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

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

2008-04-10

arxiv.org/abs/0804.1763v1

Mathematics

Rings and Algebras

20 pages

Scientific paper

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.

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