Assembling Lie Algebras from Lieons

TL;DR

This paper introduces a method to construct any finite-dimensional Lie algebra over real or complex numbers by assembling simpler building blocks called lieons, revealing new structural insights and classification techniques.

Contribution

It demonstrates that all finite-dimensional Lie algebras can be assembled from two basic lieons, providing a new perspective on their structure and classification.

Findings

Any finite-dimensional Lie algebra can be assembled from two types of lieons.

Classical Lie algebras can be constructed from lieons using disassembling techniques.

Complete classification of Lie algebras assembled from lieons is achieved.

Abstract

If a Lie algebra structures $\gG$ on a vector space is the sum of a family of mutually compatible Lie algebra structures $\gG_i$, we say that $\gG$ is \emph{simply assembled} from $\gG_s$'s. By repeating this procedure several times one gets a family of Lie algebras \emph{assembled} from $\gG_s$'s. The central result of this paper is that any finite dimensional Lie algebra over $\R$ or $\C$ can be assembled from two constituents, called $\between$- and $\pitchfork$-\emph{lieons}. A lieon is the direct sum of an abelian Lie algebra with a 2-dmensional nonabelian Lie algebra or with the 3-dimensional Heisenberg algebra. Some techniques of disassembling Lie algebras are introduced and various results concerning assembling-disassembling procedures are obtained. In particular, it is shown how classical Lie algebras are assembled from lieons and is obtained the complete list of Lie algebras, which can be simply assembled from lieons.