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

2.2. Singular quadratic Lie algebras
2.2.2 The dup number of a quadratic Lie algebra
Let V be a vector space and I ∈ A k (V ), for k ≥ 1. We introduce two subspaces of V ∗ :
Proposition 2.2.3. [Bou58]
Let I ∈ A k (V ), I = 0. Then:
VI = {α ∈ V ∗ | α ∧ I = 0},
WI = {v ∈ V | ιv(I) = 0} ⊥∗ .
(1) VI ⊂ WI, dim(VI) ≤ k and dim(WI) ≥ k.
(2) If {α1,...,αr} is a basis of VI, then α1 ∧ ··· ∧ αr divides I. Moreover, I belongs to the
k-th exterior power of WI, also denoted by k (WI).
(3) I is decomposable if and only if dim(VI) = k or dim(WI) = k. In this case, VI = WI and
if {α1,...,αk} is a basis of VI, there is some non-zero λ ∈ C such that:
Proof. First, we need the following lemmas:
I = λα1 ∧ ··· ∧ αk.
Lemma 2.2.4. Let l ≤ k and α1,...,αl be linear forms independent in V ∗ and satisfying αi ∧I =
0, for all i = 1,...,l. One has:
(1) if l ≤ k−1 then there exists a multilinear form β ∈ A k−l (V ) such that I = α1 ∧···∧αl ∧β,
(2) if l = k then there exists a non-zero complex ξ such that I = ξ α1 ∧ ··· ∧ αk.
Proof. Since α1,...,αl are linearly independent in V ∗ then we can complete this system by
vectors to get a basis {α1,...,αn} of V ∗ . In this basis, assume that I is as follows:
I = ∑
J⊂[[1,n]],|J|=k
ξJα j1 ∧ ··· ∧ α jk ,
where ξJ ∈ C and the indices meant J = ( j1,..., jk) ∈ N k with 1 ≤ j1 < ··· < jk ≤ n.
One has α1 ∧ I = 0 then ξJ = 0 if j1 = 1. Therefore, we obtain
I = α1 ∧ ∑ ξ1, j2,..., jk
2≤ j2