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

(2) g g ′ if and only if g i g ′ .
2.2. Singular quadratic Lie algebras
Proof. We assume that g g ′ . Then by Corollary 2.2.31, there exists an invertible P : q → q ′
and a non-zero λ ∈ C such that C ′ = λ P C P −1 , so q ′ N = P(qN) and q ′ I = P(qI), then dim(q ′ N ) =
dim(qN) and dim(q ′ I ) = dim(qI). Thus, there exist isometries FN : q ′ N → qN and FI : q ′ I → qI and
we can define an isometry F : q ′ → q by F(X ′ N + X ′ I ) = FN(X ′ N ) + FI(X ′ I ), for all X ′ N ∈ q′ N and
X ′ I ∈ q′ I . We now define F : g′ → g by F(X ′ 1 ) = X1, F(Y ′ 1 ) = Y1, F| q ′ = F and a new Lie bracket
on g :
[X,Y ] ′′ = F [F −1 (X),F −1 (Y )] ′ , ∀ X,Y ∈ g.
Call g ′′ this new quadratic Lie algebra. We have g ′′ = (CX1 ⊕ CY1) ⊥
⊕ q, i.e., q ′′ = q and
C ′′ = F C ′ F −1 . So q ′′ N = F(q′ N ) = qN and q ′′
I = F(q′ I ) = qI. But g g ′′ , so there exists an
invertible Q : q → q such that C ′′ = λ Q C Q −1 for some non-zero λ ∈ C (Corollary 2.2.31). It
follows that q ′′ N = Q(qN) and q ′′
I = Q(qI), so Q(qN) = qN and Q(qI) = qI.
Moreover, we have Q ∗ Q C = C (Corollary 2.2.31), so Q ∗ Q C k = C k for all k. There exists k
such that qI = Im(C k ) and (Q ∗ Q C k )(X) = C k (X), for all X ∈ g. So Q ∗ Q|qI = IdqI and QI = Q|qI
is an isometry. Since C ′′
I = λ Q I CI Q −1
I , then gI
i
g ′′
I (Corollary 2.2.31).
Let QN = Q|qN . Then C′′ N = λ QN CN Q −1
N and Q∗ N QN CN = CN, so by Corollary 2.2.31,
gN g ′′ N . Since gN and g ′′ N are nilpotent, then g′′ i
N gN by Theorem 2.2.37.
Conversely, assume that gN g ′ N and gI g ′ I . Then gN
i
g ′ N and gI
i
g ′ I by Theorem 2.2.37
and Lemma 2.2.42.
So, there exist isometries PN : gN → g ′ N , PI : gI → g ′ I and non-zero λN and λI ∈ C such that
C ′
N = λN PN CN P −1
N and C ′
I = λI PI CI P −1
I . By Lemma 2.2.36, since gN and g ′ N are nilpotent,
we can assume that λN = λI = λ. Now we define P : q → q ′ by P(XN + XI) = PN(XN) + PI(XI),
for all XN ∈ qN, XI ∈ qI, so P is an isometry. Moreover, since C(XN + XI) = CN(XN) +CI(XI),
for all XN ∈ qN, XI ∈ qI and C ′ (X ′ N + X ′ I ) = C′ N(X ′ N ) + C′ I(X ′ I ), for all X ′ N ∈ qN, X ′ I ∈ qI, we
conclude C ′ = λ P CP−1 and finally, g i g ′ , by Corollary 2.2.31.
Remark 2.2.53. The class of solvable singular quadratic Lie algebras has the remarkable property
that two Lie algebras in this class are isomorphic if and only if they are i-isomorphic. In
addition, the Fitting components do not depend on the realizations of the Lie algebra as a double
extension and they completely characterize the Lie algebra (up to isomorphisms).
It results that the classfification of Ss(n + 2) can be reduced from the classification of O(n)orbits
of o(n) in Chapter 1. We set an action of the group C ∗ on D(n) by:
µ · ([d],T ) = ([d], µ · T ) , ∀ µ ∈ C ∗ , ([d],T ) ∈ D(n).
Then, we have the classification result of Ss(n + 2) as follows:
Theorem 2.2.54. The set Ss(n + 2) is in bijection with D(n)/C ∗ .
Proof. By Theorems 2.2.35 and 2.2.52, there is a bijection between Ss(n + 2) and
P 1 (o(n)). It
needs only to show that there is a bijection between
P 1 (o(n)) and D(n)/C ∗ . Let C ∈ o(n) then
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