Proposition 4

Tag 05Z1, permanent. Cite it as [Bourbaki, Tag 05Z1].

Algebra, chapter III, § 2. Examples of algebras

Let E be an A-algebra and s and s' antiautomorphisms of E such that (E, s) and (E, s') are Cayley algebras. If E admits a basis containing the unit element e, then s' = s.

Clearly $s'(u) = s(u) = u$ for all $u \in Ae$ . If T, N (resp. T', N') are the trace and norm functions for (E, s) (resp. (E, s')), it follows from (13) that

(T(u) - T'(u)).u - (N(u) - N'(u)) = 0.

Let B be a basis of E containing e and u an element of B distinct from e; then $T(u) - T'(u) = 0$ , whence $s(u) = s'(u)$ . As s and s' coincide on B, they are equal.

In what follows, we shall write $\bar{u} = s(u)$ , so that

\left\{\begin{array}{lll} u + \bar{u} = T(u), & u\bar{u} = \bar{u}u = N(u), & \bar{u} = u \\ u + v = \bar{u} + \bar{v}, & \alpha u = \alpha \bar{u}, & uv = \bar{v}.\bar{u} \end{array}\right.

for $u, v$ in E, $\alpha \in A$ ; moreover

T(e) = 2e, \quad N(e) = e.

From the formula

T(uv) = uv + \overline{uv} = uv + \bar{v}.\bar{u} = uv + (T(v) - v)(T(u) - u),

we deduce that

uv + vu = T(u)v + T(v)u + (T(uv) - T(u)T(v))

whence, exchanging $u$ and $v$ ,

T(vu) = T(uv).

On the other hand, $N(u + v) = (u + v)(\bar{u} + \bar{v}) = N(u) + N(v) + T(u\bar{v})$ , whence

(18)

T(v\bar{u}) = T(u\bar{v}) = N(u + v) - N(u) - N(v).

Now, (16) applied with $u$ replaced by $\bar{u}$ gives

T(u\bar{v}) = T(u)T(v) + \bar{u}v + v\bar{u} - T(u)v - T(v)\bar{u} = T(u)T(v) - uv - \bar{v}.\bar{u};

whence

(19)

T(v\bar{u}) = T(u\bar{v}) = N(u + v) - N(u) - N(v) = T(u)T(v) - T(uv).

Finally, clearly for all $\alpha \in A$ ,

(20)

N(\alpha u) = \alpha^2 N(u);

in particular

N(2u) = 4N(u)

, so that formula (19) gives

(21)

(T(u))^2 - T(u^2) = 2N(u).

Clearly $T$ is a linear form on the $(Ae)$ -module $E$ . As $(u, v) \mapsto T(v\bar{u})$ is a bilinear form on this module, it follows from (18) and (20) that $N$ is a quadratic form (cf. IX, § 3, no. 4).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

IX, § 3, no. 4 not yet on this site: A IX