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

Draper, Cristina; Martín, Cándido; Viruel, Antonio

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

Authors

Draper, Cristina

Martín, Cándido

Viruel, Antonio

Description

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.
20 pages