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