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

Appendix A
In this appendix, we recall some facts on skew-symmetric maps used in this thesis. Nothing
here is new, but short proofs are given for the sake of completeness.
Throughout this section, let V be a vector space endowed with a non-degenerate bilinear
form Bε (quadratic or symplectic) and C be a skew-symmetric map in gε then one has the useful
identity ker(C) = (Im(C)) ⊥ .
Lemma A.1. There exist subspaces W and N of V such that:
(1) N ⊂ ker(C), C(W) ⊂ W and V = W ⊥
⊕ N.
(2) Let BW = Bε|W×W and CW = C|W . Then BW is non-degenerate, CW ∈ gε(W,BW ) and
ker(CW ) ⊂ Im(CW ) = Im(C).
Proof. We follow the proof of Proposition 2.1.5. Let N0 = ker(C) ∩ Im(C) and let N be a
complementary subspace of N0 in ker(C), ker(C) = N0 ⊕N. Since ker(C) = (Im(C)) ⊥ , we have
Bε(N0,N) = {0} and N ∩ N⊥ = {0}. So, if W = N⊥ , one has V = W ⊥
⊕ N. From C(N) = {0},
we deduce that C(W) ⊂ W.
It is clear that Bε is non-degenerate on W and that CW ∈ gε(W). Moreover, since C(W) ⊂ W
and C(N) = {0}, then Im(C) = Im(CW ). It is immediate that ker(CW ) = N0, so ker(CW ) ⊂
Im(CW ).
Lemma A.2. Assume that ker(C) ⊂ Im(C). Denote by L = ker(C). Let {L1,...,Lr} be a basis
of L.
(1) If dim(V ) is even, there exist subspaces L ′ with basis {L ′ 1 ,...,L′ r}, U with basis {U1,...,Us}
and U ′ with basis {U ′ 1 ,...,U′ s} such that Bε(Li,L ′ j ) = δi j, for all 1 ≤ i, j ≤ r, L and L ′ are
totally isotropic, Bε(Ui,U ′ j ) = δi j, for all 1 ≤ i, j ≤ s, U and U ′ are totally isotropic and
V = (L ⊕ L ′ ) ⊥
⊕ (U ⊕U ′ ).
Moreover Im(C) = L ⊥
⊕ (U ⊕U ′ ⊥
) and C : L ′ ⊕ (U ⊕U ′ ) → L ⊥
⊕ (U ⊕U ′ ) is a bijection.
(2) If ε = 1 and dim(V ) is odd, there exist subspaces L ′ , U and U ′ as in (1) and v ∈ V such
that Bε(v,v) = 1 and
V = (L ⊕ L ′ ) ⊥
⊕ Cv ⊥
⊕ (U ⊕U ′ ).
129