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