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

(3) The subspace
is the nucleus of J.
4.1. Preliminaries
N(J) = {x ∈ J | (x,y,z) = (y,x,z) = (y,z,x) = 0, ∀ y,z ∈ J}
Proposition 4.1.12. If (J,B) is a pseudo-Euclidean Jordan algebra then
(1) the nucleus N(J) coincide with the center Z(J) of J where Z(J) = {x ∈ N(J) | xy =
yx, ∀ y ∈ J}, that is, the set of all elements x that commute and associate with all elements
of J. Therefore
(2) Z(J) ⊥ = (J,J,J).
(3) (Ann(J)) ⊥ = J 2 .
N(J) = Z(J) = {x ∈ J | (x,y,z) = 0, ∀ y,z ∈ J}.
Proof. Since B((x,y,z),t) = B((y,x,t),z) = B((z,t,x),y) = B((t,z,y),x), for all x,y,z,t ∈ J then
we get (1) and (2). The statement (3) is gained by B non-degenerate and associative.
Definition 4.1.13. A pseudo-Euclidean Jordan algebra J is reduced if
(1) J = {0},
(2) Ann(J) is totally isotropic.
Proposition 4.1.14. Let J be non-Abelian pseudo-Euclidean Jordan algebra. Then J = z ⊥
⊕ l
where z ⊂ Ann(J) and l is reduced.
Proof. The proof is completely similar to Proposition 2.1.5. Let z0 = Ann(J) ∩ J 2 , z be a complementary
subspace of z0 in Ann(J) and l = z ⊥ . If x is an element in z such that B(x,z) =
0 then B(x,J 2 ) = 0 since Ann(J) = (J 2 ) ⊥ . As a consequence, B(x,z0) = 0 and therefore
B(x,Ann(J)) = 0. That implies x ∈ J 2 . Hence, x = 0 and the restriction of B to z is nondegenerate.
Moreover, z is an ideal, then it is easy to check that the restriction of B to l is also a
non-degenerate and that z ∩ l = {0}.
Since J is non-Abelian then l is non-Abelian and l 2 = J 2 . Moreover, z0 = Ann(l) and the
result follows.
Next, we will define some extensions of a Jordan algebra and introduce the notion of a
double extension of a pseudo-Euclidean Jordan algebra [BB].
Definition 4.1.15. Let J1 and J2 be Jordan algebras and π : J1 → End(J2) be a representation
of J1 on J2. We call π an admissible representation if it satisfies the following conditions:
(1) π(x 2 )(yy ′ ) + 2(π(x)y ′ )(π(x)y) + (π(x)y ′ )y 2 + 2(yy ′ )(π(x)y)
= 2π(x)(y ′ (π(x)y)) + π(x)(y ′ y 2 ) + (π(x 2 )y ′ )y + 2(y ′ (π(x)y))y,
(2) (π(x)y)y 2 = (π(x)y 2 )y,
102