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

4.1. Preliminaries
(3) π(xx ′ )y 2 + 2(π(x ′ )y)(π(x)y) = π(x)π(x ′ )y 2 + 2(π(x ′ )π(x)y)y,
for all x,x ′ ∈ J1,y,y ′ ∈ J2. In this case, the vector space J = J1 ⊕ J2 with the product defined
by:
(x + y)(x ′ + y ′ ) = xx ′ + π(x)y ′ + π(x ′ )y + yy ′ , ∀ x,x ′ ∈ J1,y,y ′ ∈ J2
becomes a Jordan algebra.
Definition 4.1.16. Let (J,B) be a pseudo-Euclidean Jordan algebra and C be an endomorphism
of J. We say that C is symmetric if
B(C(x),y) = B(x,C(y)), ∀ x,y ∈ J.
Denote by Ends(J) the space of symmetric endomorphisms of J.
The definition below was introduced in [BB], Theorem 3.8.
Definition 4.1.17. Let (J1,B1) be a pseudo-Euclidean Jordan algebra and let J2 be an arbitrary
Jordan algebra. Let π : J2 → Ends(J1) be an admissible representation. Define a symmetric
bilinear map ϕ : J1 ×J1 → J ∗ 2 by: ϕ(y,y′ )(x) = B1(π(x)y,y ′ ), for all x ∈ J2,y,y ′ ∈ J1. Consider
the vector space
J = J2 ⊕ J1 ⊕ J ∗ 2
endowed with the product:
(x + y + f )(x ′ + y ′ + f ′ ) = xx ′ + yy ′ + π(x)y ′ + π(x ′ )y + f ′ ◦ Rx + f ◦ R x ′ + ϕ(y,y ′ )
for all x,x ′ ∈ J2, y,y ′ ∈ J1, f , f ′ ∈ J∗ 2 . Then J is a Jordan algebra. Moreover, define a bilinear
form B on J by:
B(x + y + f ,x ′ + y ′ + f ′ ) = B1(y,y ′ ) + f (x ′ ) + f ′ (x), ∀ x,x ′ ∈ J2,y,y ′ ∈ J1, f , f ′ ∈ J ∗ 2.
Then J is a pseudo-Euclidean Jordan algebra. The Jordan algebra (J,B) is called the double
extension of J1 by J2 by means of π.
Remark 4.1.18. If γ is an associative bilinear form (not necessarily non-degenerate) on J2 then
J is again pseudo-Euclidean thanks to the bilinear form
Bγ(x + y + f ,x ′ + y ′ + f ′ ) = γ(x,x ′ ) + B1(y,y ′ ) + f (x ′ ) + f ′ (x)
for all x, x ′ ∈ J2, y, y ′ ∈ J1, f , f ′ ∈ J ∗ 2 .
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