Abdenacer Makhlouf - Algebra and Applications 1

The non-degeneracy of the restriction of n to implies that is associative. In particular, any proper subalgebra of with non-degenerate restricted norm is associative.

Conversely, given an associative Hurwitz algebra with non-degenerate n, and a non-zero scalar , consider the direct sum of two copies of , with multiplication and norm given by [2.7], extending those on . The arguments above show that ( , ∙, n) is again a Hurwitz algebra, which is said to be obtained by the Cayley–Dickson doubling process from ( , ∙, n) and α . This algebra is denoted by .

REMARK 2.1.– is associative if and only if is commutative. This follows from x ∙ ( y ∙ u ) = ( y ∙ x ) ∙ u . If the algebra is associative, this equals ( x ∙ y ) ∙ u , and it forces x ∙ y = y ∙ x for any x , . The converse is an easy exercise.

We arrive at the main result of this section.

THEOREM 2.1 (Generalized Hurwitz theorem).– Every Hurwitz algebra over a field is isomorphic to one of the following:

1 1) the ground field ;

2 2) a two-dimensional separable commutative and associative algebra: , with v∙2 = v + μ1, with 4μ +1 ≠ 0, and n(∊ + δv) = ∊2 − μδ2 + ∊δ, for ∊, ;

3 3) a quaternion algebra for as in (2) and ;

4 4) a Cayley (or octonion) algebra , for as in (3) and .

In particular, the dimension of a Hurwitz algebra is restricted to 1, 2, 4 or 8.

PROOF.– The only Hurwitz algebra of dimension 1 is, up to isomorphism, the ground field. If ( , ∙, n) is a Hurwitz algebra and , there is an element such that n( v , 1) = 1 and is non-degenerate. The Cayley–Hamilton equation shows that v ∙2− v + n( v )1= 0, so v ∙2= v + μ 1, with μ = −n( v ). The non-degeneracy condition is equivalent to the condition 4 μ + 1 ≠ 0. Then is a Hurwitz subalgebra of and, if , we are done.

If , we may take an element with n( u ) = − β ≠ 0, and hence the subspace is a subalgebra of isomorphic to . By the previous remark, is associative (as is commutative), but it fails to be commutative, as . If , we are done.

Finally, if , we may take an element with n( u ʹ) = − γ ≠ 0, and hence the subspace is a subalgebra of isomorphic to , which is not associative by remark 2.1, so it is necessarily the whole . □

Note that if char , the restriction of n to is non-degenerate, so we could have used the same argument for dimension > 1 in the proof above than the one used for > 2. Hence, we get: