Alexander Skutin - Academia.edu (original) (raw)

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Papers by Alexander Skutin

In this short note we confirm an analog of a conjecture of James Wiegold for finite dimensional n... more In this short note we confirm an analog of a conjecture of James Wiegold for finite dimensional nilpotent Lie algebras.

Sbornik: Mathematics

Let be a nilpotent Lie algebra. By the breadth of an element of we mean the number . Vaughan-Lee ... more Let be a nilpotent Lie algebra. By the breadth of an element of we mean the number . Vaughan-Lee showed that if the breadth of all elements of the Lie algebra is bounded by a number , then the dimension of the commutator subalgebra of the Lie algebra does not exceed . We show that if for some nonnegative , then the Lie algebra is generated by the elements of breadth , and thus we prove a conjecture due to Wiegold (Question 4.69 in the Kourovka Notebook) in the case of nilpotent Lie algebras. Bibliography: 4 titles.

Sbornik: Mathematics

We study maximal Lie subalgebras among locally nilpotent derivations of the polynomial algebra. F... more We study maximal Lie subalgebras among locally nilpotent derivations of the polynomial algebra. Freudenburg conjectured that the triangular Lie algebra of locally nilpotent derivations of the polynomial algebra is a maximal Lie algebra contained in the set of locally nilpotent derivations, and that every maximal Lie algebra contained in the set of locally nilpotent derivations is conjugate to the triangular Lie algebra. In this paper we prove the first part of the conjecture and present a counterexample to the second part. We also show that under a certain natural condition imposed on a maximal Lie algebra there is a conjugation taking this Lie algebra to the triangular Lie algebra. Bibliography: 2 titles.

arXiv: Commutative Algebra, 2020

It has been conjectured by Gene Freudenburg that for a polynomial ring, the triangular Lie algebr... more It has been conjectured by Gene Freudenburg that for a polynomial ring, the triangular Lie algebra is the maximal Lie algebra which lies in the set of locally nilpotent derivations of the ring. Also it was conjectured that each other maximal Lie algebra which lies in the set of locally nilpotent derivations of the polynomial ring is conjugated to the triangular Lie algebra. In the present work we prove the first part of this conjecture and provide the counterexample to the second part. Also we show that the second part of the conjecture holds for the maximal Lie algebras among locally nilpotent derivations with some natural additional properties.

arXiv: Metric Geometry, 2019

This paper focuses on a new approach to plane geometry and develops important concepts that can a... more This paper focuses on a new approach to plane geometry and develops important concepts that can allow researchers to unite and observe plane geometry from a new, meaningful perspective.

The number of φ,ψ-twisted conjugacy classes is described for every pair of endomorphisms φ,ψ of a... more The number of φ,ψ-twisted conjugacy classes is described for every pair of endomorphisms φ,ψ of a finetly generated nilpotent Lie algebra.

Исследуются максимальные подалгебры Ли среди локально нильпотентных дифференцирований алгебры мно... more Исследуются максимальные подалгебры Ли среди локально нильпотентных дифференцирований алгебры многочленов. Дж. Фройденбургом была высказана гипотеза о том, что треугольная алгебра Ли локально нильпотентных дифференцирований алгебры многочленов является максимальной алгеброй Ли, содержащейся в множестве локально нильпотентных дифференцирований, и гипотеза о том, что каждая максимальная алгебра Ли, содержащаяся в множестве локально нильпотентных дифференцирований, сопряжена треугольной алгебре Ли. В настоящей работе мы доказываем справедливость первой части гипотезы и приводим контрпример ко второй ее части. Также мы покажем, что при некотором естественном условии, наложенном на максимальную алгебру Ли, существует сопряжение, переводящее эту алгебру Ли в треугольную алгебру Ли. Библиография: 2 названия.

In this article we will represent some ideas and a lot of new theorems in Euclidean plane geometry.

In this short note we confirm an analog of a conjecture of James Wiegold for finite dimensional n... more In this short note we confirm an analog of a conjecture of James Wiegold for finite dimensional nilpotent Lie algebras.

Sbornik: Mathematics

Let be a nilpotent Lie algebra. By the breadth of an element of we mean the number . Vaughan-Lee ... more Let be a nilpotent Lie algebra. By the breadth of an element of we mean the number . Vaughan-Lee showed that if the breadth of all elements of the Lie algebra is bounded by a number , then the dimension of the commutator subalgebra of the Lie algebra does not exceed . We show that if for some nonnegative , then the Lie algebra is generated by the elements of breadth , and thus we prove a conjecture due to Wiegold (Question 4.69 in the Kourovka Notebook) in the case of nilpotent Lie algebras. Bibliography: 4 titles.

Sbornik: Mathematics

We study maximal Lie subalgebras among locally nilpotent derivations of the polynomial algebra. F... more We study maximal Lie subalgebras among locally nilpotent derivations of the polynomial algebra. Freudenburg conjectured that the triangular Lie algebra of locally nilpotent derivations of the polynomial algebra is a maximal Lie algebra contained in the set of locally nilpotent derivations, and that every maximal Lie algebra contained in the set of locally nilpotent derivations is conjugate to the triangular Lie algebra. In this paper we prove the first part of the conjecture and present a counterexample to the second part. We also show that under a certain natural condition imposed on a maximal Lie algebra there is a conjugation taking this Lie algebra to the triangular Lie algebra. Bibliography: 2 titles.

arXiv: Commutative Algebra, 2020

It has been conjectured by Gene Freudenburg that for a polynomial ring, the triangular Lie algebr... more It has been conjectured by Gene Freudenburg that for a polynomial ring, the triangular Lie algebra is the maximal Lie algebra which lies in the set of locally nilpotent derivations of the ring. Also it was conjectured that each other maximal Lie algebra which lies in the set of locally nilpotent derivations of the polynomial ring is conjugated to the triangular Lie algebra. In the present work we prove the first part of this conjecture and provide the counterexample to the second part. Also we show that the second part of the conjecture holds for the maximal Lie algebras among locally nilpotent derivations with some natural additional properties.

arXiv: Metric Geometry, 2019

This paper focuses on a new approach to plane geometry and develops important concepts that can a... more This paper focuses on a new approach to plane geometry and develops important concepts that can allow researchers to unite and observe plane geometry from a new, meaningful perspective.

The number of φ,ψ-twisted conjugacy classes is described for every pair of endomorphisms φ,ψ of a... more The number of φ,ψ-twisted conjugacy classes is described for every pair of endomorphisms φ,ψ of a finetly generated nilpotent Lie algebra.

Исследуются максимальные подалгебры Ли среди локально нильпотентных дифференцирований алгебры мно... more Исследуются максимальные подалгебры Ли среди локально нильпотентных дифференцирований алгебры многочленов. Дж. Фройденбургом была высказана гипотеза о том, что треугольная алгебра Ли локально нильпотентных дифференцирований алгебры многочленов является максимальной алгеброй Ли, содержащейся в множестве локально нильпотентных дифференцирований, и гипотеза о том, что каждая максимальная алгебра Ли, содержащаяся в множестве локально нильпотентных дифференцирований, сопряжена треугольной алгебре Ли. В настоящей работе мы доказываем справедливость первой части гипотезы и приводим контрпример ко второй ее части. Также мы покажем, что при некотором естественном условии, наложенном на максимальную алгебру Ли, существует сопряжение, переводящее эту алгебру Ли в треугольную алгебру Ли. Библиография: 2 названия.

In this article we will represent some ideas and a lot of new theorems in Euclidean plane geometry.