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

2.3. Quadratic dimension of quadratic Lie algebras
Corollary 2.3.10. Let (g,B) be a quadratic Lie algebra and δ ∈ Dera(g). Let g be the double
extension of g by means of δ. Then the endomorphism D of g defined by:
D|g = xId, D(X1) = xX1, D(Y1) = xY1 + yX1
with x ∈ C ∗ ,y ∈ C is an invertible endomorphism of g. Consequently, dq(g) ≥ 2.
Proof. The result can be obtained from the previous proposition by setting D = xId,x ∈ C ∗ and
U = 0.
Keep the notations as in Proposition 2.3.9 and define the set:
E(g,B,δ) = {(x,y,U,D) ∈ C × C × kerδ × C(g) | δD = Dδ = xδ + adg(U)}.
Therefore, if dq(g) = 2 then E(g,B,δ) = {(x,y,0,xId) | x,y ∈ C} [BB07].
Proposition 2.3.11. Let g be a Lie algebra and D : g → g be an invertible linear map satisfying
D[X,Y ] = [D(X),Y ], for all X,Y ∈ g. Assume that there exists a cyclic 2-cocycle θ : g × g → g ∗
such that θ(D(X),Y ) = θ(X,D(Y )), for all X,Y ∈ g. Denote by T ∗
means of θ then the endomorphism D of T ∗
θ
is an invertible centromorphism of T ∗
θ (g).
(g) defined by:
D(X + f ) = D(X) + f ◦ D, ∀ X ∈ g, f ∈ g ∗
Proof. Since D is invertible, so is D. Let X + f ,Y + g ∈ T ∗
θ (g), one has:
θ (g) the T ∗ -extension of g by
B(D(X + f ),Y + g) = f ◦ D(Y ) + g ◦ D(X) = B(X + f ,D(Y + g)),
D[X + f ,Y + g] = D[X,Y ]g + θ(X,Y ) ◦ D + f ◦ adg(Y ) ◦ D − g ◦ adg(X) ◦ D
and [D(X + f ),Y + g] = [D(X),Y ]g + θ(D(X),Y ) + f ◦ D ◦ adg(Y ) − g ◦ D ◦ adg(X).
Remark that the condition D[X,Y ] = [D(X),Y ], for all X,Y ∈ g is equivalent to D◦adg(X) =
adg(X) ◦ D, for all X ∈ g. Since θ is cyclic then θ(X,Y ) ◦ D = θ(D(X),Y ). Therefore D[X +
f ,Y + g] = [D(X + f ),Y + g], ∀ X + f ,Y + g ∈ T ∗
θ (g) and so D is a centromorphism of T ∗
θ (g).
A more general result is given in the proposition below:
Proposition 2.3.12. Let g be a Lie algebra endowed with an invariant symmetric bilinear
form ω (not necessarily non-degenerate) and D : g → g be the invertible linear map satisfying
D[X,Y ] = [D(X),Y ], for all X,Y ∈ g. Assume that there exists a cyclic 2-cocycle θ : g × g → g∗ such that θ(D(X),Y ) = θ(X,D(Y )), for all X,Y ∈ g then the endomorphism D of T ∗
θ (g) defined
by:
D(X + f ) = D(X) + ϕ(X) + f ◦ D, ∀ X ∈ g, f ∈ g ∗
is an invertible centromorphism of T ∗
θ (g) where ϕ : g → g∗ is defined by ϕ(X) = ω(X,.), for all
X ∈ g.
Proof. It is easy to see that D is invertible. Since ω is symmetric then D is also symmetric.
Prove similarly to Proposition 2.3.11 and note that the condition invariance of ω is equivalent
to ϕ([X,Y ]g) = ϕ(X) ◦ adg(Y ), for all X,Y ∈ g, we get the result.
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