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

2.1. Preliminaries
Corollary 2.1.6. Let (g,B) be a non-Abelian solvable quadratic Lie algebra. Then there exists
a central element X of g such that X is isotropic.
Proof. By the above proposition, g can be decomposed by g = z ⊥
⊕ l where l is non-Abelian and
Z(l) is totally isotropic. Since g is solvable then l is solvable. Moreover, l is a quadratic Lie
algebra then Z(l) = 0 and the result follows.
Definition 2.1.7. A quadratic Lie algebra g is reduced if:
(1) g = {0}
(2) Z(g) is totally isotropic.
Notice that a reduced quadratic Lie algebra is necessarily non-Abelian.
Definition 2.1.8. Let (g,B) be a quadratic Lie algebra and C be an endomorphism of g. We say
that C is skew-symmetric (or B-antisymmetric) if B(C(X),Y ) = −B(X,C(Y )), for all X,Y ∈ g.
Denote by Enda(g) (resp. Dera(g)) space of skew-symmetric endomorphisms (resp. derivations)
of g.
Next, we recall two effective methods to construct quadratic Lie algebras: double extensions
and T ∗ -extensions. The former method is initiated by V. Kac for the solvable case ([Kac85] and
[FS87]), after that developed generally by A. Medina and Ph. Revoy [MR85]; the later is given
by M. Bordemann [Bor97].
Definition 2.1.9. Let (g,B) be a quadratic Lie algebra, h be another Lie algebra and π : h →
Dera(g) be a representation of h by means of skew-symmetric derivations of g. Define the map
ϕ : g×g → h ∗ by ϕ(X,Y )Z = B(π(Z)X,Y ), for all X,Y ∈ g, Z ∈ h. Denote by ad ∗ the coadjoint
representation of h. Then the vector space g = h ⊕ g ⊕ h ∗ with the product:
[X +Y + f ,X ′ +Y ′ + f ′ ] = [X,X ′ ]h + [Y,Y ′ ]g + π(X)Y ′ − π(X ′ )Y + ad ∗ (X) f ′
−ad ∗ (X ′ ) f + ϕ(Y,Y ′ )
for all X,X ′ ∈ h, Y,Y ′ ∈ g, f , f ′ ∈ h ∗ is a Lie algebra and it is called the double extension of g by
h by means of π. It is easy to show that g is also a quadratic Lie algebra with the bilinear form
B defined by:
B(X +Y + f ,X ′ +Y ′ + f ′ ) = B(Y,Y ′ ) + f (X ′ ) + f ′ (X)
for all X,X ′ ∈ h, Y,Y ′ ∈ g, f , f ′ ∈ h ∗ .
If there is an invariant symmetric bilinear form γ on h (not necessarily non-degenerate) then
g is also a quadratic Lie algebra with the bilinear form Bγ as follows:
Bγ(X +Y + f ,X ′ +Y ′ + f ′ ) = B(Y,Y ′ ) + γ(X,X ′ ) + f (X ′ ) + f ′ (X)
for all X,X ′ ∈ h, Y,Y ′ ∈ g, f , f ′ ∈ h ∗ .
Proposition 2.1.10. ([Kac85], 2.11, [MR85], Theorem I)
Let (g,B) be an indecomposable quadratic Lie algebra (see Definition 2.2.17) such that it is
not simple nor one-dimensional. Then g is the double extension of a quadratic Lie algebra by a
simple or one-dimensional algebra.
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