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

2.2. Singular quadratic Lie algebras
Keep the notations as in Chapter 1. To describe the set of semisimple O(n)-orbits in
P 1 (o(n)), we need to add maps (λ1,...,λp) ↦→ λ(λ1,...,λp), for all λ ∈ C, λ = 0 to the group
Gp. We obtain a group denoted by Hp. As a consequence of Section 1.3, we have the classification
result for the diagonalizable case:
Theorem 2.2.41. There is a bijection between Di (n+2) and Λp/Hp with n = 2p or n = 2p+1.
Moreover, if n = 2p + 1, Di red (n + 2) = /0 and if n = 2p, then Di red (2p + 2) is in bijection with
Λ + p /Hp where Λ + p = {(λ1,...,λp) | λi ∈ C,λi = 0, ∀ i }.
To go further in the study of diagonalizable reduced case, we need the following Lemma:
Lemma 2.2.42.
Let g ′ and g ′′ be solvable singular quadratic Lie algebras, g ′ = (CX ′ 1 ⊕ CY ′ 1
) ⊥
⊕ q ′ a decom-
position of g ′ as a double extension and C ′ = ad(Y ′ 1 )| q ′. We assume that C′ is invertible. Then g ′
and g ′′ are isomorphic if and only if they are i-isomorphic.
Proof. Write g ′′ = (CX ′′
⊥
1 ⊕ CY ′′
1 ) ⊕ q ′′ a decomposition of g ′′ as a double extension and C ′′ =
ad(Y ′′
1 )| q ′′.
Assume that g ′ and g ′′ are isomorphic. By Corollary 2.2.31, there exist Q : q ′ → q ′′ and
P ∈ L (q ′ ) such that Q P −1 is an isometry, P ∗ P C ′ = C ′ and C ′′ = λ Q C ′ Q −1 for some nonzero
λ ∈ C. But C ′ is invertible, so P ∗ P = Idq ′. Therefore, P is an isometry of q ′ and then Q is
an isometry from q ′ to q ′′ . The conditions of Corollary 2.2.31 (1) are satisfied, so g ′ and g ′′ are
i-isomorphic.
Corollary 2.2.43. One has:
Dred(2p + 2) = D i red (2p + 2), ∀ p ≥ 1.
Next, we describe diagonalizable reduced singular quadratic Lie algebras using the amalgamated
products. First, let g4(λ) be the double extension of q = C2
λ 0
by C = , λ = 0.
0 −λ
By Lemma 2.2.27, g4(λ) is i-isomorphic to g4(1), call it g4.
Proposition 2.2.44. Let (g,B) be a diagonalizable reduced singular quadratic Lie algebra.
Then g is an amalgamated product of singular quadratic Lie algebras all i-isomorphic to g4.
Proof. We write g = (CX0 ⊕CY0) ⊥
⊕ q, C = ad(Y0), C =C|q and B = Bq×q. Then C is a diagonalizable
invertible element of o(q,B). Apply Appendix A to obtain a basis {e1,...,ep, f1,..., fp}
of q and λ1,...,λp ∈ C, all non-zero, such that B(ei,ej) = B( fi, f j) = 0, B(ei, f j) = δi j and
C(ei) = λiei, C( fi) = −λi fi, for all 1 ≤ i, j ≤ p. Let qi = span{ei, fi}, 1 ≤ i ≤ p. Then
q = ⊥
⊕ qi.
1≤i≤p
Furthermore, hi = (CX0 ⊕ CY0) ⊥
⊕ qi is a Lie subalgebra of g for all 1 ≤ i ≤ p and
g = h1 × a h2 × a ... × a hp with hi
42
i
g4(λi) i g4.