Lie Algebras with Prescribed sl3 Decomposition

TL;DR

This paper classifies Lie algebras containing an sl3 subalgebra with a specific module decomposition, linking their structure to structurable algebras through symmetry considerations.

Contribution

It explicitly determines the multiplication in such Lie algebras and connects their structure to structurable algebras using S4 symmetry.

Findings

Classification of Lie algebras with sl3 decomposition

Explicit multiplication formulas derived

Connections established with structurable algebras

Abstract

In this work, we consider Lie algebras L containing a subalgebra isomorphic to sl3 and such that L decomposes as a module for that sl3 subalgebra into copies of the adjoint module, the natural 3-dimensional module and its dual, and the trivial one-dimensional module. We determine the multiplication in L and establish connections with structurable algebras by exploiting symmetry relative to the symmetric group S4.