Lie Algebras with Nilpotent Length Greater than that of each of their Subalgebras (original) (raw)

Abelian subalgebras and ideals of maximal dimension in supersolvable and nilpotent lie algebras

Linear and Multilinear Algebra, 2020

In this paper, we continue the study of abelian subalgebras and ideals of maximal dimension for finite-dimensional supersolvable and nilpotent Lie algebras. We show that supersolvable Lie algebras with an abelian subalgebra of codimension 3 contain an abelian ideal with the same dimension, provided that the characteristic of the underlying field is not two, and that the same is true for nilpotent Lie algebras with an abelian subalgebra of codimension 4, provided that the characteristic of the field is greater than five.

Nilpotent subspaces of maximal dimension in semi-simple Lie algebras

Compositio Mathematica, 2006

We show that a linear subspace of a reductive Lie algebra g that consists of nilpotent elements has dimension at most 1 2 (dim g−rk g), and that any nilpotent subspace attaining this upper bound is equal to the nilradical of a Borel subalgebra of g. This generalizes a classical theorem of Gerstenhaber, which states this fact for the algebra of (n×n)-matrices. 1. Results and method A classical theorem of Gerstenhaber [Ger58] states that any vector space consisting of nilpotent (n × n)-matrices has dimension at most n 2 , and that any such space attaining this maximal possible dimension is conjugate to the space of upper triangular matrices. We will prove the following generalization of this result. Main Theorem. Let V be a linear subspace of a complex semi-simple Lie algebra g consisting of ad-nilpotent elements. Then dim V 1 2 (dim g − rk g) and, if equality holds, then V is the nilradical of a Borel subalgebra of g.

On maximal solvable extensions of a pure non-characteristically nilpotent Lie algebra

Cornell University - arXiv, 2021

In this paper we introduce the notion of pure non-characteristically nilpotent Lie algebra and under a condition we prove that a complex maximal extension of a finite-dimensional pure noncharacteristically nilpotent Lie algebra is isomorphic to a semidirect sum of the nilradical and its maximal torus. We also prove that such solvable Lie algebras are complete and we specify a subclass of the maximal solvable extensions of pure non-characteristically nilpotent Lie algebras that have trivial cohomology group. Some comparisons with the results obtained earlier are given.

On abelian subalgebras and ideals of maximal dimension in supersolvable Lie algebras

Journal of Pure and Applied Algebra, 2014

In this paper, the main objective is to compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional supersolvable Lie algebras. We characterise the maximal abelian subalgebras of solvable Lie algebras and study solvable Lie algebras containing an abelian subalgebra of codimension 2. Finally, we prove that nilpotent Lie algebras with an abelian subalgebra of codimension 3 contain an abelian ideal with the same dimension, provided that the characteristic of the underlying field is not two. Throughout the paper, we also give several examples to clarify some results.

Maximal solvable extensions of a pure non-characteristically nilpotent Lie algebra

2021

In the paper we prove that a complex maximal extension of a non-characteristically nilpotent finite-dimensional nilpotent Lie algebra is isomorphic to a semidirect sum of the nilradical and its maximal torus, where the nilpotent Lie algebra is either non-split and non-characteristically nilpotent or in the split case it does not contain a characteristically nilpotent direct factor. We also prove that such solvable Lie algebras are complete. It is also shown that among all complex solvable Lie algebras only the maximal extensions of nilpotent algebras do not have outer derivations. We specify a subclass of the maximal solvable extensions of nilpotent Lie algebras that have trivial cohomology group. Some comparisons with the results obtained earlier are given.

Lie algebras all of whose maximal subalgebras have codimension one

Proceedings of the Edinburgh Mathematical Society, 1981

Let denote the class of finite-dimensional Lie algebras L (over a fixed, but arbitrary, field F) all of whose maximal subalgebras have codimension 1 in L. In (2) Barnes proved that the solvable algebras in are precisely the supersolvable ones. The purpose of this paper is to extend this result and to give a characterisation of all of the algebras in . Throughout we shall place no restrictions on the underlying field of the Lie algebra.

Capability of Nilpotent Lie Algebras of Small Dimension

Bulletin of the Iranian Mathematical Society

Given a nilpotent Lie algebra L of dimension ≤ 6 on an arbitrary field of characteristic = 2, we show a direct method which allows us to detect the capability of L via computations on the size of its nonabelian exterior square L ∧ L. For dimensions higher than 6, we show a result of general nature, based on the evidences of the low dimensional case, focusing on generalized Heisenberg algebras. Indeed we detect the capability of L ∧ L via the size of the Schur multiplier M (L/Z ∧ (L)) of L/Z ∧ (L), where Z ∧ (L) denotes the exterior center of L. 1. Statement of the results and terminology Throughout this paper all Lie algebras are considered over a prescribed field F and [ , ] denotes the Lie bracket. Let L and K be two Lie algebras. Following [6, 7], an action of L on K is an F-bilinear map L × K → K given by (l, k) → l k satisfying [l,l ′ ] k = l (l ′ k) − l ′ (l k) and l [k, k ′ ] = [ l k, k ′ ] + [k, l k ′ ], for all l, l ′ ∈ L and k, k ′ ∈ K. These actions are compatible, if k l k ′ = k ′ l k and l k l ′ = l ′ k l for all l, l ′ ∈ L and k, k ′ ∈ K. Now L ⊗ K is the Lie algebra generated by the symbols l ⊗ k subject to the relations c(l ⊗ k) = cl ⊗ k = l ⊗ ck, (l + l ′) ⊗ k = l ⊗ k + l ′ ⊗ k,

Solvable Lie Algebras and Maximal Abelian Dimensions

2008

In this paper some results on the structure of finite-dimensional Lie algebras are obtained by means of the concept of maximal abelian dimension. More concretely, a sucient condition is given for the solvability in finite-dimensional Lie algebras by using maximal abelian dimensions. Besides, a necessary condition for the nilpotency is also stated for such Lie algebras. Finally, the maximal abelian