Gradings on Octonions

The octonions form an Azumaya algebra in certain braided linear Gr-categories

“…Then by pulling back along F, we get the desired set of braidings of Vec ∂F G as presented in Theorem 1.4. Finally, we remark that it is possible to classify all the braided linear Gr-categories in which the octonion algebra is Azumaya by our method with a help of the result of Elduque [6] which provides all the gradings on the octonions by finite groups and thus all the possibilities of O as an algebra in braided linear Gr-categories. As this is only a matter of computations, we do not include further details.…”

Fine gradings and gradings by root systems on simple Lie algebras

“…The only gradings by root systems of rank ≥ 3 in Table 1 are the Cartan gradings, the gradings by the root system of type F 4 of E r , r = 6, 7, 8, and the grading by the root system of type C 3 of E 7 . In the second case, the coordinate algebra is H = K, H or O respectively, and the only grading on these algebras with neutral component equal to F1 are the gradings obtained by the Cayley-Dickson doubling process (see [Eld98] or [EK12]), whose universal groups are Z 2 , Z 2 2 and Z 3 2 respectively. The computation of the types is straightforward using the model T (H, A).…”

Gradings on the Albert algebra and on $\mathfrak{f}_4$

“…It has been first proved in [15], and then in [13,Subsection 3.3], that up to equivalence the unique nontoral grading on C is the Z …”

“…The precise meaning of this theorem is that there is some φ ∈ F 4 such that A := φAφ −1 is in some of the cases above. So we can replace at any moment A with some of its conjugated subgroups in the group • If some of the σ j 's is in the orbit of σ 15 we can suppose that f 1 = σ j t.…”