Non-solvable contractions of semisimple Lie algebras in low dimension

TL;DR

This paper investigates non-solvable contractions of low-dimensional semisimple Lie algebras, linking the problem to embeddings and branching rules, and classifies all such contractions for algebras of dimension up to 8.

Contribution

It provides a complete classification of non-solvable contractions of low-dimensional semisimple Lie algebras using a stability theorem and representation theory.

Findings

Classified all deformations of Lie algebras with Levi decomposition in dimension ≤8.

Determined all non-solvable contractions for these low-dimensional algebras.

Linked contractions to embeddings and branching rules among semisimple Lie algebras.

Abstract

The problem of non-solvable contractions of Lie algebras is analyzed. By means of a stability theorem, the problem is shown to be deeply related to the embeddings among semisimple Lie algebras and the resulting branching rules for representations. With this procedure, we determine all deformations of indecomposable Lie algebras having a nontrivial Levi decomposition onto semisimple Lie algebras of dimension $n\leq 8$, and obtain the non-solvable contractions of the latter class of algebras.