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

3.4. Quadratic Lie superalgebras with 2-dimensional even part
(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 by Definition 3.4.6.
(2) Write g = t ⊥
⊕ q = g ′ . Denote by [·,·] ′ the Lie super-bracket on g ′ . Define A : g → g ′ by
A(X 0) = λX 0, A(Y 0) = 1
λ Y 0 and A|q = Idq. Then A([Y 0,X]) = C(X) = [A(Y 0),A(X)] ′ and
A([X,Y ]) = [A(X),A(Y )] ′ , for all X,Y ∈ q. So A is an i-isomorphism.
Theorem 3.4.8.
(1) Let g be a non-Abelian quadratic Lie superalgebra with 2-dimensional even part. Keep
the notations as in Proposition 3.4.4. Then g is the double extension of q = (CX 0 ⊕
CY 0 ) ⊥ = g 1 by C = ad(Y 0 )|q.
(2) Let g be the double extension of a symplectic vector space q by a map C = 0. Then g is a
singular solvable quadratic Lie superalgebra with 2-dimensional even part. Moreover:
(i) g is reduced if and only if ker(C) ⊂ Im(C).
(ii) g is nilpotent if and only if C is nilpotent.
(3) Let (g,B) be a quadratic Lie superalgebra. Let g ′ be the double extension of a symplectic
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 ′.
Proof. The assertions (1) and (2) follow Proposition 3.4.4 and Lemma 3.4.7. For (3), since A
is i-isomorphic then g has also 2-dimensional even part. Write g ′ = (CX ′
⊥
⊕ CY ′ ) ⊕ q
0 0 ′ . Let
X0 = A(X ′
0 ) and Y0 = A(Y ′
0 ). Then g = (CX0 ⊕ CY ⊥
0 ) ⊕ q and one has:
This proves the result.
[Y 0 ,X] = (AC ′ A −1 )(X), ∀ X ∈ q, and
[X,Y ] = B((AC ′ A −1 )(X),Y )X 0 , ∀ X,Y ∈ q.
Example 3.4.9. For reduced elementary quadratic Lie superalgebras with 2-dimensional even
part, then
79