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

3.1. Application of Z × Z2-graded Lie superalgebras to quadratic Lie superalgebras
Proof. Since ιX(I)(g,g) = B(X,[g,g]) and Z(g) = [g,g] ⊥ then one has ιX(I) = 0 if and only if
X ∈ Z(g).
Definition 3.1.21. Let (g,B) and (g ′ ,B ′ ) be two quadratic Lie superalgebras. We say that (g,B)
and (g ′ ,B ′ ) are isometrically isomorphic (or i-isomorphic) if there exists a Lie superalgebra
isomorphism A from g onto g ′ satisfying
B ′ (A(X),A(Y )) = B(X,Y ), ∀ X,Y ∈ g.
In other words, A is an i-isomorphism if it is a (necessarily even) Lie superalgebra isomorphism
and an isometry. We write g i g ′ .
Note that two isomorphic quadratic Lie superalgebras (g,B) and (g ′ ,B ′ ) are not necessarily
i-isomorphic by the example below:
Example 3.1.22. Let g = osp(1,2) and B its Killing form. Recall that g 0 = o(3). Consider
another bilinear form B ′ = λB, λ ∈ C, λ = 0. In this case, (g,B) and (g,λB) cannot be iisomorphic
if λ = 1 since (g 0 ,B) and (g 0 ,λB) are not i-isomorphic (see Example 2.1.4).
69