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David Towers

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I'm Professor of Pure Mathematics at Lancaster University. My research interests lie in the areas of General nonassociative algebras (Frattini subalgebra, Generators, Lie triple systems, Leibniz algebras, Genetic algebras), Lie algebras (Solvable and Nilpotent Lie algebras, subalgebra lattice, Maximal subalgebras, A-algebras, Elementary algebras, Complemented algebras, Decomposition results, Formation Theory), Group Theory (Areas related to my interests in Lie algebras) and Mathematical Education.

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Papers by David Towers

Supplements to maximal subalgebras of Lie algebras

arXiv (Cornell University), Jul 28, 2010

Quasi-ideals of Leibniz algebras

arXiv (Cornell University), Apr 8, 2020

On the Lengths of Certain Chains of Subalgebras in Lie Algebras

arXiv (Cornell University), Dec 10, 2011

Zinbiel algebras are Nilpotent

Journal of Algebra and Its Applications, Apr 30, 2022

In this paper we show that every finite-dimensional Zinbiel algebra over an arbitrary field is ni... more

Abelian subalgebras and ideals of maximal dimension in Zinbiel algebras

Communications in Algebra, Oct 20, 2022

Chief factors of Lie algebras

arXiv (Cornell University), Dec 29, 2015

Abelian subalgebras and ideals of maximal dimension in Leibniz algebras

arXiv (Cornell University), May 14, 2021

Classes of Algebras and closure operations

arXiv (Cornell University), Nov 20, 2020

Further results on elementary Lie algebras and Lie A-algebras

arXiv (Cornell University), Apr 20, 2009

Lie algebras with nilpotent length greater than that of each of their subalgebras

arXiv (Cornell University), Aug 24, 2016

Complements of Intervals and Prefrattini Subalgebras of Solvable Lie Algebras

arXiv (Cornell University), Apr 19, 2011

The index complex of a maximal subalgebra of a Lie algebra

arXiv (Cornell University), Jun 29, 2010

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

arXiv (Cornell University), Oct 11, 2011

Leibniz A-algebras

arXiv (Cornell University), May 24, 2019

A finite-dimensional Lie algebra is called an A-algebra if all of its nilpotent subalgebras are a... more A finite-dimensional Lie algebra is called an A-algebra if all of its nilpotent subalgebras are abelian. These arise in the study of constant Yang-Mills potentials and have also been particularly important in relation to the problem of describing residually finite varieties. They have been studied by several authors, including Bakhturin, Dallmer, Drensky, Sheina, Premet, Semenov, Towers and Varea. In this paper we establish generalisations of many of these results to Leibniz algebras.

Almost-reductive and almost-algebraic Leibniz algebra

International Electronic Journal of Algebra, Jan 7, 2024

Guide to Abstract Algebra

The Mathematical Gazette, 1989

Higher-order Bernstein algebras given by symmetric bilinear forms

Linear Algebra and its Applications, 1997

Let (A, w) be a k th-order Bernstein algebra and let N be the kernel of W. This article studies t... more Let (A, w) be a k th-order Bernstein algebra and let N be the kernel of W. This article studies the structure of such algebras in which N2 has dimension one. The algebras are of two types, I and II, according as N 2 C_ U or N s g U. A characteriza tion of the algebras of type I is given. Power associative k th-order Bernstein algebras with dim N s = 1 are then considered: they turn out to be Bernstein algebras of at most second order, and multiplication tables for these algebras over the real field are given. Finally, second-order Bernstein algebras of type II are examined and a structure theorem for them is obtained.

On n-maximal subalgebras of Lie algebras

arXiv (Cornell University), Feb 10, 2015

A chain S 0 < S 1 <. .. < S n = L is a maximal chain if each S i is a maximal subalgebra of S i+1... more A chain S 0 < S 1 <. .. < S n = L is a maximal chain if each S i is a maximal subalgebra of S i+1. The subalgebra S 0 in such a series is called an n-maximal subalgebra. There are many interesting results concerning the question of what certain intrinsic properties of the maximal subalgebras of a Lie algebra L imply about the structure of L itself. Here we consider whether similar results can be obtained by imposing conditions on the n-maximal subalgebras of L, where n > 1.

Lattice isomorphisms of Lie algebras (addendum)

Lemma 4-1, which appears on p. 289 of [2], is false. Nevertheless, Theorem 4-2, the only result d... more Lemma 4-1, which appears on p. 289 of [2], is false. Nevertheless, Theorem 4-2, the only result dependent upon it, is true. The following supplies the necessary modification to the proof. Replace Lemma 4-1 by LEMMA 4-1. Let S be a three-dimensional non-split simple Lie algebra, and let R be an irreducible S-module. Then, for any seS, R has an ad s-invariant svbspace of dimension less than or equal to two.

Leibniz $A$-algebras

summary:A finite-dimensional Lie algebra is called an $A$-algebra if all of its nilpotent subalge... more summary:A finite-dimensional Lie algebra is called an $A$-algebra if all of its nilpotent subalgebras are abelian. These arise in the study of constant Yang-Mills potentials and have also been particularly important in relation to the problem of describing residually finite varieties. They have been studied by several authors, including Bakhturin, Dallmer, Drensky, Sheina, Premet, Semenov, Towers and Varea. In this paper we establish generalisations of many of these results to Leibniz algebras

Supplements to maximal subalgebras of Lie algebras

arXiv (Cornell University), Jul 28, 2010

Quasi-ideals of Leibniz algebras

arXiv (Cornell University), Apr 8, 2020

On the Lengths of Certain Chains of Subalgebras in Lie Algebras

arXiv (Cornell University), Dec 10, 2011

Zinbiel algebras are Nilpotent

Journal of Algebra and Its Applications, Apr 30, 2022

In this paper we show that every finite-dimensional Zinbiel algebra over an arbitrary field is ni... more

Abelian subalgebras and ideals of maximal dimension in Zinbiel algebras

Communications in Algebra, Oct 20, 2022

Chief factors of Lie algebras

arXiv (Cornell University), Dec 29, 2015

Abelian subalgebras and ideals of maximal dimension in Leibniz algebras

arXiv (Cornell University), May 14, 2021

Classes of Algebras and closure operations

arXiv (Cornell University), Nov 20, 2020

Further results on elementary Lie algebras and Lie A-algebras

arXiv (Cornell University), Apr 20, 2009

Lie algebras with nilpotent length greater than that of each of their subalgebras

arXiv (Cornell University), Aug 24, 2016

Complements of Intervals and Prefrattini Subalgebras of Solvable Lie Algebras

arXiv (Cornell University), Apr 19, 2011

The index complex of a maximal subalgebra of a Lie algebra

arXiv (Cornell University), Jun 29, 2010

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

arXiv (Cornell University), Oct 11, 2011

Leibniz A-algebras

arXiv (Cornell University), May 24, 2019

A finite-dimensional Lie algebra is called an A-algebra if all of its nilpotent subalgebras are a... more A finite-dimensional Lie algebra is called an A-algebra if all of its nilpotent subalgebras are abelian. These arise in the study of constant Yang-Mills potentials and have also been particularly important in relation to the problem of describing residually finite varieties. They have been studied by several authors, including Bakhturin, Dallmer, Drensky, Sheina, Premet, Semenov, Towers and Varea. In this paper we establish generalisations of many of these results to Leibniz algebras.

Almost-reductive and almost-algebraic Leibniz algebra

International Electronic Journal of Algebra, Jan 7, 2024

Guide to Abstract Algebra

The Mathematical Gazette, 1989

Higher-order Bernstein algebras given by symmetric bilinear forms

Linear Algebra and its Applications, 1997

Let (A, w) be a k th-order Bernstein algebra and let N be the kernel of W. This article studies t... more Let (A, w) be a k th-order Bernstein algebra and let N be the kernel of W. This article studies the structure of such algebras in which N2 has dimension one. The algebras are of two types, I and II, according as N 2 C_ U or N s g U. A characteriza tion of the algebras of type I is given. Power associative k th-order Bernstein algebras with dim N s = 1 are then considered: they turn out to be Bernstein algebras of at most second order, and multiplication tables for these algebras over the real field are given. Finally, second-order Bernstein algebras of type II are examined and a structure theorem for them is obtained.

On n-maximal subalgebras of Lie algebras

arXiv (Cornell University), Feb 10, 2015

A chain S 0 < S 1 <. .. < S n = L is a maximal chain if each S i is a maximal subalgebra of S i+1... more A chain S 0 < S 1 <. .. < S n = L is a maximal chain if each S i is a maximal subalgebra of S i+1. The subalgebra S 0 in such a series is called an n-maximal subalgebra. There are many interesting results concerning the question of what certain intrinsic properties of the maximal subalgebras of a Lie algebra L imply about the structure of L itself. Here we consider whether similar results can be obtained by imposing conditions on the n-maximal subalgebras of L, where n > 1.

Lattice isomorphisms of Lie algebras (addendum)

Lemma 4-1, which appears on p. 289 of [2], is false. Nevertheless, Theorem 4-2, the only result d... more Lemma 4-1, which appears on p. 289 of [2], is false. Nevertheless, Theorem 4-2, the only result dependent upon it, is true. The following supplies the necessary modification to the proof. Replace Lemma 4-1 by LEMMA 4-1. Let S be a three-dimensional non-split simple Lie algebra, and let R be an irreducible S-module. Then, for any seS, R has an ad s-invariant svbspace of dimension less than or equal to two.

Leibniz $A$-algebras

summary:A finite-dimensional Lie algebra is called an $A$-algebra if all of its nilpotent subalge... more summary:A finite-dimensional Lie algebra is called an $A$-algebra if all of its nilpotent subalgebras are abelian. These arise in the study of constant Yang-Mills potentials and have also been particularly important in relation to the problem of describing residually finite varieties. They have been studied by several authors, including Bakhturin, Dallmer, Drensky, Sheina, Premet, Semenov, Towers and Varea. In this paper we establish generalisations of many of these results to Leibniz algebras