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

2.2. Singular quadratic Lie algebras
Proof. Let g = (CX0 ⊕ CY0) ⊥
⊕ q, B be the bilinear form of g, B = B|q×q, C = ad(Y0) and C =
C|q ∈ o(q,B). We decompose C into its semisimple and nilpotent parts, C = S + N. It is well
known that S and N ∈ o(q,B).
Let Λ ⊂ C \ {0} be the spectrum of S. We have that λ ∈ Λ if and only if −λ ∈ Λ (see
Appendix A). Let Vλ be the eigenspace corresponding to the eigenvalue λ. We have dim(Vλ ) =
dim(V−λ ). Denote by q(λ) the direct sum q(λ) = Vλ ⊕V−λ . If µ ∈ Λ, µ = ±λ, then q(λ) and
q(µ) are orthogonal (Appendix A). Choose Λ+ such that Λ = Λ+∪(−Λ+) and Λ+∩(−Λ+) = /0.
We have (see Appendix A):
q = ⊥
⊕ q(λ).
λ∈Λ+
So the restriction B λ = B| q(λ)×q(λ) is non-degenerate. Moreover, V λ and V −λ are maximal
isotropic subspaces in q(λ).
Now, consider the map Ψ : V −λ → V ∗ λ defined by Ψ(u)(v) = B λ (u,v), for all u ∈ V −λ ,
v ∈ V λ . Then Ψ is an isomorphism. Given any basis B(λ) = {e1(λ),...,en λ (λ)} of V λ , there
is a basis B(−λ) = {e1(−λ),...,en λ (−λ)} of V −λ such that B λ (ei(λ),ej(−λ)) = δi j, for all
1 ≤ i, j ≤ n λ : simply define ei(−λ) = ψ −1 (ei(λ) ∗ ), for all 1 ≤ i ≤ n λ .
Remark that N and S commute, so N(V λ ) ⊂ V λ , for all λ ∈ Λ. Define N λ = N| q(λ), then
N λ ∈ o(q(λ),B λ ). Hence, if N λ |V λ has a matrix M λ with respect to B(λ), then N λ |V −λ has
a matrix − t M λ with respect to B(−λ). We choose the basis B(λ) such that M λ is of Jordan
type, i.e.
B(λ) = B(λ,1) ∪ ··· ∪ B(λ,r λ ),
the multiplicity m λ of λ is m λ = ∑ r λ
i=1 dλ (i) where dλ (i) = ♯B(λ,i) and
Jdλ (1),...,J dλ (rλ ) .
M λ = diag rλ
The matrix of C| q(λ) written on the basis B(λ) ∪ B(−λ) is:
diagnλ Jdλ (1)(λ),...,J dλ (rλ )(λ),− t Jdλ (1)(λ),...,− t Jdλ (rλ )(λ) .
Let q(λ,i) be the subspace generated by B(λ,i) ∪ B(−λ,i), for all 1 ≤ i ≤ r λ and let
C(λ,i) = C| q(λ,i). We have
q(λ) = ⊥
⊕
1≤i≤rλ q(λ,i).
The matrix of C(λ,i) written on the basis of q(λ,i) is C J 2d λ (i) (λ). Let g(λ,i), λ ∈ Λ+, 1 ≤ i ≤ r λ
be the double extension of q(λ,i) by C(λ,i). Then g(λ,i) is i-isomorphic to j 2dλ (i)(λ). But
q = ⊥
⊕
λ∈Λ+
1≤i≤r λ
Therefore, g is the amalgamated product
q(λ,i) and C| q(λ,i) = C(λ,i).
g = ×
a
λ∈Λ+
1≤i≤r
λ
g(λ,i).
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