LOCALLY NILPOTENT IDEALS OF SPECIAL LIE ALGEBRAS

2001 ◽  
Vol 29 (9) ◽  
pp. 3781-3786 ◽  
Author(s):  
S. A. Pikhtilkov
Keyword(s):  
Lie Algebras ◽  
Locally Nilpotent

scholarly journals The Nowicki conjecture for free metabelian Lie algebras

2019 ◽  
Vol 19 (05) ◽  
pp. 2050095
Author(s):  
Vesselin Drensky ◽  
Şehmus Fındık
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Polynomial Algebra ◽  
Vector Subspace ◽  
Classical Theorem ◽  
Finitely Generated ◽  
Commutator Ideal ◽  
Ideal Formula ◽  
Locally Nilpotent ◽  
Derivation Formula

Let [Formula: see text] be the polynomial algebra in [Formula: see text] variables over a field [Formula: see text] of characteristic 0. The classical theorem of Weitzenböck from 1932 states that for linear locally nilpotent derivations [Formula: see text] (known as Weitzenböck derivations), the algebra of constants [Formula: see text] is finitely generated. When the Weitzenböck derivation [Formula: see text] acts on the polynomial algebra [Formula: see text] in [Formula: see text] variables by [Formula: see text], [Formula: see text], [Formula: see text], Nowicki conjectured that [Formula: see text] is generated by [Formula: see text] and [Formula: see text] for all [Formula: see text]. There are several proofs based on different ideas confirming this conjecture. Considering arbitrary Weitzenböck derivations of the free [Formula: see text]-generated metabelian Lie algebra [Formula: see text], with few trivial exceptions, the algebra [Formula: see text] is not finitely generated. However, the vector subspace [Formula: see text] of the commutator ideal [Formula: see text] of [Formula: see text] is finitely generated as a [Formula: see text]-module. In this paper, we study an analogue of the Nowicki conjecture in the Lie algebra setting and give an explicit set of generators of the [Formula: see text]-module [Formula: see text].

scholarly journals On a locally nilpotent radical Jacobson for special Lie algebras

2021 ◽  
Vol 22 (1) ◽  
pp. 234-272
Author(s):  
Olga Alexandrovna Pikhtilkova ◽  
Elena Vladimirovna Mescherina ◽  
Anna Nikolaevna Blagovisnava ◽  
Elena Vladislavovna Pronina ◽  
Olga Alekseevna Evseeva
Keyword(s):  
Lie Algebras ◽  
Nilpotent Radical ◽  
Locally Nilpotent

Vector Fields and Automorphism Groups of Danielewski Surfaces

2020 ◽  
Author(s):  
Matthias Leuenberger ◽  
Andriy Regeta
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Vector Fields ◽  
Automorphism Groups ◽  
Locally Nilpotent

Abstract In this paper, we study the Lie algebra of vector fields ${\operatorname{Vec}}(\textrm{D}_p)$ of a smooth Danielewski surface $\textrm{D}_p$. We prove that the Lie subalgebra $\langle{\operatorname{LNV}}(\textrm{D}_p) \rangle$ of ${\operatorname{Vec}}(\textrm{D}_p)$ generated by locally nilpotent vector fields is simple. Moreover, if the two Lie algebras $\langle{\operatorname{LNV}}(\textrm{D}_p) \rangle$ and $\langle{\operatorname{LNV}}(\textrm{D}_q) \rangle$ of two Danielewski surfaces $\textrm{D}_p$ and $\textrm{D}_q$ are isomorphic, then the surfaces $\textrm{D}_p$ and $\textrm{D}_q$ are isomorphic. As an application we prove that the ind-groups ${\operatorname{Aut}}(\textrm{D}_p)$ and ${\operatorname{Aut}}(\textrm{D}_q)$ are isomorphic if and only if $\textrm{D}_p \simeq \textrm{D}_q$ as a variety. We also show that any automorphism of the ind-group ${\operatorname{Aut}}^\circ (\textrm{D}_p)$ is inner.

scholarly journals Maximal Lie algebras among locally nilpotent derivations

2021 ◽  
Vol 212 (2) ◽  
Author(s):  
Alexander Andreevich Skutin
Keyword(s):  
Lie Algebras ◽  
Locally Nilpotent

scholarly journals Nilpotent Lie algebras of derivations with the center of small corank

2020 ◽  
Vol 12 (1) ◽  
pp. 189-198
Author(s):  
Y.Y. Chapovskyi ◽  
L.Z. Mashchenko ◽  
A.P. Petravchuk
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Integral Domain ◽  
Natural Basis ◽  
Nilpotent Lie Algebras ◽  
Locally Nilpotent ◽  
Nilpotent Subalgebra

Let $\mathbb K$ be a field of characteristic zero, $A$ be an integral domain over $\mathbb K$ with the field of fractions $R=Frac(A),$ and $Der_{\mathbb K}A$ be the Lie algebra of all $\mathbb K$-derivations on $A$. Let $W(A):=RDer_{\mathbb K} A$ and $L$ be a nilpotent subalgebra of rank $n$ over $R$ of the Lie algebra $W(A).$ We prove that if the center $Z=Z(L)$ is of rank $\geq n-2$ over $R$ and $F=F(L)$ is the field of constants for $L$ in $R,$ then the Lie algebra $FL$ is contained in a locally nilpotent subalgebra of $ W(A)$ of rank $n$ over $R$ with a natural basis over the field $R.$ It is also proved that the Lie algebra $FL$ can be isomorphically embedded (as an abstract Lie algebra) into the triangular Lie algebra $u_n(F)$, which was studied early by other authors.

A generalized Noetherian condition for Lie algebras

2019 ◽  
Vol 18 (08) ◽  
pp. 1950146
Author(s):  
Falih A. M. Aldosray ◽  
Ian Stewart
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
General Class ◽  
Maximal Condition ◽  
Locally Nilpotent

A Lie algebra (over any field and of any dimension) is Noetherian if it satisfies the maximal condition on ideals. We introduce a new and more general class of quasi-Noetherian Lie algebras that possess several of the main properties of Noetherian Lie algebras. This class is shown to be closed under quotients and extensions. We obtain conditions under which a quasi-Noetherian Lie algebra is Noetherian. Next, we consider various questions about locally nilpotent and soluble radicals of quasi-Noetherian Lie algebras. We show that there exists a semisimple quasi-Noetherian Lie algebra that is not Noetherian. Finally, we consider some analogous results for groups and prove that a quasi-Noetherian group is countably recognizable.

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