A property of subgroups of free groups
cites work: Q3947798
cites work: Q5821489
cites work: Q5825747
cites work: Recognizing automorphisms of the free groups
cites work: On the Rank of an Element of a Free Lie Algebra
cites work: On monomorphisms of free groups

Primitive, almost primitive, test, and \(\Delta\)-primitive elements of free algebras with the Nielsen-Schreier property (cites work)
cites work: Test elements for endomorphisms of free groups and algebras

Primitive, almost primitive, test, and \(\Delta\)-primitive elements of free algebras with the Nielsen-Schreier property (cites work)
cites work: Generalized primitive elements of a free group

Primitive, almost primitive, test, and \(\Delta\)-primitive elements of free algebras with the Nielsen-Schreier property (cites work)
cites work: Polynomial automorphisms and Gröbner reductions

Primitive, almost primitive, test, and \(\Delta\)-primitive elements of free algebras with the Nielsen-Schreier property (cites work)
cites work: On generators of polynomial algebras in two commuting or non-commuting variables

Primitive, almost primitive, test, and \(\Delta\)-primitive elements of free algebras with the Nielsen-Schreier property (cites work)
cites work: Polynomial Retracts and the Jacobian Conjecture

Primitive, almost primitive, test, and \(\Delta\)-primitive elements of free algebras with the Nielsen-Schreier property (cites work)
cites work: Q3717247

Primitive, almost primitive, test, and \(\Delta\)-primitive elements of free algebras with the Nielsen-Schreier property (cites work)
cites work: Q4124942

Primitive, almost primitive, test, and \(\Delta\)-primitive elements of free algebras with the Nielsen-Schreier property (cites work)
cites work: Test Words for Automorphisms of Free Groups

Primitive, almost primitive, test, and \(\Delta\)-primitive elements of free algebras with the Nielsen-Schreier property (cites work)
cites work: Generalized analytic functions

Primitive, almost primitive, test, and \(\Delta\)-primitive elements of free algebras with the Nielsen-Schreier property (cites work)
cites work: Schreier varieties of algebras

Primitive, almost primitive, test, and \(\Delta\)-primitive elements of free algebras with the Nielsen-Schreier property (cites work)
cites work: Q4875144

Primitive, almost primitive, test, and \(\Delta\)-primitive elements of free algebras with the Nielsen-Schreier property (cites work)
cites work: Q4396596

Primitive, almost primitive, test, and \(\Delta\)-primitive elements of free algebras with the Nielsen-Schreier property (cites work)
cites work: Partitioning a primitive word into a generating set

Primitive, almost primitive, test, and \(\Delta\)-primitive elements of free algebras with the Nielsen-Schreier property (cites work)
cites work: On Certain Sets of Elements in a Free Group

Primitive, almost primitive, test, and \(\Delta\)-primitive elements of free algebras with the Nielsen-Schreier property (cites work)
cites work: Die Unterringe der freien Lieschen Ringe

Primitive, almost primitive, test, and \(\Delta\)-primitive elements of free algebras with the Nielsen-Schreier property (cites work)
cites work: Alternierende Produkte in freien Gruppen

Primitive, almost primitive, test, and \(\Delta\)-primitive elements of free algebras with the Nielsen-Schreier property (cites work)
cites work: Über Automorphismen ebener diskontinuierlicher Gruppen

Primitive, almost primitive, test, and \(\Delta\)-primitive elements of free algebras with the Nielsen-Schreier property (DOI)
DOI: 10.1006/JABR.2000.8288