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

2.2. Singular quadratic Lie algebras
A natural consequence of formulas in Proposition 2.2.22 (1) and Lemma 2.2.27 is given by
the proposition below:
Proposition 2.2.28.
(1) Consider the notation in Subection 2.2.3, Remark 2.2.25. Let g be a singular quadratic
Lie algebra of type S1 (that is, dup(g) = 1). Then g is the double extension of q = (CX0 ⊕
CY0) ⊥ by C = ad(Y0)|q.
(2) Let (g,B) be a quadratic Lie algebra. Let g ′ be the double extension of a quadratic vector
space (q ′ ,B ′ ) by a map C ′ . Let A be an i-isomorphism of g ′ onto g and write q = A(q ′ ).
Then g is the double extension of (q,B|q×q) by the map C = A C ′ A −1 where A = A| q ′.
(3) Let g be the double extension of a quadratic vector space q by a map C = 0. Then g is a
solvable singular quadratic Lie algebra. Moreover:
Proof.
(i) g is of type S3 if and only if rank(C) = 2.
(ii) g is of type S1 if and only if rank(C) ≥ 4.
(iii) g is reduced if and only if ker(C) ⊂ Im(C).
(iv) g is nilpotent if and only if C is nilpotent.
(1) Let b = CX0 ⊕CY0. Then B|b×b is non-degenerate and g = b⊕q. Since ad(Y0)(b) ⊂ b and
ad(Y0) is skew-symmetric, we have ad(Y0)(q) ⊂ q. By Proposition 2.2.22 (1), we have
Set X1 = X0 and Y1 = Y0 to obtain the result.
(2) Write g ′ = (CX ′ 1 ⊕CY ′ 1
[X,X ′ ] = B(C(X),X ′ )X0, ∀ X,X ′ ∈ q.
⊥
) ⊕ q ′ . Let X1 = A(X ′ 1 ) and Y1 = A(Y ′ 1 ). Then g = (CX1 ⊕CY1) ⊥
⊕ q
since A is i-isomorphic. One has:
[Y1,X] = A[Y ′ 1,A −1 (X)] = (AC ′ A −1 )(X), ∀ X ∈ q, and
[X,Y ] = A[A −1 (X),A −1 (Y )] = B((AC ′ A −1 )(X),Y )X1, ∀ X,Y ∈ q.
Hence, this proves the result.
(3) Let g = (CX1 ⊕CY1) ⊥
⊕ q, C = ad(Y1), α = φ(X1), Ω(X,Y ) = B(C(X),Y ), for all X, Y ∈ g
and I be the 3-form associated to g. Then the formula for the Lie bracket in Lemma 2.2.27
(1) can be translated as I = α ∧ Ω, hence dup(g) ≥ 1 and g is singular.
Let WΩ be the set WΩ = {ιX(Ω) = φ(C(X)),X ∈ g} then WΩ = φ(Im(C)). Therefore
rank(C) ≥ 2 by Proposition 2.2.3 and Ω is decomposable if and only if rank(C) = 2.
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