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

2.2. Singular quadratic Lie algebras
In a canonical basis of the quadratic vector space q = C2p consider the map C J
2p with matrix
Jp 0
0 −t
and in a canonical basis of the quadratic vector space q = C
Jp
2p+1 consider the
map C J
Jp+1 M
2p+1 with matrix
0 −t
where Jp is the Jordan block of size p, M = (mi j)
Jp
denotes the (p + 1) × p-matrix with mp+1,p = −1 and mi j = 0 otherwise. Then C J
2p ∈ o(2p)
and C J
2p+1 ∈ o(2p+1). Denote by j2p the double extension of q by C J
2p and by j2p+1 the double
extension of q by C J
2p+1. So j2p ∈ N(2p + 2) and j2p+1 ∈ N(2p + 3). Lie algebras j2p or j2p+1
will be called nilpotent Jordan-type Lie algebras.
Keep the notations in Chapter 1, each [d] ∈ P1(n) can be written as (p1, p1, p2, p2,..., pk, pk,
2q1 + 1,...2qℓ + 1) with all pi even, p1 ≥ p2 ≥ ··· ≥ pk and q1 ≥ q2 ≥ ··· ≥ qℓ. We associate
a map C [d] ∈ o(n) with the matrix
diag k+ℓ (C J
2p1 ,CJ
2p2 ,...,CJ
2pk ,CJ
2q1+1,...,C J
2qℓ+1)
in a canonical basis of C n and denote by g [d] the double extension of C n by C [d]. Then
g [d] ∈ N(n+2) and g [d] is an amalgamated product of nilpotent Jordan-type Lie algebras. More
precisely,
g [d] = j2p1 × a j2p2 × a ... × a j2pk × a j2q1+1 × a ... × a j2qℓ+1.
By Proposition 1.2.10, the map [d] ↦→ C [d] from P1(n) to o(n) induces a bijection from
P1(n) onto N (n). Therefore, combined with Theorem 2.2.37, we deduce:
Theorem 2.2.38.
(1) The map [d] ↦→ g [d] from P1(n) to N(n+2) induces a bijection from P1(n) onto N(n+2).
(2) Each nilpotent singular n + 2-dimensional Lie algebra is i-isomorphic to a unique amalgamated
product g [d], [d] ∈ P1(n) of nilpotent Jordan-type Lie algebras.
Definition 2.2.39. Let g be a solvable singular quadratic Lie algebra and write g = (CX0 ⊕
CY0) ⊥
⊕ q a decomposition of g as a double extension (Proposition 2.2.28). Let C = ad(Y0)|q.
We say that g is diagonalizable if C is diagonalizable.
We denote by D(n + 2) the set of such structures on the quadratic vector space Cn+2 , by
Dred(n+2) the reduced ones, by D(n+2), Di (n+2), Dred(n+2), Di red (n+2) the corresponding
sets of isomorphic and i-isomorphic classes of elements in D(n + 2) and Dred(n + 2).
Remark that the property of being diagonalizable does not depend on the chosen decomposition
of g (see Remark 2.2.32) and a diagonalizable C satisfies ker(C) ⊂ Im(C) if and only if
ker(C) = {0}. By Corollary 2.2.31 and using a proof completely similar to Theorem 2.2.35 or
Theorem 2.2.37, we conclude:
Proposition 2.2.40. There is a bijection between Di (n + 2) and the set of semisimple O(n)orbits
in P1 (o(n)). The same result holds for Di red (n + 2) and semisimple invertible orbits in
P1 (o(n)).
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