scholarly journals Normal automorphisms of the metabelian product of free abelian Lie algebras

2020 ◽  
Vol 30 (2) ◽  
pp. 230-234
Author(s):  
N. Ş. Öğüşlü ◽  
Keyword(s):  
Lie Algebras ◽  
Finite Rank ◽  
Normal Automorphism ◽  
Metabelian Product

Let M be the metabelian product of free abelian Lie algebras of finite rank. In this study we prove that every normal automorphism of M is an IA-automorphism and acts identically on M′.

Central automorphisms of free nilpotent Lie algebras

2017 ◽  
Vol 16 (11) ◽  
pp. 1750205
Author(s):  
Özge Öztekin ◽  
Naime Ekici
Keyword(s):  
Lie Algebras ◽  
Finite Rank ◽  
Characteristic Zero ◽  
Sufficient Conditions ◽  
Nilpotency Class ◽  
Nilpotent Lie Algebra ◽  
Central Automorphism ◽  
Nilpotent Lie Algebras ◽  
Central Automorphisms

Let [Formula: see text] be the free nilpotent Lie algebra of finite rank [Formula: see text] [Formula: see text] and nilpotency class [Formula: see text] over a field of characteristic zero. We give a characterization of central automorphisms of [Formula: see text] and we find sufficient conditions for an automorphism of [Formula: see text] to be a central automorphism.

The test rank of a solvable product of free abelian Lie algebras

2019 ◽  
Vol 18 (02) ◽  
pp. 1950025
Author(s):  
Nazar Şahin Öğüşlü ◽  
Naime Ekici
Keyword(s):  
Lie Algebras ◽  
Finite Rank ◽  
Test Set ◽  
Test Rank ◽  
Solvable Product

Let [Formula: see text] be the [Formula: see text]th solvable product of free abelian Lie algebras of finite rank. We prove that the test rank of [Formula: see text] is one less than the number of the factors. We also give a test set for endomorphisms of [Formula: see text].

scholarly journals Semidomains and Metabelian Product of Metabelian Lie Algebras

2005 ◽  
Vol 131 (6) ◽  
pp. 6015-6022 ◽  
Author(s):  
E. Yu. Daniyarova ◽  
I. V. Kazatchkov ◽  
V. N. Remeslennikov
Keyword(s):  
Lie Algebras ◽  
Metabelian Product

scholarly journals EXTENSIONS OF JOHNSON'S AND MORITA'S HOMOMORPHISMS THAT MAP TO FINITELY GENERATED ABELIAN GROUPS

2013 ◽  
Vol 05 (01) ◽  
pp. 57-85 ◽  
Author(s):  
MATTHEW B. DAY
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Lie Algebras ◽  
Mapping Class Group ◽  
Finite Rank ◽  
Class Group ◽  
Homogeneous Spaces ◽  
Group Cohomology ◽  
Mapping Class ◽  
Nilpotent Lie Algebras

We extend each higher Johnson homomorphism to a crossed homomorphism from the automorphism group of a finite-rank free group to a finite-rank abelian group. We also extend each Morita homomorphism to a crossed homomorphism from the mapping class group of once-bounded surface to a finite-rank abelian group. This improves on the author's previous results [5]. To prove the first result, we express the higher Johnson homomorphisms as coboundary maps in group cohomology. Our methods involve the topology of nilpotent homogeneous spaces and lattices in nilpotent Lie algebras. In particular, we develop a notion of the "polynomial straightening" of a singular homology chain in a nilpotent homogeneous space.

scholarly journals ON THE DIMENSION OF PRODUCTS OF HOMOGENEOUS SUBSPACES IN FREE LIE ALGEBRAS

2013 ◽  
Vol 23 (01) ◽  
pp. 205-213 ◽  
Author(s):  
NIL MANSUROǦLU ◽  
RALPH STÖHR
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Rank ◽  
Homogeneous Component ◽  
Free Lie Algebra ◽  
Dimension Function ◽  
Free Lie Algebras ◽  
Explicit Formulae ◽  
Characteristic 2

Let L be a free Lie algebra of finite rank over a field K and let Ln denote the degree n homogeneous component of L. Formulae for the dimension of the subspaces [Lm, Ln] for all m and n were obtained by the second author and Michael Vaughan-Lee. In this note we consider subspaces of the form [Lm, Ln, Lk] = [[Lm, Ln], Lk]. Surprisingly, in contrast to the case of a product of two homogeneous components, the dimension of such products may depend on the characteristic of the field K. For example, the dimension of [L2, L2, L1] over fields of characteristic 2 is different from the dimension over fields of characteristic other than 2. Our main results are formulae for the dimension of [Lm, Ln, Lk]. Under certain conditions on m, n and k they lead to explicit formulae that do not depend on the characteristic of K, and express the dimension of [Lm, Ln, Lk] in terms of Witt's dimension function.

scholarly journals On Marginal Automorphisms of Free Nilpotent Lie Algebras

2020 ◽  
Vol 2020 ◽  
pp. 1-4
Author(s):  
Özge Öztekin
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Rank ◽  
Characteristic Zero ◽  
Nilpotent Lie Algebra ◽  
Nilpotent Lie Algebras

Let L be the free nilpotent Lie algebra of finite rank over a field of characteristic zero. We define the concepts of marginal ideals and marginal automorphisms of L, and we give some results on marginal automorphisms.

PRODUCTS OF HOMOGENEOUS SUBSPACES IN FREE LIE ALGEBRAS

2009 ◽  
Vol 19 (05) ◽  
pp. 699-703 ◽  
Author(s):  
RALPH STÖHR ◽  
MICHAEL VAUGHAN-LEE
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Rank ◽  
Free Lie Algebra ◽  
Free Lie Algebras ◽  
Free Generators

Let L be a free Lie algebra of finite rank. If n ≥ 1 then we let Ln denote the homogeneous subspace of L spanned by Lie products of weight n in the free generators of L. We obtain formulae for the dimensions of the subspaces [Lm,Ln] for all m and n.

TEST RANK OF F/R′ LIE ALGEBRAS

2006 ◽  
Vol 16 (04) ◽  
pp. 817-825 ◽  
Author(s):  
ZERRIN ESMERLIGIL ◽  
DILEK KAHYALAR ◽  
NAIME EKICI
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Rank ◽  
Integral Domain ◽  
Universal Enveloping Algebra ◽  
Free Lie Algebra ◽  
Enveloping Algebra ◽  
Test Rank ◽  
Ore Condition

Let F be a free Lie algebra of finite rank n and R be an ideal of F such that the universal enveloping algebra U(F/R) for F/R is an integral domain satisfying the Ore condition. We show that test rank for the Lie algebras of the form F/R′ is equal to n - 1 or n.

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