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