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

2.1. Preliminaries
Proof. Let I be a minimal ideal of g. Since I ∩ I ⊥ is also an ideal of g then we must have
I ∩ I ⊥ = I or I ∩ I ⊥ = {0}. But g is indecomposable so the second case does not happen. It
means that I ⊂ I ⊥ , i.e. I is totally isotropic. So B([I,I ⊥ ],g) = B(I,[I ⊥ ,g]) = 0. Therefore,
[I,I ⊥ ] = 0 by the non-degeneracy of B.
Consider two exact sequences of Lie algebras
0 → I → I ⊥ → I ⊥ /I → 0,
0 → I ⊥ → g → g/I ⊥ → 0.
Denote by h = g/I ⊥ then h is simple or one-dimensional since I ⊥ is a maximal ideal of
g. We can identify h with a subalgebra of g and g is the semi-direct product of I ⊥ by h. Let
W = I ⊥ /I, p : I ⊥ → W be the canonical projection and define on W the bilinear form T that is
the restriction of B on W. Then (W,T ) is a quadratic Lie algebra. Since the subspace H = I ⊕ h
is non-degenerate, we can identify W (regarded as a subspace of g) with H ⊥ , i.e. g = I ⊕W ⊕h.
Now, we will define an action π of h on W as follows. Let x ∈ W, regarded as an element
of H ⊥ , and take h ∈ h. We will show that [h,x] ∈ H ⊥ . Indeed, let h ′ ∈ h then B(h ′ ,[h,x]) =
B([h ′ ,h],x) = 0 since [h ′ ,h] ∈ h. Therefore, [h,x] ∈ h ⊥ . Let y ∈ I then B([x,h],y) = B(x,[h,y]) =
0. Hence, [h,x] must be in H ⊥ . We set π : h → Dera(W) by π(h)(x) = [h,x], for all h ∈ h, x ∈ W.
For all x,y ∈ W one has:
[x,y]g = [x,y]W + ϕ(x,y),
where ϕ : W ×W → I satisfies B(ϕ(x,y),z) = B(π(z)x,y), for all x,y ∈ W, z ∈ h.
Finally, since I ⊕ h is non-degenerate and I is a totally isotropic subspace of I ⊕ h, we can
identify I with h ∗ and the adjoint action of h into I becomes the coadjoint action ad ∗ of h into
h ∗ . Then g is the double extension of W by h by means of π.
Corollary 2.1.11. [FS87]
Let (g,B) be a non-Abelian solvable quadratic Lie algebra then g is the double extension of
a quadratic Lie algebra of dimension dim(g) − 2 by a one-dimensional algebra.
Proof. By Corollary 2.1.6, there is a totally isotropic central ideal CX of g. Let Y ∈ g isotropic
such that B(X,Y ) = 1 then g is the double extension of (CX ⊕ CY ) ⊥ by the one-dimensional
algebra CY .
Let us present now the second construction of quadratic Lie algebras which is given by M.
Bordemann in [Bor97] as follows:
Definition 2.1.12. Let g be a Lie algebra over C, V be a complex vector space and ρ : g →
End(V ) be a representation of g in V . That means
ρ([X,Y ]) = ρ(X)ρ(Y ) − ρ(Y )ρ(X), ∀ X,Y ∈ g.
In this case, V is also called a g-module. For an integer k ≥ 0, denote by C k (g,V ) the space
of alternating k-linear maps from g × ... × g into V if k ≥ 1 and C 0 (g,V ) = V . Let C(g,V ) =
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