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

Introduction
This result can be obtained by checking Jordan identity for the algebra J. However, it
can be seen as a particular case of the general theory of double extension on pseudo-Euclidean
(commutative) Jordan algebras given by A. Baklouti and S. Benayadi in [BB], that is the double
extension of an Abelian algebra by one dimensional Jordan algebra. By the similar method as
in Chapter 2 and Chapter 3, we obtain the classification result (Theorem 4.2.5, Theorem 4.2.8
and Corollary 4.2.9):
THEOREM 15:
(1) Let J = q ⊥
⊕ (Cx1 ⊕Cy1) and J ′ = q ⊥
⊕ (Cx ′ 1 ⊕Cy′ 1 ) be nilpotent double extensions of q by
symmetric maps C and C ′ , respectively. Then there exists a Jordan algebra isomorphism
A : J → J ′ such that A(q ⊕ Cx1) = q ⊕ Cx ′ 1 if and only if there exist an invertible map
P ∈ End(q) and a non-zero λ ∈ C such that λC ′ = PCP−1 and P∗PC = C where P∗ is the
adjoint map of P with respect to B. In this case A i-isomorphic then P ∈ O(q).
(2) Let J = q ⊥
⊕ (Cx1 ⊕ Cy1) and J ′ = q ⊥
⊕ (Cx ′ 1 ⊕ Cy′ 1 ) be diagonalizable double extensions
of q by symmetric maps C and C ′ , respectively. Then J and J ′ are isomorphic if and only
if they are i-isomorphic. In this case, C and C ′ have the same spectrum.
The next part of Chapter 4 can be regarded as the symmetric version of 2-step nilpotent
quadratic Lie algebras, that is the class of 2-step nilpotent pseudo-Euclidean Jordan algebras.
We introduce the notion of generalized double extension but with a restricting condition for 2step
nilpotent pseudo-Euclidean Jordan algebras. As a consequence, we obtain in this way the
inductive characterization of those algebras (Proposition 4.3.11): a non-Abelian 2-step nilpotent
pseudo-Euclidean Jordan algebra is obtained from an Abelian algebra by a sequence of
generalized double extensions.
To characterize (up to isomorphisms and i-isomorphisms) 2-step nilpotent pseudo-Euclidean
Jordan algebras we need to use again the concept of a T ∗ -extension as above with a little change.
Given a complex vector space a and a non-degenerate cyclic symmetric bilinear map θ : a×a →
a ∗ . On the vector space J = a ⊕ a ∗ we define the product
(x + f )(y + g) = θ(x,y).
Then J is a 2-step nilpotent pseudo-Euclidean Jordan algebra and it is called the T ∗ -extension of
a by θ (or T ∗ -extension, simply). Moreover, every reduced 2-step nilpotent pseudo-Euclidean
Jordan algebra is i-isomorphic to some T ∗ -extension (Proposition 4.3.14). An i-isomorphic and
isomorphic characterization of T ∗ -extensions is given in Theorem 4.3.15 as follows:
THEOREM 16:
Let J1 and J2 be 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.
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