On the structures of split $\delta$ Jordan-Lie algebras

TL;DR

This paper analyzes the structure of split δ Jordan-Lie algebras with symmetric root systems, showing they decompose into well-described ideals and characterizing conditions for simplicity.

Contribution

It provides a structural decomposition of split δ Jordan-Lie algebras and characterizes their simplicity based on root system properties.

Findings

Algebras decompose into a direct sum of ideals with specific properties.

Conditions for simplicity are explicitly characterized.

Each minimal ideal is a simple split δ Jordan-Lie algebra with connected roots.

Abstract

We study the structures of arbitrary split $\delta$ Jordan-Lie algebras with symmetric root systems. We show that any of such algebras $L$ is of the form $L = U + \sum\limits_{[j] \in \Lambda/\sim}I_{[j]}$ with $U$ a subspace of $H$ and any $I_{[j]}$, a well described ideal of $L$, satisfying $[I_{[j]}, I_{[k]}] = 0$ if $[j]\neq [k]$. Under certain conditions, the simplicity of $L$ is characterized and it is shown that $L$ is the direct sum of the family of its minimal ideals, each one being a simple split $\delta$ Jordan-Lie algebra with a symmetric root system and having all its nonzero roots connected.