Algebra and Applications 1. Abdenacer Makhlouf

, for any x, y, z, so that (, ∙, n) is a symmetric composition algebra (note that for any x: 1 is a para-unit of (, ∙, n)).

       – Okubo algebras: assume char (the case of char requires a different definition), and let be a primitive cubic root of 1. Let be a central simple associative algebra of degree 3 with trace tr, and let . For any , the quadratic form make sense even if char (check this!). Now define a multiplication and norm on by:

      Then, for any x, :

      But if tr(x) = 0, then , so

      Since , we have .

      Therefore, (, ∗, n) is a symmetric composition algebra.

      In case , take and a central simple associative algebra of degree 3 over endowed with a -involution of second kind J. Then take (this is an -subspace) and use the same formulas above to define the multiplication and the norm.

      REMARK 2.5.– For , take , and then there appears the Okubo algebra (, ∗, n) with (x∗ denotes the conjugate transpose of x). This algebra was termed the algebra of pseudo-octonions by Okubo (1978), who studied these algebras and classified them, under some restrictions, in joint work with Osborn Okubo and Osborn (1981a,b).

      The name Okubo algebras was given in Elduque and Myung (1990). Faulkner (1988) discovered Okubo’s construction independently, in a more general setting, related to separable alternative algebras of degree 3, and gave the key idea for the classification of the symmetric composition algebras in Elduque and Myung (1993) (char ). A different, less elegant, classification was given in Elduque and Myung (1991), based on the fact that Okubo algebras are Lie-admissible.

      The term symmetric composition algebra was given in Knus et al. (1998, Chapter VIII).

      REMARK 2.6.– Given an Okubo algebra, note that for any x, ,

      so that

      and

[2.8]

      so the product in is determined by the product in the Okubo algebra.

      Also, as noted by Faulkner, the construction above is valid for separable alternative algebras of degree 3.

      THEOREM 2.4 (Elduque and Myung (1991, 1993)).– Let be a field of characteristic not 3.

       – If contains a primitive cubic root ω of 1, then the symmetric composition algebras of dimension ≥ 2 are, up to isomorphism, the algebras (, ∗, n) for a separable alternative algebra of degree 3.

      Two such symmetric composition algebras are isomorphic if and only if the corresponding alternative algebras are too.

       – If does not contain primitive cubic roots of 1, then the symmetric composition algebras of dimension ≥ 2 are, up to isomorphism, the algebras (K(, J)0, ∗, n) for a separable alternative algebra of degree 3 over , and J a -involution of the second kind.

      Two such symmetric composition algebras are isomorphic if and only if the corresponding alternative algebras, as algebras with involution, are too.

Sketch of proof : we can go in the reverse direction of Okubo’s construction. Given a symmetric composition algebra (, ∗, n) over a field containing ω, define the algebra with multiplication determined by formula [2.8]. Then turns out to be a separable alternative algebra of degree 3.

In case , then we must consider , with the same formula for the product. In , we have the Galois automorphism ωτ = ω². Then the conditions J(1) = 1 and J(s) = −s for any induce a -involution of the second kind in . □

COROLLARY 2.4.– The algebras in examples 2.1 essentially exhaust, up to isomorphism, the symmetric composition algebras over a field