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

(1) g i g ′ if and only if
⎧
⎨
⎩
i
gN g ′ N
i
gI g ′ I
.
Introduction
The result remains valid if we replace i by .
(2) g g ′ if and only if g i g ′ . Therefore Ss(n + 2) = i
Ss (n + 2).
Theorem 5 is a really interesting and unexpected property of solvable singular quadratic Lie
algebras.
From the above facts, in order to describe more precisely the set Ss(n + 2), we turn to the
problem of classification of O(n)-adjoint orbits in o(n). Since the study of the nilpotent orbits is
complete, we begin with the invertible case. Let I (n) be the set of invertible elements in o(n)
and I (n) be the set of O(n)-adjoint orbits of elements in I (n). Notice that I (2p + 1) = /0
(Appendix A) then we consider n = 2p. Define the set
D =
{(d1,...,dr) ∈ N r | d1 ≥ d2 ≥ ··· ≥ dr ≥ 1}
r∈N ∗
and the map Φ : D → N defined by Φ(d1,...,dr) = ∑ r i=1 di. We introduce the set Jp of all
triples (Λ,m,d) such that:
(1) Λ is a subset of C \ {0} with ♯Λ ≤ 2p and λ ∈ Λ if and only if −λ ∈ Λ.
(2) m : Λ → N∗ satisfies m(λ) = m(−λ), for all λ ∈ Λ and ∑ m(λ) = 2p.
λ∈Λ
(3) d : Λ → D satisfies d(λ) = d(−λ), for all λ ∈ Λ and Φ ◦ d = m.
To every C ∈ I (2p), we can associate an element (Λ,m,d) of Jp as follows: write C =
S + N as a sum of its semisimple and nilpotent parts. Then Λ is the spectrum of S, m is the
multiplicity map on Λ and d gives the size of the Jordan blocks of N. Therefore, we obtain a
map i : I (2p) → Jp and we prove:
THEOREM 6:
The map i : I (2p) → Jp induces a bijection from
I (2p) onto Jp.
As a corollary, we deduce a bijection from Sinv(2p + 2) onto Jp/C ∗ (Theorem 2.2.50)
where the action of µ ∈ C ∗ = C \ {0} on Jp is defined by
µ · (Λ,m,d) = (µΛ,m ′ ,d ′ ), with m ′ (µλ) = m(λ) and d ′ (µλ) = d(λ), ∀ λ ∈ Λ.
Combined with the previous theorems, we obtain a complete classification of Ss(n + 2) as
follows. Let D(n) be the set of all pairs ([d],T ) such that [d] ∈ P1(m), the set of partitions of
m in which even parts occur with even multiplicity, and T ∈ Jℓ satisfying m + 2ℓ = n. We set
an action of the multiplicative group C ∗ on D(n) by:
µ · ([d],T ) = ([d], µ · T ) , ∀ µ ∈ C ∗ , ([d],T ) ∈ D(n).
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