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

2.2. Singular quadratic Lie algebras
Proof. Assume that A is an i-isomorphism from (g,B) onto (g ′ ,B ′ ) then
B ′ ([A(X),A(Y )],A(Z)) = B ′ (A[X,Y ],A(Z)) = B([X,Y ],Z), ∀ X,Y,Z ∈ g.
That means I = t A(I ′ ).
Let α ′ ∈ V I ′ then α ′ ∧I ′ = 0. So t A(α ′ ∧I ′ ) = t A(α ′ )∧ t A(I ′ ) = 0. It means that t A(V I ′) ⊂ VI.
Similarly, t A −1 (VI) ⊂ V I ′. Therefore, VI = t A(V I ′).
For all X ∈ g, Y,Z ∈ g ′ one has ι A(X)(I ′ )(Y,Z) = ιX(I)(A −1 (Y ),A −1 (Z)). Therefore, the
restriction of A to the subspace {v ∈ g | ιv(I) = 0} is an i-isomorphism from {v ∈ g | ιv(I) = 0}
onto {v ∈ g ′ | ιv(I ′ ) = 0} then WI = t A(W I ′).
It results from the previous lemma that dim(VI) and dim(WI) are invariant under
i-isomorphisms. This is not new for dim(WI) since dim(WI) = dim([g,g]). Actually, dim(WI)
is invariant under isomorphisms.
For dim(VI), to our knowledge this fact was not remarked up to now, so we introduce the
following definition:
Definition 2.2.9. Let g be a quadratic Lie algebra. The dup number dup(g) is defined by
dup(g) = dim(VI).
Remark 2.2.10. By Corollary 2.2.6, when g is non-Abelian, one has dup(g) ∈ {0,1,3} and
dim([g,g]) ≥ 3. Moreover, I is decomposable if and only if dup(g) = dim([g,g]) = 3, a simple
but rather interesting remark. Finally, if g is decomposed by g = z ⊥
⊕ l as in Proposition 2.1.5
then dup(g) = dup(l) since I ∈ 3 (WI) and WI = φ([g,g]) = φ([l,l]).
We separate non-Abelian quadratic Lie algebras as follows:
Definition 2.2.11. Let g be a non-Abelian quadratic Lie algebra.
(1) g is an ordinary quadratic Lie algebra if dup(g) = 0.
(2) g is a singular quadratic Lie algebra if dup(g) ≥ 1.
(i) g is a singular quadratic Lie algebra of type S1 if dup(g) = 1.
(ii) g is a singular quadratic Lie algebra of type S3 if dup(g) = 3.
Now, given a non-Abelian n-dimensional quadratic Lie algebra g, we can assume, up to
i-isomorphisms, that g is regarded as the quadratic vector space C n equipped with its canonical
bilinear form B. Our problem is considering Lie algebra structures on g such that B is invariant.
So we introduce the following sets:
Definition 2.2.12. For n ≥ 1:
(1) Q(n) is the set of non-Abelian quadratic Lie algebra structures on C n .
(2) O(n) is the set of ordinary quadratic Lie algebra structures on C n .
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