A $\mathbb{Z}_4^3$-grading on a 56-dimensional simple structurable algebra and related fine gradings on the simple Lie algebras of type E

TL;DR

This paper constructs a specific $Z_4^3$-grading on a 56-dimensional simple structurable algebra, the Brown algebra, and demonstrates how it induces new fine gradings on exceptional Lie algebras E6, E7, and E8.

Contribution

It introduces a novel $Z_4^3$-grading on the Brown algebra and links it to new fine gradings on key exceptional Lie algebras.

Findings

Constructed a $Z_4^3$-grading on the Brown algebra.

Derived new fine gradings on E6, E7, and E8.

Established connections between structurable algebra gradings and Lie algebra gradings.

Abstract

We describe two constructions of a certain $\mathbb{Z}_4^3$-grading on the so-called Brown algebra (a simple structurable algebra of dimension 56 and skew-dimension 1) over an algebraically closed field of characteristic different from 2 and 3. We also show how this grading gives rise to several interesting fine gradings on exceptional simple Lie algebras of types E6, E7 and E8.