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

2.2. Singular quadratic Lie algebras
Denote by Sinv(2p + 2) the set of invertible singular quadratic Lie algebra structures on
C2p+2 , by Sinv(2p + 2) the set of isomorphism (or i-isomorphism) classes of Sinv(2p + 2). The
classification of Sinv(2p + 2) can be deduced from the classification of the set of orbits I (2p)
by Jp as follows (see Chapter 1): introduce an action of the multiplicative group C ∗ = C\{0}
on Jp by
for all µ ∈ C ∗ , µ · (Λ,m,d) = (µΛ,m ′ ,d ′ ), ∀ (Λ,m,d) ∈ Jp,λ ∈ Λ,
where m ′ (µλ) = m(λ),d ′ (µλ) = d(λ), for all λ ∈ Λ. Since i(µC) = µi(C), for all C ∈ I (2p)
and µ ∈ C∗ where i : I (2p) → Jp the bijection is defined as in Chapter 1, then there is a
bijection i : P1 ( I (2p)) → Jp/C∗ given by i([C]) = [i(C)], if [C] is the class of C ∈ I (2p) and
[(Λ,m,d)] is the class of (Λ,m,d) ∈ Jp.
Theorem 2.2.50. The set Sinv(2p + 2) is in bijection with Jp/C ∗ .
Proof. By Theorem 2.2.35, there is a bijection between i
Ss (2p + 2) and
P1 (o(2p)). By restriction,
that induces a bijection between i
Sinv (2p+2) and P1 (I (2p)).
By Lemma 2.2.42, we have
i
Sinv (2p + 2) = Sinv(2p + 2). Then, the result follows: given g ∈ Sinv(2p + 2) and an associated
C ∈ I (2p), the bijection maps g to [i(C)] where g is the isomorphism class of g.
Let g be a solvable singular quadratic Lie algebra. We fix a realization of g as a double
extension, g = (CX0 ⊕ CY0) ⊥
⊕ q (Proposition 2.2.28 and Lemma 2.2.34). Let C = ad(Y0), C =
C|q and B = B|q×q. We consider the Fitting decomposition of C:
q = qN ⊕ qI,
where qN and qI are C-stable, CN = C|qN is nilpotent and CI = C|qI is invertible.
We recall the facts in Section 1.5, one has qI = q⊥ N , the restrictions BN = B|qN×qN and BI =
B|qI×qI are non-degenerate, CN and CI are skew-symmetric and [qI,qN] = 0. Let gN = (CX0 ⊕
CY0) ⊥
⊕ qN and gI = (CX0 ⊕ CY0) ⊥
⊕ qI. Then gN and gI are Lie subalgebras of g, gN is the
double extension of qN by CN, gI is the double extension of qI by CI and gN is a nilpotent
singular quadratic Lie algebra. Moreover, we have
g = gN × a gI.
Definition 2.2.51. The Lie subalgebras gN and gI are respectively the nilpotent and invertible
Fitting components of g.
This definition is justified by:
Theorem 2.2.52. Let g and g ′ be solvable singular quadratic Lie algebras and gN, gI, g ′ N , g′ I
be their Fitting components. Then
(1) g i g ′ if and only if gN
i
g ′ N and gI
i
g ′ I . The result remains valid if we replace i by .
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