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].

Weitzenböck derivations of free metabelian associative algebras

2017 ◽  
Vol 16 (03) ◽  
pp. 1750041 ◽  
Author(s):  
Rumen Dangovski ◽  
Vesselin Drensky ◽  
Şehmus Fındık
Keyword(s):  
Hilbert Series ◽  
Polynomial Algebra ◽  
Finitely Generated ◽  
Associative Algebras ◽  
Triangular Matrices ◽  
Commutator Ideal ◽  
Several Variables ◽  
Locally Nilpotent ◽  
Derivation Formula ◽  
Infinite Set

By the classical theorem of Weitzenböck the algebra of constants [Formula: see text] of a nonzero locally nilpotent linear derivation [Formula: see text] of the polynomial algebra [Formula: see text] in several variables over a field [Formula: see text] of characteristic 0 is finitely generated. As a noncommutative generalization one considers the algebra of constants [Formula: see text] of a locally nilpotent linear derivation [Formula: see text] of a finitely generated relatively free algebra [Formula: see text] in a variety [Formula: see text] of unitary associative algebras over [Formula: see text]. It is known that [Formula: see text] is finitely generated if and only if [Formula: see text] satisfies a polynomial identity which does not hold for the algebra [Formula: see text] of [Formula: see text] upper triangular matrices. Hence the free metabelian associative algebra [Formula: see text] is a crucial object to study. We show that the vector space of the constants [Formula: see text] in the commutator ideal [Formula: see text] is a finitely generated [Formula: see text]-module, where [Formula: see text] acts on [Formula: see text] and [Formula: see text] in the same way as on [Formula: see text]. For small [Formula: see text], we calculate the Hilbert series of [Formula: see text] and find the generators of the [Formula: see text]-module [Formula: see text]. This gives also an (infinite) set of generators of the algebra [Formula: see text].

scholarly journals FAITHFUL REPRESENTATIONS OF MINIMAL DIMENSION OF CURRENT HEISENBERG LIE ALGEBRAS

2009 ◽  
Vol 20 (11) ◽  
pp. 1347-1362 ◽  
Author(s):  
LEANDRO CAGLIERO ◽  
NADINA ROJAS
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Polynomial Algebra ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Abelian Subalgebras ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Minimal Dimension

Given a Lie algebra 𝔤 over a field of characteristic zero k, let μ(𝔤) = min{dim π : π is a faithful representation of 𝔤}. Let 𝔥m be the Heisenberg Lie algebra of dimension 2m + 1 over k and let k [t] be the polynomial algebra in one variable. Given m ∈ ℕ and p ∈ k [t], let 𝔥m, p = 𝔥m ⊗ k [t]/(p) be the current Lie algebra associated to 𝔥m and k [t]/(p), where (p) is the principal ideal in k [t] generated by p. In this paper we prove that [Formula: see text]. We also prove a result that gives information about the structure of a commuting family of operators on a finite dimensional vector space. From it is derived the well-known theorem of Schur on maximal abelian subalgebras of 𝔤𝔩(n, k ).

The Schreier Technique for Subalgebras of a Free Lie Algebra

1997 ◽  
Vol 49 (3) ◽  
pp. 600-616 ◽  
Author(s):  
Shmuel Rosset ◽  
Alon Wasserman
Keyword(s):  
Group Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Free Group ◽  
Finite Codimension ◽  
Free Lie Algebra ◽  
Finitely Generated ◽  
Free Lie Algebras

AbstractIn group theory Schreier's technique provides a basis for a subgroup of a free group. In this paper an analogue is developed for free Lie algebras. It hinges on the idea of cutting a Hall set into two parts. Using it, we show that proper subalgebras of finite codimension are not finitely generated and, following M. Hall, that a finitely generated subalgebra is a free factor of a subalgebra of finite codimension.

Complements of Minimal Ideals in Solvable Lie Rings

1980 ◽  
Vol 23 (3) ◽  
pp. 363-366
Author(s):  
Ernest L. Stitzinger
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Noetherian Ring ◽  
Minimal Ideal ◽  
Finitely Generated ◽  
Lie Rings ◽  
Solvable Lie Algebra ◽  
Solvable Lie Algebras

AbstractConditions for the existence and conjugacy of complements of certain minimal ideals of solvable Lie algebras over a Noetherian ring R are considered. Let L be a solvable Lie algebra and A be a minimal ideal of L. If L/A is nilpotent and L is not nilpotent then A has a complement in L, all such complements are conjugate and self-normalizing and if C is a complement then there exists an x∈L such that C = {y∈L; yadnx = 0 for some n = 1, 2,…}. A similar result holds if A is self-centralizing and a finitely generated R-module.

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 The invariant polynomial algebras for the groups IU(n) and ISO(n)

1984 ◽  
Vol 94 ◽  
pp. 43-59 ◽  
Author(s):  
Hitoshi Kaneta
Keyword(s):  
Lie Algebras ◽  
Lie Group ◽  
Dual Space ◽  
Polynomial Algebra ◽  
Invariant Polynomial ◽  
Coadjoint Representation ◽  
Finitely Generated ◽  
Polynomial Algebras ◽  
Enveloping Algebras ◽  
Connected Lie Group

By the coadjoint representation of a connected Lie group G with the Lie algebra g we mean the representation CoAd(g) = tAd(g-1) in the dual space g*. Imitating Chevalley’s argument for complex semi-simple Lie algebras, we shall show that the CoAd (G)-invariant polynomial algebra on g* is finitely generated by algebraically independent polynomials when G is the inhomogeneous linear group IU(n) or ISO(n). In view of a well-known theorem [8, p. 183] our results imply that the centers of the enveloping algebras for the (or the complexified) Lie algebras of these groups are also finitely generated. Recently much more inhomogeneous groups have been studied in a similar context [2]. Our results, however, are further reaching as far as the groups IU(n) and ISO(n) are concerned [cf. 3, 4, 6, 7, 9].

ON ISOMORPHISM OF LIE ALGEBRAS WITH ONE DEFINING RELATION

2004 ◽  
Vol 14 (03) ◽  
pp. 389-393 ◽  
Author(s):  
ALEXANDER A. MIKHALEV ◽  
VLADIMIR SHPILRAIN ◽  
UALBAI U. UMIRBAEV
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Free Lie Algebra ◽  
Finitely Generated ◽  
Defining Relation

Let L be a finitely generated free Lie algebra. We construct an example of two elements u and v of L such that the factor algebras L/(u) and L/(v) are isomorphic, where (u) and (v) are ideals of L generated by u and v, respectively, but there is no automorphism φ of L such that φ(u)=v.

GROWTH OF SUBALGEBRAS FOR RESTRICTED LIE ALGEBRAS AND TRANSITIVE ACTIONS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 1151-1168 ◽  
Author(s):  
V. M. PETROGRADSKY
Keyword(s):  
Group Theory ◽  
Finite Field ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Finitely Generated ◽  
Subgroup Growth ◽  
Restricted Lie Algebra ◽  
Restricted Lie Algebras ◽  
Transitive Actions

We study a growth of subalgebras for restricted Lie algebras over a finite field 𝔽q. This kind of growth is an analog of the subgroup growth in the group theory. Let L be a finitely generated restricted Lie algebra. Then an(L) is the number of restricted subalgebras H ⊂ L such that dim 𝔽q L/H = n, n ≥ 0. We compute the numbers an(Fd) explicitly and find asymptotics, where Fd is the free restricted Lie algebra of rank d, d ≥ 1. As an important instrument, we use the notion of transitive L-action on coalgebras and algebras.

scholarly journals The group of automorphisms of the Lie algebra of derivations of a polynomial algebra

2017 ◽  
Vol 16 (05) ◽  
pp. 1750088 ◽  
Author(s):  
V. V. Bavula
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Short Proof ◽  
Polynomial Algebra ◽  
Group Of Automorphisms ◽  
Cartan Type

A short proof is given of Rudakov’s result (announced in [A. N. Rudakov, Subalgebras and automorphisms of Lie algebras of Cartan type, Funktsional. Anal. i Prilozhen. 20(1) (1986) 83–84]), that the group of automorphisms of the Lie algebra [Formula: see text] of derivations of a polynomial algebra [Formula: see text], over a field of characteristic zero is canonically isomorphic to the group of automorphisms of the polynomial algebra [Formula: see text].

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.

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