A Survey on Albert Algebras

Albert algebras over rings and related torsors

“…We need to verify the identities (1) of [P, 5.2] in all scalar extensions of R. It suffices to do this upon replacing R by a faithfully flat R-ring S. By Proposition 3.6, we can choose S so that M ⊗ S ≃ M C for some symmetric composition Salgebra C. Extending any isomorphism M ⊗ S ≃ M C by the identity on S 3 , we obtain a linear bijection H(M, Γ) ⊗ S = H(M ⊗ S, Γ) → H(C, Γ) that transforms the expressions for the norm, adjoint and trace into those of Proposition 3.2, from which the first two statements therefore follow. We conclude with [P,Theorem 17] and Proposition 3.2. Remark 3.9.…”

“…Over fields, a celebrated theorem that goes back, in its essential form, to Albert and Jacobson [AJ] (see [P,Theorem 21] for a more general statement) implies that for two octonion algebras C and C ′ , we have…”

“…However, by e.g. [P,Proposition 20], any isotope of the split Albert algebra over a field K is isomorphic it. As the next result shows, this is not the case over rings.…”

R‐triviality of groups of type F4 arising from the first Tits construction

“…It is clear that every isotopy is a norm similarity of . This holds for Albert algebras over commutative rings (see [16] for a discussion of such algebras) and implies that the structure group is a subgroup of the group of norm similarities. By [11, VI Theorem 7], if is a field with more than three elements, then every norm similarity is an isotopy.…”

“…Second proof By [2, Theorem 2.5], this kernel parametrizes the isomorphism classes of isotopes of . But from [16, Corollary 60], it follows that the invariants mod 2 and 3 of any isotope of the split Albert algebra are trivial. Thus all such isotopes are isomorphic, and the above kernel is trivial, as desired.…”