Fine Gradings on the exceptional Lie algebra $\mathfrak d_4$

TL;DR

This paper classifies all fine group gradings on the exceptional Lie algebra d4, identifying fourteen distinct gradings through computational and algebraic methods, including triality and outer automorphisms.

Contribution

It provides a complete classification of fine gradings on d4, combining computational and algebraic approaches to identify all possible gradings.

Findings

Fourteen fine gradings identified

Two complementary methods used: computational and algebraic

Emphasis on gradings involving outer automorphisms and triality

Abstract

We describe all the fine group gradings, up to equivalence, on the Lie algebra $\mathfrak d_4$. This problem is equivalent to finding the maximal abelian diagonalizable subgroups of the automorphism group of $\mathfrak d_4$. We prove that there are fourteen by using two different viewpoints. The first approach is computational: we get a full description of the gradings by using a particular implementation of the automorphism group of the Dynkin diagram of $\mathfrak d_4$ and some algebraic groups stuff. The second approach, more qualitative, emphasizes some algebraic aspects, as triality, and it is mostly devoted to gradings involving the outer automorphisms of order three.