General properties of the expansion methods of Lie algebras

TL;DR

This paper explores the properties and criteria of Lie algebra expansion methods, providing theorems and examples to understand how these expansions relate different Lie algebras and their implications in physics.

Contribution

The paper develops new theorems and criteria for understanding property preservation in Lie algebra expansions, enhancing the theoretical framework for their application in physics.

Findings

Identified conditions under which properties are preserved during algebra expansion

Derived criteria to determine when two algebras are related by expansion

Provided an explicit example illustrating the theoretical results

Abstract

The study of the relation between Lie algebras and groups, and especially the derivation of new algebras from them, is a problem of great interest in mathematics and physics, because finding a new Lie group from an already known one also means that a new physical theory can be obtained from a known one. One of the procedures that allow to do so is called expansion of Lie algebras, and has been recently used in different physical applications - particularly in gauge theories of gravity. Here we report on further developments of this method, required to understand in a deeper way their consequences in physical theories. We have found theorems related to the preservation of some properties of the algebras under expansions that can be used as criteria and, more specifically, as necessary conditions to know if two arbitrary Lie algebras can be related by the some expansion mechanism. Formal aspects, such as the Cartan decomposition of the expanded algebras, are also discussed. Finally, an instructive example that allows to check explicitly all our theoretical results is also provided.