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