Quasi-classical Lie algebras and their contractions

TL;DR

This paper classifies low-dimensional quasi-classical Lie algebras, explores their contractions, and demonstrates how contractions can generate families of Lagrangians and recover Yang-Mills equations.

Contribution

It provides new conditions for contractions of quasi-classical Lie algebras and illustrates their applications in physics, such as Yang-Mills theories.

Findings

Classification of low-dimensional indecomposable quasi-classical Lie algebras

Existence of non-reductive stable quasi-classical Lie algebras

Contractions can generate parameterized families of Lagrangians

Abstract

After classifying indecomposable quasi-classical Lie algebras in low dimension, and showing the existence of non-reductive stable quasi-classical Lie algebras, we focus on the problem of obtaining sufficient conditions for a quasi-classical Lie algebras to be the contraction of another quasi-classical algebra. It is illustrated how this allows to recover the Yang-Mills equations of a contraction by a limiting process, and how the contractions of an algebra may generate a parameterized families of Lagrangians for pairwise non-isomorphic Lie algebras.