Particle-like Structure of Lie algebras

TL;DR

This paper introduces a novel framework for decomposing and assembling finite-dimensional Lie algebras using elementary building blocks called dyons and triadons, providing a new perspective on their structure.

Contribution

It develops a modular disassembling and assembling approach for Lie algebras, showing any finite-dimensional Lie algebra can be constructed from dyons and triadons.

Findings

Any finite-dimensional Lie algebra can be assembled from dyons and triadons.

The disassembling process separates a Lie algebra into unimodular and non-unimodular parts.

Classical Lie algebras can be constructed from triadons.

Abstract

If a Lie algebra structure g on a vector space is the sum of a family of mutually compatible Lie algebra structures g_i's, we say that g is simply assembled from the g_i's. Repeating this procedure with a number of Lie algebras, themselves simply assembled from the g_i's, one obtains a Lie algebra assembled in two steps from the g_i's, and so on. We describe the process of modular disassembling of a Lie algebra into a unimodular and a non-unimodular part. We then study two inverse questions: which Lie algebras can be assembled from a given family of Lie algebras, and from which Lie algebras can a given Lie algebra be assembled? We develop some basic assembling and disassembling techniques that constitute the elements of a new approach to the general theory of Lie algebras. The main result of our theory is that any finite-dimensional Lie algebra over an algebraically closed field of characteristic zero or over R can be assembled in a finite number of steps from two elementary constituents, which we call dyons and triadons. Up to an abelian summand, a dyon is a Lie algebra structure isomorphic to the non-abelian 2-dimensional Lie algebra, while a triadon is isomorphic to the 3-dimensional Heisenberg Lie algebra. As an example, we describe constructions of classical Lie algebras from triadons.