Fine group gradings of the real forms of $sl(4,\C)$, $sp(4,\C)$, and $o(4,\C)$

TL;DR

This paper explicitly classifies all fine group gradings of the 12 real forms of the semisimple Lie algebras $sl(4, ext{C})$, $sp(4, ext{C})$, and $o(4, ext{C})$, totaling 44 gradings.

Contribution

It provides a comprehensive and explicit description of all fine group gradings for these real Lie algebras, utilizing their matrix representations and inclusion relations.

Findings

44 fine group gradings identified

Systematic use of faithful matrix representations

Clarification of inclusion relations among algebras

Abstract

We present an explicit description of the 'fine group gradings' (i.e. group gradings which cannot be further refined) of the real forms of the semisimple Lie algebras $sl(4,\C)$, $sp(4,\C)$, and $o(4,\C)$. All together 12 real Lie algebras are considered, and the total of 44 of their fine group gradings are listed. The inclusions $sl(4,\C)\supset sp(4,\C)\supset o(4,\C)$ are an important tool in our presentation. Systematic use is made of the faithful representations of the three Lie algebras by $4\times 4$ matrices.