Particle-like structure of coaxial Lie algebras

Abstract

This paper is a natural continuation of Vinogradov [J. Math. Phys. 58, 071703 (2017)] where we proved that any Lie algebra over an algebraically closed field or over R can be assembled in a number of steps from two elementary constituents, called dyons and triadons. Here we consider the problems of the construction and classification of those Lie algebras which can be assembled in one step from base dyons and triadons, called coaxial Lie algebras. The base dyons and triadons are Lie algebra structures that have only one non-trivial structure constant in a given basis, while coaxial Lie algebras are linear combinations of pairwise compatible base dyons and triadons. We describe the maximal families of pairwise compatible base dyons and triadons called clusters, and, as a consequence, we give a complete description of the coaxial Lie algebras. The remarkable fact is that dyons and triadons in clusters are self-organised in structural groups which are surrounded by casings and linked by connectives. We discuss generalisations and applications to the theory of deformations of Lie algebras.

Particle-like, dyx-coaxial and trix-coaxial Lie algebra structures for a multi-dimensional continuous Toda type system

“…He considered the problems of the construction and classification of those Lie algebras which can be assembled in one step from base dyons and triadons, called coaxial Lie algebras. The base dyons and triadons are Lie algebra structures that have only one non-trivial structure constant in a given basis, while coaxial Lie algebras are linear combinations of pairwise compatible base dyons and triadons [24]. Here for the convenience of the reader we recall some basic facts of the theory.…”

“…We now recall Proposition 3.1 of [24] stating some necessary and sufficient conditions for the compatibility or incompatibility of particle-like Lie algebra structures:…”

“…In this note we show that such (otherwise discarded) symmetries deserve a more careful study. We take a (2 + 1)-dimensional Toda type system as a study case and show that it posses algebraic properties related to the recently introduced concept of particle-like Lie algebra structures [23,24].…”

Particle-like structure of Lie algebras

“…The problem of an explicit description of the Lie algebras of the first two levels seems to be attainable. Some useful suggestions on how to solve it can be found in [14] where a simplified version of this problem is solved in the case of first-level Lie algebras. A solution of this problem would significantly enrich the assembling techniques, since it presumes a systematic analysis of obstructions to compatibility.…”

Alexandre Mikhailovich Vinogradov (obituary)

“…Чем бы ни занимался А. М. Виноградов -геометрией уравнений, скобками Схоутена и Нийенхейса [22], [23], математическими вопросами теории гравитации [30]- [32], n-арными обобщениями алгебр Ли [25]- [27] или структурным анализом последних [33], [34], -все его работы объединяет неортодоксальность подхода, глубина и нетривиальность результатов. Его печатное наследие составляют более сотни статей и десять монографий.…”