Structure and Representation Theory of basic simple $\mathbb{Z}_2\times \mathbb{Z}_2$-graded color Lie algebras

TL;DR

This paper develops a root theory for basic simple $ ext{Z}_2 imes ext{Z}_2$-graded color Lie algebras, classifies their finite-dimensional representations, and proves key theorems assuming the Cartan subalgebra is self-centralizing.

Contribution

It introduces a root theory for basic simple color Lie algebras and classifies their finite-dimensional representations with new theorems.

Findings

Established a root theory for basic simple color Lie algebras.

Classified all finite-dimensional representations via highest weight and reducibility theorems.

Provided a framework for understanding representations assuming self-centralizing Cartan subalgebra.

Abstract

We adapt methods from the theory of complex semisimple Lie algebras to develop a root theory for a class of simple $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded (color) Lie algebras, which we call basic. As an application, assuming that the Cartan subalgebra is self-centralizing, we classify all finite-dimensional representations of these algebras by proving a highest weight theorem and a complete reducibility theorem.