Algebra and Applications 1. Abdenacer Makhlouf

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alt="image"/>, so that .

      Fix a basis {u₁, u₂, u₃} of with n(u₁ ∙ u₂, u₃₎ = 1 and take v₁ := u₂ ∙ u₃, v₂ := u₃ ∙ u₁, v₃ := u₁ ∙ u₂. Then {v₁, v₂, v_3} is the dual basis in relative to the norm, and the multiplication of the basis {e₁, e₂, u₁, u₂, u₃, v₁, v₂, v₃} is completely determined. For instance, v₁ ∙ v₂ = v₁ ∙ (u₃ ∙ u₁₎ = −u₃ ∙ (v₁ ∙ u₁₎ = −u₃ ∙ (−n(v₁, u₁₎e₂) = u₃, …

The multiplication table is given in Figure 2.1.

      The Cayley algebra with this multiplication table is called the split Cayley algebra and denoted by . The subalgebra spanned by e₁, e₂, u₁, v₁ is isomorphic to the algebra of 2 × 2 matrices.

      We summarize the above arguments in the next result.

      THEOREM 2.2.– There are, up to isomorphism, only three Hurwitz algebras with isotropic norm: × , and .

      COROLLARY 2.3.– The real Hurwitz algebras are, up to isomorphism, the following algebras:

       – the classical division algebras ℝ, ℂ, ℍ, and ;

       – the algebras ℝ × ℝ, and .

Figure 2.1. Multiplication table of the split Cayley algebra

      PROOF.– It is enough to take into account that a non-degenerate quadratic form over ℝ is either isotropic or definite. Hence, the norm of a real Hurwitz algebra is either isotropic or positive definite (as n(1) = 1). □

2.4. Symmetric composition algebras

      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

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