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4.3. Pseudo-Euclidean 2-step nilpotent Jordan algebras
4.3.2 T ∗ -extensions of pseudo-Euclidean 2-step nilpotent
Given a 2SN-Jordan algebra J and a symmetric bilinear map θ : J × J → J ∗ such that
R ∗ (z)(θ(x,y)) + θ(xy,z) = 0, for all x,y,z ∈ J, then by Proposition 4.3.2, J = J ⊕ J ∗ is also a
2SN-Jordan algebra. Moreover, if θ is cyclic (that is, θ(x,y)(z) = θ(y,z)(x), for all x,y,z ∈ J),
then J is a pseudo-Euclidean Jordan algebra with the bilinear form defined by
B(x + f ,y + g) = f (y) + g(x), ∀ x,y ∈ J, f ,g ∈ J ∗ .
In a more general framework, we can define:
Definition 4.3.12. Let a be a complex vector space and θ : a × a → a ∗ a cyclic symmetric
bilinear map. Assume that θ is non-degenerate, i.e. if θ(x,a) = 0 then x = 0. Consider the
vector space J = a ⊕ a ∗ equipped with the product
and the bilinear form
(x + f )(y + g) = θ(x,y)
B(x + f ,y + g) = f (y) + g(x)
for all x + f ,y + g ∈ J. Then (J,B) is a 2SNPE-Jordan algebra and it is called the T ∗ -extension
of a by θ.
Lemma 4.3.13. Let J be a T ∗ -extension of a by θ. If J = {0} then J is reduced.
Proof. Since θ is non-degenerate, it is easy to check that Ann(J) = a ∗ is totally isotropic by the
above definition.
Proposition 4.3.14. Let (J,B) be a 2SNPE-Jordan algebra. If J is reduced then J is i-isomorphic
to some T ∗ -extension.
Proof. Assume that J is a reduced 2SNPE-Jordan algebra. Then one has Ann(J) = J2 , so
dim(J2 ) = 1 2 dim(J). Let J = Ann(J) ⊕ a where a is a totally isotropic subspace of J. Then
a J/J2 as an Abelian algebra. Since a and Ann(J) are maximal totally isotropic subspaces of
J, we can identify Ann(J) to a∗ by the isomorphism: ϕ : Ann(J) → a∗ , ϕ(x)(y) = B(x,y), for
all x ∈ Ann(J), y ∈ a. Define θ : a × a → a∗ by θ(x,y) = ϕ(xy), for all x,y ∈ a.
Now, set α : J → a ⊕ a∗ by α(x) = p1(x) + ϕ(p2(x)), for all x ∈ J where p1 : J → a and
p2 : J → Ann(J) are canonical projections. Then α is i-isomorphic.
Theorem 4.3.15. Let J1 and J2 be two T ∗ -extensions of a by θ1 and θ2 respectively. Then:
(1) there exists a Jordan algebra isomorphism between J1 and J2 if and only 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.
(2) there exists an i-isomorphism between J1 and J2 if and only if there exists an isomorphism
A1 of a satisfying
θ1(x,y) = θ2(A1(x),A1(y)) ◦ A1, ∀ x,y ∈ a.
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