alt="image"/>, then we must consider COROLLARY 2.4.– The algebras in examples 2.1 essentially exhaust, up to isomorphism, the symmetric composition algebras over a field
Sketch of proof: let ω be a primitive cubic root of 1 in an algebraic closure of
– a central simple associative algebra, and hence we obtain the Okubo algebras in examples 2.1;
– for a Hurwitz algebra , in which case (, ∗, n) is shown to be isomorphic to the para-Hurwitz algebra attached to if , and (K (K (, J)0, ∗, n) to the para-Hurwitz algebra attached to if ;
– , for a cubic field extension of (if ), in which case the symmetric composition algebra is shown to be a twisted form of a two-dimensional para-Hurwitz algebra. □
One of the clues to understand symmetric composition algebras over fields of characteristic 3 is the following result of Petersson (1969) (dealing with char
THEOREM 2.5.– Let
for any x, y, z is, up to isomorphism, one of the following:
– the algebra (, ∙), where (, ∙, n) is a Hurwitz algebra and (that is, a para-Hurwitz algebra);
– the algebra (, ∗), where is the split Cayley algebra, and , where φ is a precise order 3 automorphism of given, in the basis in Figure 2.1 by
where ω is a primitive cubic root of 1.
Note that any symmetric composition algebra (
This results in the next definition:
DEFINITION 2.3 (Knus et al. (1998, §34.b)).– Let (
is called a Petersson algebra, and denoted by
In case φ = id, the Petersson algebra is the para-Hurwitz algebra associated with (
Modifying the automorphism in theorem 2.5, consider the order 3 automorphism φ of the split Cayley algebra given by:
With this automorphism, we may define Okubo algebras over arbitrary fields (see Elduque and Pérez (1996)).
DEFINITION 2.4.– Let (
Its twisted forms (i.e. those composition algebras (
In the basis in Figure 2.1, the multiplication table of the split Okubo algebra is given in Figure 2.2.
Over fields of characteristic ≠ 3, our new definition of Okubo algebras coincide with the definition in examples 2.1, due to corollary 2.4. Okubo and Osborn (1981b) had given an ad hoc definition of the Okubo algebra over an algebraically closed field of characteristic 3.
Note that the split Okubo algebra does not contain any non-zero element that commutes with every other element, that is, its commutative center is trivial. This is not so for the para-Hurwitz algebra, where the para-unit lies in the commutative center.
Let
