Gradings on modules over Lie algebras of E types

TL;DR

This paper classifies all possible gradings of simple modules over the exceptional Lie algebras of types E6 and E7, providing a complete understanding of their graded structures and invariants.

Contribution

It computes graded Brauer invariants for all simple modules over E6 and E7, completing the classification of G-gradings on these Lie algebras.

Findings

Classification of G-graded simple modules for E6 and E7

Explicit computation of graded Brauer invariants

Conditions for modules to admit compatible G-gradings

Abstract

For any grading by an abelian group $G$ on the exceptional simple Lie algebra $\mathcal{L}$ of type $E_6$ or $E_7$ over an algebraically closed field of characteristic zero, we compute the graded Brauer invariants of simple finite-dimensional modules, thus completing the computation of these invariants for simple finite-dimensional Lie algebras. This yields the classification of $G$-graded simple $\mathcal{L}$-modules, as well as necessary and sufficient conditions for an $\mathcal{L}$-module to admit a $G$-grading compatible with the given $G$-grading on $\mathcal{L}$.