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In this section, a new important family of composition algebras will be described.

DEFINITION 2.2.– A composition algebra (, ∗, n) is said to be a symmetric composition algebra if for any (that is, n(x ∗ y, z) = n(x, y ∗ z) for any x, y, ).

THEOREM 2.3.– Let (, ∗, n) be a composition algebra. The following conditions are equivalent:

1 a) (, ∗, n) is symmetric;

2 b) for any x, , (x ∗ y) ∗ x = x ∗ (y ∗ x) = n(x)y.

The dimension of any symmetric composition algebra is finite, and hence restricted to 1, 2, 4 or 8.

PROOF.– If (, ∗, n) is symmetric, then for any x, y, ,

so that . Also

whence (b), since n is non-singular.

Conversely, take x, y, with n(y) ≠ 0, so that Ly and Ry are bijective, and hence there is an element with z = zʹ ∗ y. Then:

This proves (a) assuming n(y) ≠ 0, but any isotropic element is the sum of two non-isotropic elements, so (a) follows.

Finally, we can use a modified version of Kaplansky’s trick (see corollary 2.2) as follows. Let be a norm 1 element and define a new product on by:

for any x, . Then (, ◊, n) is also a composition algebra. Let e = a^∗2. Then, using (b) we have e ◊ x = a ∗ (a ∗ a)) ∗ (x ∗ a) = a ∗ (x ∗ a) = x, and similarly x ◊ e = x for any x. Thus, (, ◊, n) is a Hurwitz algebra with unity e, and hence it is finite-dimensional. □

REMARK 2.4.– Condition (b) above implies that ((x ∗ y) ∗ x) ∗ (x ∗ y) = n(x ∗ y)x, but also ((x ∗ y) ∗ x) ∗ (x ∗ y) = n(x)y ∗ (x ∗ y) = n(x)n(y)x, so that condition (b) already forces the quadratic form n to be multiplicative.

EXAMPLES 2.1 (Okubo (1978)).–

 – Para-Hurwitz algebras: let (, ∙, n) be a Hurwitz algebra and consider the composition algebra (, ∙, n) with the new product given by

Then , 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 of characteristic not 3.

Sketch 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 of the symmetric composition algebras in characteristic 3, which completes the classification of symmetric composition algebras over fields, is as follows (Elduque (1997), see also Chernousov et al. (2013)):

THEOREM 2.6.– Any symmetric composition algebra (, ∗, n) over a field of characteristic 3 is either:

 – a para-Hurwitz algebra. Two such algebras are isomorphic if and only if the associated Hurwitz algebras are too;

 – a two-dimensional algebra with a basis {u, v} and multiplication given by

for a non-zero scalar . These algebras do not contain idempotents and are twisted forms of the para-Hurwitz algebras.

Algebras corresponding to the scalars λ and λʹ are isomorphic if and only if .

 – Isomorphic to for some 0 ≠ α, . Moreover, is isomorphic or anti-isomorphic to if and only if .

A more precise statement for the isomorphism condition in the last item is given in (Elduque 1997). A key point in the proof of this theorem is the study of idempotents on Okubo algebras. If there are non-zero idempotents, then these algebras are Petersson algebras. The most difficult case appears in the absence of idempotents. This is only possible if the ground field is not perfect.