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

2.3. Quadratic dimension of quadratic Lie algebras
A symmetric map D satisfying (II) is called a centromorphism of g. The equality (II) is
equivalent to
D ◦ ad(X) = ad(X) ◦ D = ad(D(X)), ∀ X ∈ g.
Denote by C(g) the space of centromorphisms of g and by CI(g) the subspace spanned by
invertible centromorphisms in C(g). We recall Lemma 2.1 in [BB97] as follows.
Lemma 2.3.2. One has C(g) = CI(g).
Proof. Let D be an invertible centromorphism and ϕ ∈ C(g). Fix B a basis of g. Denote by
M(D) and M(ϕ) respectively the associated matrices of D and ϕ in B. Consider the polynomial
P(x) = det(M(ϕ) − xM(D)). Since P(x) is a non-zero polynomial so there exists λ ∈ C such
that P(λ) = 0. It means that ϕ − λD is invertible and thus ϕ = (ϕ − λD) + λD ∈ CI(g). It
shows that C(g) = CI(g).
Therefore the space of invariant symmetric bilinear forms on g and the subspace generated
by non-degenerated ones are the same. Let us denote it by B(g). The dimension of B(g) is
called the quadratic dimension of g and denoted by dq(g). As a consequence of the previous
lemmas, one has dq(g) = dim(C(g)). Moreover, one has other properties of C(g) as follows:
Proposition 2.3.3. Let D ∈ C(g) then
(1) D n ∈ C(g) for all n ≥ 1. Furthermore, if D is invertible then D −1 ∈ C(g).
(2) Z(g) and [g,g] are stable subspaces under D.
Proposition 2.3.4. Let δ ∈ Dera(g) be a skew-symmetric derivation of g. Assume D ∈ C(g)
such that D and δ commute. Then D ◦ δ is also a skew-symmetric derivation of g.
Proof. Since (D ◦ δ)[X,Y ] = D[δ(X),Y ] + D[X,δ(Y )] = [(D ◦ δ)(X),Y ] + [X,(D ◦ δ)(Y )] and
B((D ◦ δ)(X),Y ) = −B(X,(δ ◦ D)(Y )) = −B(X,(D ◦ δ)(Y )) for all X,Y ∈ g, one has D ◦ δ ∈
Dera(g).
Corollary 2.3.5. For all X ∈ g and D ∈ C(g), D ◦ ad(X) ∈ Dera(g).
It is known that dq(g) = 1 if g is simple or one-dimensional Lie algebra. If g is reductive,
but neither simple, nor one-dimensional, then
dq(g) = s(g) +
dim(Z(g))(1 + dim(Z(g)))
2
where Z(g) is the center of g and s(g) is the number of simple ideals of a Levi factor of g
(Corollary 2.1 in [Ben03], see also in [BB07]). A general formula for dq(g) is not known. Next,
we give a formula of dq(g) for reduced singular quadratic Lie algebras.
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