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

Introduction
by I. Bajo, S. Benayadi and M. Bordemann in [BBB], we prove that (Corollary 3.6.5 and Theorem
3.6.8)
THEOREM 13:
A quasi-singular quadratic Lie superalgebra is a generalized double extension of a quadratic
Z2-graded vector space. This superalgebra is 2-nilpotent.
The algebras obtained in Chapter 2 and Chapter 3 lead us to the general framework as
follows: let q be a complex vector space equipped with a non-degenerate bilinear form Bq and
C : q → q be a linear map. We associate a vector space :
J = q ⊥
⊕ t
to the triple (q,Bq,C) where (t = span{x1,y1},Bt) is a 2-dimensional vector space and Bt :
t × t → C is the bilinear form defined by
Bt(x1,x1) = Bt(y1,y1) = 0, Bt(x1,y1) = 1.
Define a product ⋆ on the vector space J such that t is a subalgebra of J,
y1 ⋆ x = C(x), x1 ⋆ x = 0, x ⋆ y = Bq(C(x),y)x1
for all x,y ∈ q and such that the bilinear form BJ = Bq + Bt is associative. We call J the double
extension of q by C. It can be completely characterized by the pair (Bq,C). Solvable singular
quadratic Lie algebras and singular quadratic Lie superalgebras are only particular cases of this
notion. Therefore, it is natural to consider similar algebras corresponding to the remaining
different cases of the pair (Bq,C). In Chapter 4 we give a condition that J is a pseudo-Euclidean
(commutative) Jordan algebra (i.e a Jordan algebra endowed with a non-degenerate associative
symmetric bilinear form). Consequently, the bilinear forms Bq, Bt are symmetric, C must be
also symmetric (with respect to Bq) and the product ⋆ is defined by:
(x + λx1 + µy1) ⋆ (y + λ ′ x1 + µ ′ y1) = µC(y) + µ ′ C(x) + Bq(C(x),y)x1
+ε λ µ ′ + λ ′ µ x1 + µµ ′
y1 ,
ε ∈ {0,1}, for all x,y ∈ q, λ,µ,λ ′ , µ ′ ∈ C.
Since there exist only two one-dimensional Jordan algebras, one Abelian and one simple,
then we have two types of extensions called respectively nilpotent double extension and diagonalizable
double extension. The first result (Proposition 4.2.1, Corollary 4.2.2, Lemma 4.2.7
and Appendix D) is the following:
THEOREM 14:
(1) If J is the nilpotent double extension of q by C then C 3 = 0, J is k-step nilpotent, k ≤ 3,
and t is an Abelian subalgebra of J.
(2) If J is the diagonalizable double extension of q by C then 3C 2 = 2C 3 +C, J is not solvable
and t ⋆ t = t. In the reduced case, y1 acts diagonally on J with eigenvalues 1 and 1 2 .
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