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

2.3. Quadratic dimension of quadratic Lie algebras
We start with the case dup B (g) = 3. This is true if and only if dim([g,g]) = 3 by Remark
2.2.10. Then dup B ′(g) = 3.
If dup B (g) = 1, then g is of type S1 with respect to B. We apply Proposition 2.3.6 to obtain an
invertible centromorphism D = µ Id+Z for a non-zero µ ∈ C, Z : g → Z(g) satisfying Z| [g,g] =
0 and such that B ′ (X,Y ) = B(D(X),Y ), for all X,Y ∈ g. Then I ′ (X,Y,Z) = B ′ ([X,Y ],Z) =
B([D(X),Y ],Z) = µB([X,Y ],Z) = µI(X,Y,Z), for all X, Y , Z ∈ g. So I ′ = µI and dup B ′(g) =
dup B (g).
Finally, if dup B (g) = 0, then from the previous cases, g cannot be of type S3 or S1 with
respect to B ′ , so dup B ′(g) = 0.
2.3.3 Centromorphisms and extensions of a quadratic Lie algebra
First we recall the definition of double extension of a quadratic Lie algebra by a onedimensional
algebra as follows:
Definition 2.3.8. Let (g,B) be a quadratic Lie algebra and δ ∈ Dera(g) the space of skewsymmetric
derivations of g. Denote by g the Lie algebra defined by g = (CX1 ⊕ CY1) ⊥
⊕ g with
the bracket:
[αX1 + βY1 + X,α ′ X1 + β ′ Y1 +Y ] = [X,Y ]g + βδ(Y ) − β ′ δ(X) + B(δ(X),Y )X1
and the non-degenerate invariant symmetric bilinear form B on g is extended on g by:
B(αX1 + βY1 + X,α ′ X1 + β ′ Y1 +Y ) = B(X,Y ) + αβ ′ + α ′ β,
for all X,Y ∈ g, α,α ′ ,β,β ′ ∈ C. Then the quadratic Lie algebra (g,B) is called the double
extension of (g,B) by means of δ.
Proposition 2.3.9. Let (g,B) be a quadratic Lie algebra and D ∈ CI(g). Assume that there exist
a derivation δ ∈ Dera(g), a non-zero x ∈ C and an element U ∈ ker(δ) such that δD = Dδ =
xδ + adg(U). Let g be the double extension of g by means of δ. Then the endomorphism D of g
defined by:
D|g = D + φ(U) ⊗ X1, D(X1) = xX1, D(Y1) = xY1 +U + yX1
with y ∈ C is an invertible centromorphism of g.
Proof. It is obvious that D is symmetric and invertible. Let αX1 +βY1 +X and α ′ X1 +β ′ Y1 +Y
be elements in g, one has:
D[αX1 + βY1 + X,α ′ X1 + β ′ Y1 +Y ] = D[X,Y ]g + B(U,[X,Y ]g)X1 + βDδ(Y )
Furthermore, we get:
−β ′ Dδ(X) + xB(δ(X),Y )X1.
D(αX1 + βY1 + X),α ′ X1 + β ′ Y1 +Y ] = [D(X),Y ]g + β[U,Y ]g + βxδ(Y )
−β ′ δD(X) + B(δD(X),Y )X1.
Therefore, since δD = Dδ = xδ + adg(U) we obtain D a centromorphism of g.
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