On a locally nilpotent radical Jacobson for special Lie algebras

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].

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A locally nilpotent radical of nonassociative rings

Algebra and Logic ◽

10.1007/bf02219808 ◽

1971 ◽

Vol 10 (4) ◽

pp. 219-224 ◽

Cited By ~ 16

Author(s):

G. V. Dorofeev

Keyword(s):

Nilpotent Radical ◽

Locally Nilpotent

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LOCALLY NILPOTENT IDEALS OF SPECIAL LIE ALGEBRAS

Communications in Algebra ◽

10.1081/agb-100105974 ◽

2001 ◽

Vol 29 (9) ◽

pp. 3781-3786 ◽

Cited By ~ 1

Author(s):

S. A. Pikhtilkov

Keyword(s):

Lie Algebras ◽

Locally Nilpotent

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Classical Radicals of Invariants of Algebraic Skew Derivations

Algebra Colloquium ◽

10.1142/s1005386722000050 ◽

2022 ◽

Vol 29 (01) ◽

pp. 53-66

Author(s):

Jeffrey Bergen ◽

Piotr Grzeszczuk

Keyword(s):

Hopf Algebra ◽

Lie Algebra ◽

Jacobson Radical ◽

Prime Radical ◽

Nilpotent Radical ◽

Enveloping Algebra ◽

Locally Nilpotent ◽

Nil Radical

Let [Formula: see text] be an automorphism and[Formula: see text] be a [Formula: see text]-skew [Formula: see text]-derivation of an [Formula: see text]-algebra [Formula: see text]. We prove that if [Formula: see text] is semiprimitive and [Formula: see text] is algebraic, then the subalgebra [Formula: see text] has nilpotent Jacobson radical. Using this result, we obtain similar relations for the Baer prime radical, the Levitzki locally nilpotent radical, and the Köthe nil radical when the field [Formula: see text] is uncountable. Then we apply it to actions of the [Formula: see text]-dimensional Taft Hopf algebra [Formula: see text] and the [Formula: see text]-analogue [Formula: see text] of the enveloping algebra of the Lie algebra [Formula: see text].

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Lie algebras with nilpotent radical

Ukrainian Mathematical Journal ◽

10.1007/bf01058491 ◽

1986 ◽

Vol 38 (2) ◽

pp. 223-223

Author(s):

Li Sun Gen

Keyword(s):

Lie Algebras ◽

Nilpotent Radical

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Classification of linked indecomposable modules of a family of solvable Lie algebras over an arbitrary field of characteristic 0

Journal of Algebra and Its Applications ◽

10.1142/s0219498816500298 ◽

2015 ◽

Vol 15 (02) ◽

pp. 1650029 ◽

Cited By ~ 1

Author(s):

Leandro Cagliero ◽

Fernando Szechtman

Keyword(s):

Lie Algebra ◽

Lie Algebras ◽

Weight Space ◽

Arbitrary Field ◽

Nilpotent Radical ◽

Indecomposable Modules ◽

Finite Dimensional ◽

A Chain ◽

Algebra Over A Field

Let 𝔤 be a finite-dimensional Lie algebra over a field of characteristic 0, with solvable radical 𝔯 and nilpotent radical 𝔫 = [𝔤, 𝔯]. Given a finite-dimensional 𝔤-module U, its nilpotency series 0 ⊂ U(1) ⊂ ⋯ ⊂ U(m) = U is defined so that U(1) is the 0-weight space of 𝔫 in U, U(2)/U(1) is the 0-weight space of 𝔫 in U/U(1), and so on. We say that U is linked if each factor of its nilpotency series is a uniserial 𝔤/𝔫-module, i.e. its 𝔤/𝔫-submodules form a chain. Every uniserial 𝔤-module is linked, every linked 𝔤-module is indecomposable with irreducible socle, and both converses fail. In this paper, we classify all linked 𝔤-modules when 𝔤 = 〈x〉 ⋉ 𝔞 and ad x acts diagonalizably on the abelian Lie algebra 𝔞. Moreover, we identify and classify all uniserial 𝔤-modules amongst them.

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Locally nilpotent ideals of a Lie algebra

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100041177 ◽

1967 ◽

Vol 63 (2) ◽

pp. 257-272 ◽

Cited By ~ 16

Author(s):

B. Hartley

Keyword(s):

Lie Algebra ◽

Characteristic Zero ◽

Nilpotent Radical ◽

Finite Dimensional ◽

Locally Nilpotent

The purpose of this paper is to investigate the locally nilpotent radical of a Lie algebra L over a field of characteristic zero, its behaviour under derivations of L, and its behaviour with regard to finite-dimensional nilpotent subinvariant and ascendant subalgebras of L.

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Left 3-Engel elements in groups of exponent 60

International Journal of Algebra and Computation ◽

10.1142/s0218196718500303 ◽

2018 ◽

Vol 28 (04) ◽

pp. 673-695 ◽

Cited By ~ 2

Author(s):

Gareth Tracey ◽

Gunnar Traustason

Keyword(s):

Nilpotent Radical ◽

Engel Elements ◽

Locally Nilpotent ◽

Engel Element

Let [Formula: see text] be a group and let [Formula: see text] be a left [Formula: see text]-Engel element of order dividing [Formula: see text]. Suppose furthermore that [Formula: see text] has no elements of order [Formula: see text], [Formula: see text] and [Formula: see text]. We show that [Formula: see text] is then contained in the locally nilpotent radical of [Formula: see text]. In particular, all the left [Formula: see text]-Engel elements of a group of exponent [Formula: see text] are contained in the locally nilpotent radical.

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