Symmetries on the real form $\mathfrak e_{6,-26}$

TL;DR

This paper classifies four specific fine gradings on the real Lie algebra _{6,-26}, linking them to gradings on its complex form and detailing their universal grading groups.

Contribution

It identifies and describes four fine gradings on _{6,-26} with explicit universal grading groups, expanding understanding of its symmetry structure.

Findings

Four fine gradings on _{6,-26} are characterized.

Each grading's universal group is explicitly determined.

Gradings correspond to those on the complex algebra _6.

Abstract

We describe four fine gradings on the real form $\mathfrak e_{6,-26}$. They are precisely the gradings whose complexifications are fine gradings on the complexified algebra $\mathfrak{e}_6$. The universal grading groups are $\mathbb Z_2^6$, $\mathbb Z\times\mathbb Z_2^4$, $\mathbb Z^2\times\mathbb Z_2^3$, and $\mathbb Z_4\times\mathbb Z_2^4$.