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

4.2. Jordanian double extension of a quadratic vector space
Conversely, define A : J → J ′ by A(y1) = λy ′ 1 , A(x) = P(x), for all x ∈ q and A(x1) = 1
λ x′ 1 ,
we prove that A is a Jordan isomorphism. Indeed, for all x,y ∈ q and δ,δ ′ , µ, µ ′ ∈ C one
has:
A((δy1 + x + µx1)(δ ′ y1 + y + µ ′ x1)) = δPC(y) + δ ′ PC(x) + 1
λ B(C(x),y)x′ 1
and A(δy1 + x + µx1)A(δ ′ y1 + y + µ ′ x1) = λδC ′ P(y) + λδ ′ CP(x) + B(C ′ P(x),P(y))x ′ 1 .
Since λC ′ = PCP−1 and P∗PC = C, we obtain λC ′ P = PC and P∗C ′ P = 1
λ C. Therefore,
A is an Jordan isomorphism.
(2) If A : J → J ′ is an i-isomorphism then the isomorphism P in the proof of (1) is also an
isometry. Hence P ∈ O(q). Conversely, define A as in (1) then it is obvious that A is an
i-isomorphism.
Proposition 4.2.6. Let (q,B) be a quadratic vector space and let J = q ⊥
⊕ (Cx1 ⊕ Cy1), J ′ =
q ⊥
⊕ (Cx ′ 1 ⊕ Cy′ 1 ) be nilpotent double extensions of q, by symmetric maps C and C′ respectively.
Assume that rank(C ′ ) ≥ 3. Let A : J → J ′ be an isomorphism. Then A(q ⊕ Cx1) = q ⊕ Cx ′ 1 .
Proof. We assume that there is x ∈ q such that A(x) = y+βx ′ 1 +γy′ 1
Then for all q ∈ q and λ ∈ C, we have
A(x)(q + λx ′ 1) = γC ′ (q) + B(C ′ (y),q)x ′ 1.
Therefore, dim(A(x)(q ⊕ Cx ′ 1 )) ≥ 3. But A is an isomorphism, hence
where y ∈ q,β,γ ∈ C,γ = 0.
A(x)(q ⊕ Cx ′ 1) ⊂ A(xA −1 (q ⊕ Cx ′ 1)) ⊂ A(x(q ⊕ Cx1 ⊕ Cy1)) ⊂ A(CC(x) ⊕ Cx1).
This is a contradiction. Hence A(q ⊕ Cx1) = q ⊕ Cx ′ 1 .
4.2.2 Diagonalizable double extensions
Lemma 4.2.7. Let J = q ⊥
⊕ (Cx1 ⊕ Cy1) be a diagonalizable double extension of q by C. Then
y 2 1 = y1,y1x1 = x1,y1x = C(x),xy = B(C(x),y)x1 and x1x = x 2 1 = 0, ∀ x ∈ q.
Note that x1 /∈ Ann(J). Let x ∈ q. Then x ∈ Ann(J) if and only if x ∈ ker(C). Moreover,
J 2 = Im(C) ⊕ (Cx1 ⊕ Cy1). Therefore J is reduced if and only if ker(C) ⊂ Im(C).
Let x ∈ Im(C). Then there exists y ∈ q such that x = C(y). Since 3C 2 = 2C 3 +C, one has
3C(x) − 2C 2 (x) = x. Therefore, if J is reduced then ker(C) = {0} and C is invertible. That
implies that 3C − 2C 2 = Id and we have the following proposition:
Theorem 4.2.8. Let (q,B) be a quadratic vector space. Let J = q ⊥
⊕ (Cx1 ⊕ Cy1) and J ′ =
q ⊥
⊕ (Cx ′ 1 ⊕ Cy′ 1 ) be diagonalizable double extensions of q, by invertible maps C and C′ respectively.
Then there exists a Jordan algebra isomorphism A : J → J ′ if and only if there exists an
isometry P such that C ′ = PCP−1 . In this case, J and J ′ are also i-isomorphic.
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