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