TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...

Proof.
4.3. Pseudo-Euclidean 2-step nilpotent Jordan algebras
(1) Let A : J1 → J2 be a Jordan algebra isomorphism. Since a ∗ = Ann(J1) = Ann(J2) is
stable by A then there exist linear maps A1 : a → a, A ′ 1 : a → a∗ and A2 : a ∗ → a ∗ such
that:
A(x + f ) = A1(x) + A ′ 1(x) + A2( f ), ∀ x + f ∈ J1.
Since A is an isomorphism one has A2 also isomorphic. We show that A1 is an isomorphism
of a. Indeed, if A1(x0) = 0 with some x0 ∈ a then A(x0) = A ′ 1 (x0) and
0 = A(x0)J2 = A(x0A −1 (J2)) = A(x0J1).
That implies x0J1 = 0 and so x0 ∈ a ∗ . That means x0 = 0, i.e. A1 is an isomorphism of a.
For all x and y in a, one has A(xy) = A(θ1(x,y)) = A2(θ1(x,y)) and
A(x)A(y) = (A1(x) + A ′ 1(x))(A1(y) + A ′ 1(y)) = A1(x)A1(y) = θ2(A1(x),A1(y)).
Therefore, A2(θ1(x,y)) = θ2(A1(x),A1(y)), for all x,y ∈ a.
Conversely, if there exist an isomorphism A1 of a and an isomorphism A2 of a ∗ satisfying:
A2(θ1(x,y)) = θ2(A1(x),A1(y)), ∀ x,y ∈ a,
then we define A : J1 → J2 by A(x + f ) = A1(x) + A2( f ), for all x + f ∈ J1. It is easy to
see that A is a Jordan algebra isomorphism.
(2) Assume that A : J1 → J2 is a Jordan algebra i-isomorphism then there exist A1 and A2
defined as in (1). Let x ∈ a, f ∈ a ∗ , one has:
B ′ (A(x),A( f )) = B(x, f ) ⇒ A2( f )(A1(x)) = f (x).
Hence, A2( f ) = f ◦A −1
1 , for all f ∈ a∗ . Moreover, A2(θ1(x,y)) = θ2(A1(x),A1(y)) implies
that
θ1(x,y)) = θ2(A1(x),A1(y)) ◦ A1, ∀ x,y ∈ a.
Conversely, define A(x+ f ) = A1(x)+ f ◦A −1
1 , for all x+ f ∈ J1 then A is an i-isomorphism.
Example 4.3.16. We keep the notations as above. Let J ′ be the T ∗ -extension of a by θ ′ =
λθ,λ = 0 then J and J ′ is i-isomorphic by A : J → J ′ defined by
where α ∈ C, α 3 = λ.
A(x + f ) = 1
x + α f , ∀ x + f ∈ J
α
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