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

Introduction
What about non-solvable singular quadratic Lie algebras? Such a Lie algebra g can be
written as
g = s ⊥
⊕ z
where z is a central ideal of g and s o(3) equipped with a bilinear form λκ for some nonzero
λ ∈ C where κ is the Killing form of o(3). Note that different from the solvable case, the
notions of i-isomophism and isomorphism are not equivalent in this case.
We denote by Ss(n+2) the set of solvable singular quadratic Lie algebra structures on Cn+2 ,
by Ss(n+2) the set of isomorphism classes of elements in Ss(n+2) and by i
Ss (n+2) the set of
i-isomorphism classes. Given C ∈ o(n), there is an associated double extension gC ∈ Ss(n + 2)
(Definition 2.2.26) and then (Theorem 2.2.35):
THEOREM 2:
The map C → gC induces a bijection from P1 (o(n)) onto i
Ss (n + 2).
A weak form of Theorem 2 was stated in the paper by G. Favre and L. J. Santharoubane
[FS87], in the case of i-isomorphisms satisfying some (dispensable) conditions. A strong improvement
to Theorem 2 will be given in Theorem 5 where the i-isomorphic notion is replaced
by the isomorphic notion.
We detail Theorem 2 in some particular cases. Let N(n+2) be the set of nilpotent singular
structures on C n+2 , N i (n + 2) be the set of i-isomorphism classes and N(n + 2) be the set of
isomorphism classes of elements in N(n+2). We denote g and g ′ i-isomorphic by g i g ′ . Using
the Jacobson-Morozov theorem, we prove that (Theorem 2.2.37):
THEOREM 3:
(1) Let g and g ′ be in N(n+2). Then g i g ′ if and only if g g ′ . Thus N i (n+2) = N(n+2).
(2) Let N (n) be the set of nilpotent O(n)-adjoint orbits in o(n). Then the map C ↦→ gC induces a bijection from N (n) onto N(n + 2).
In Chapter 1, we recall the well-known classification of nilpotent O(n)-adjoint orbits in
o(n). An important ingredient is the Jacobson-Morosov and Kostant theorems on sl(2)-triples
in semisimple Lie algebras (see the book by D. H. Collingwood and W. M. McGovern [CM93]
for more details). Using this classification, we obtain a classification of N(n + 2) in term of
special partitions of n, that is, there is a one-to-one correspondence between N(n + 2) and
the set P1(n) of partitions of n in which even parts occur with even multiplicity (Theorem
2.2.38). In other words, we can parametrize the set N(n + 2) by the set of indices P1(n). This
parametrization is detailed by means of amalgamated products of nilpotent Jordan-type Lie
algebras.
Let D(n+2) be the set of diagonalizable singular structures on C n+2 (i.e. C is a semisimple
element of o(n)) and Dred(n + 2) be the set of reduced ones (see Definition 2.1.7 for the
definition of a reduced quadratic Lie algebra). Denote by D(n + 2), D i (n + 2), Dred(n + 2)
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