Algebra and Applications 1. Abdenacer Makhlouf

of proof: let ω be a primitive cubic root of 1 in an algebraic closure of , and let , so that if . A separable alternative algebra over is, up to isomorphism, one of the following:

       – 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 be an algebraically closed field of characteristic ≠ 2, 3. Then any simple finite-dimensional algebra satisfying

[2.9]

      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 (, ∗, n) satisfies [2.9] so the unique, up to isomorphism, Okubo algebra over an algebraically closed field of characteristic ≠ 2, 3 must be isomorphic to the last algebra in the theorem above, and this seems to be the first appearance of these algebras in the literature.

      This results in the next definition:

      DEFINITION 2.3 (Knus et al. (1998, §34.b)).– Let (, ∙, n) be a Hurwitz algebra, and let φ ∈ Aut(, ∙, n) be an automorphism with φ³ = id. The composition algebra (, ∗, n), with

      is called a Petersson algebra, and denoted by .

      In case φ = id, the Petersson algebra is the para-Hurwitz algebra associated with (, ∙, n).

      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 (, ∙, n) be the split Cayley algebra over an arbitrary field . The Petersson algebra is called the split Okubo algebra over .

      Its twisted forms (i.e. those composition algebras (, ∗, n) that become isomorphic to the split Okubo algebra after extending scalars to an algebraic closure) are called Okubo 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 be a field of characteristic 3 and let 0 ≠ α, . Consider the elements

      in ( being an algebraic closure of ). These elements generate, by multiplication and linear combinations over , a twisted form of the split Okubo algebra (, ∗, n). Denote by this twisted form.

Figure 2.2. Multiplication table of the split Okubo algebra

      The classification