alt="image"/> on a vector space V over a field is so, that is, if its radical V⊥ := {v ∈ V : n(v, V) = 0} is trivial. Moreover, n is said to be non-singular either if it is non-degenerate or if it satisfies that the dimension of V⊥ is 1 and n(V⊥) ≠ 0. The last possibility only occurs over fields of characteristic 2.
DEFINITION 2.1.– A composition algebra over a field
– (, ∙) is a non-associative algebra;
– is a non-singular quadratic form that is multiplicative, that is,
for any x,
The unital composition algebras are called Hurwitz algebras.
For simplicity, we will usually refer to the composition algebra
Our goal in this section is to prove that Hurwitz algebras are quite close to ℝ, ℂ, ℍ and
By linearization of [2.4], we obtain:
for any x, y, z,
PROPOSITION 2.2.– Let (
– either n is non-degenerate or char and is isomorphic to the ground field (with norm α ↦ α2);
– the map is an involution. That is, and for any x, . This involution is referred to as the standard conjugation;
– if ∗ denotes the conjugation of a linear endomorphism relative to n (i.e. n(f(x), y) = n(x, f∗(y)) for any x, y), then for the left and right multiplications by elements we have and ;
– any satisfies the Cayley–Hamilton equation:
– (, ∙) is an alternative algebra: x ∙ (x ∙ y) = x∙2 ∙ y and (y ∙ x) ∙ x = y ∙ x∙2 for any x, .
PROOF.– Plug t =1 in [2.5] to get
and symmetrically we get
Now, if char
Assuming hence that
for any x, y, z, whence
Finally, for any x, y, z, using again [2.5],
so that, using left-right symmetry:
With y = 1, this gives
2.3.1. The Cayley–Dickson doubling process and the generalized Hurwitz theorem
Let (
