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

2.2. Singular quadratic Lie algebras
equipped with a bilinear form B = Bq + Bt and define a bracket on g by
[X + λX1 + µY1,Y + λ ′ X1 + µ ′ Y1] = µC(Y ) − µ ′ C(X) + B(C(X),Y )X1,
for all X,Y ∈ q, λ,µ,λ ′ , µ ′ ∈ C. Then (g,B) is a quadratic solvable Lie algebra. We say
that g is the double extension of q by C.
(2) Let gi be double extensions of quadratic vector spaces (qi,Bi) by skew-symmetric maps
Ci ∈ L (qi), for 1 ≤ i ≤ k. The amalgamated product
is defined as follows:
g = g1 × a g2 × a ... × a gk
• consider (q,B) be the quadratic vector space with q = q1 ⊕ q2 ⊕ ··· ⊕ qk and the
bilinear form B such that B(∑ k i=I Xi,∑ k i=I Yi) = ∑ k i=I Bi(Xi,Yi), for Xi,Yi ∈ qi, 1 ≤ i ≤
k.
• the skew-symmetric map C ∈ L (q) is defined by C(∑ k i=I Xi) = ∑ k i=I Ci(Xi), for Xi ∈
qi, 1 ≤ i ≤ k.
Then g is the double extension of q by C.
Next, we will show that double extensions are highly related to solvable singular quadratic
Lie algebras and amalgamated products can be used to decompose double extensions that are
useful in the nilpotent case. However, we notice here that generally this decomposition is a bad
behavior with respect to i-isomorphisms as follows: if g1 g ′ 1 and g2 g ′ 2 , it may happen that
g1 × g2 and g
a ′ 1 × a g′ 2 are not even isomorphic. An example will be given in Remark 2.2.45.
Lemma 2.2.27. We keep the notation above.
(1) Let g be the double extension of q by C. Then
[X,Y ] = B(X1,X)C(Y ) − B(X1,Y )C(X) + B(C(X),Y )X1, ∀ X,Y ∈ g,
where C = ad(Y1). Moreover, X1 ∈ Z(g) and C|q = C.
(2) Let g ′ be the double extension of q by C ′ = λC, λ ∈ C, λ = 0. Then g and g ′ are iisomorphic.
Proof.
(1) This is a straightforward computation from the previous definition.
(2) Write g = q ⊥
⊕ t = g ′ . Denote by [·,·] ′ the Lie bracket on g ′ . Define A : g → g ′ by
A(X1) = λX1, A(Y1) = 1
λ Y1 and A|q = Idq. Then A([Y1,X]) = C(X) = [A(Y1),A(X)] ′ and
A([X,Y ]) = [A(X),A(Y )] ′ , for all X,Y ∈ q. So A is an i-isomorphism.
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