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

Introduction
and obtain the following result (Theorem 2.2.54):
THEOREM 7:
The set Ss(n + 2) is in bijection with D(n)/C ∗ .
By this process, we also obtain a complete classification of O(n)-adjoint orbits in o(n), a
result which is certainly known, but for which we have no available reference.
We close the first problem with the result as follows (Theorem 2.3.7):
THEOREM 8:
The dup-number is invariant under isomorphisms, i.e. if
g g ′ then dup(g) = dup(g ′ ).
Its proof is not really obvious as in the case of i-isomorphisms. It is obtained through a
computation of centromorphisms in the reduced singular case. Here, we use a result of I. Bajo
and S. Benayadi given in [BB97]. We also have the quadratic dimension dq(g) of g in this case
(Proposition 2.3.6):
dim(Z(g))(1 + dim(Z(g))
dq(g) = 1 + ,
2
where Z(g) is the center of g.
As we will see in Chapter 4, we are also interested in 2-step nilpotent quadratic Lie algebras.
Thanks to double extensions, the simplest case of a quadratic Lie algebra is a solvable singular
quadratic Lie algebra. We have a similar situation for 2-step nilpotent quadratic Lie algebras
in term of T ∗-extensions, a notion given by M. Bordemann [Bor97]. Such algebras with a
characterization of i-isomorphic classes and isomorphic classes were introduced in a paper of
G. Ovando [Ova07]. By studying the set of linear transformations in o(h) where h is a vector
space with a fixed inner product, the author shows that if the dimension of the vector space
h is three or greater than four, there exists a reduced 2-step nilpotent quadratic Lie algebra.
Moreover, there is only one six-dimensional reduced 2-step nilpotent quadratic Lie algebra (up
to i-isomorphisms). In this thesis, once again, we want to approach 2-step nilpotent quadratic
Lie algebras through the method of double extensions and the associated 3-form I. In term
of double extensions, we have a rather obvious result: every 2-step nilpotent quadratic Lie
algebra can be obtained from an Abelian algebra by a sequence of double extensions by onedimensional
algebra (Proposition 2.4.12).
In order to observe the appearance of the element I, we recall the notion of T ∗-extension of a Lie algebra in [Bor97] but with some restricted conditions as follows. Let h be a complex
vector space and θ : h×h → h∗ be a non-degenerate skew-symmetric bilinear map. We assume
that θ is cyclic (that means θ(x,y)z = θ(y,z)x for all x,y,z ∈ g). Let g = h ⊕ h∗ be the vector
space equipped with the bracket
and the bilinear form
[x + f ,y + g] = θ(x,y)
B(x + f ,y + g) = f (y) + g(x),
xi