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

3.4. Quadratic Lie superalgebras with 2-dimensional even part
Proof. Let X,Y and Z be homogeneous elements in g of degrees x, y and z, respectively.
(1) If B ′ is invariant then
B ′ ([X,Y ],Z) = B ′ (X,[Y,Z]).
That means B(D([X,Y ]),Z = B(D(X),[Y,Z]) = B([D(X),Y ],Z). Since B is non-degenerate,
one has D([X,Y ]) = [D(X),Y ]. As a consequence, D([X,Y ]) = −(−1) xy D([Y,X]) =
−(−1) xy [D(Y ),X] = [X,D(Y )] by D even.
Conversely, if D satisfies D([X,Y ]) = [D(X),Y ] = [X,D(Y )], for all X,Y ∈ g, it is easy to
check that B ′ is invariant.
(2) B ′ is supersymmetric if and only if B ′ (X,Y ) = (−1) xy B ′ (Y,X). Therefore, B(D(X),Y ) =
(−1) xy B(D(Y ),X) = B(X,D(Y )) by B supersymmetric.
(3) It is obvious since B is non-degenerate.
Definition 3.4.21. An even and symmetric map D ∈ L (g) satisfying Lemma 3.4.20 (1) is called
a centromorphism of g.
As in Subsection 2.3.1, for a quadratic Lie superalgebra g, the space of centromorphisms of
g and the space generated by invertible ones are the same, denote it by C(g) (the proof is similar
completely to Lemma 2.3.2). As a consequence, the space of even invariant supersymmetric
bilinear forms on g coincides with its subspace generated by non-degenerate ones. Moreover,
the dimensions of all those spaces are the same and we denote it by dq(g), in particular, dq(g) =
dim(C(g)). The following proposition gives the formula of dq(g) for reduced quadratic Lie
superalgebras with 2-dimensional even part.
Proposition 3.4.22. Let g be a reduced quadratic Lie superalgebra with 2-dimensional even
part and D ∈ L (g) be an even symmetric map. Then:
(1) D is a centromorphism if and only if there exist µ ∈ C and an even symmetric map Z :
g → Z(g) such that Z| [g,g] = 0 and D = µ Id+Z. Moreover D is invertible if and only if
µ = 0.
(2)
Proof.
dq(g) = 2 +
(dim(Z(g) − 1))(dim(Z(g) − 2)
.
2
(1) The proof follows exactly as Proposition 2.3.6. First, g can be realized as the double
extension g = (CX0 ⊕ CY0 ) ⊥
⊕ q by C = ad(Y0) where C = C|q ∈ sp(q).
Let D be an invertible centromorphism. Lemma 3.4.20 (1) implies that D ◦ ad(X) =
ad(X) ◦ D, for all X ∈ g and then DC = CD. Using Lemma 3.4.7 (1) and CD = DC, from
[D(X),Y 0 ] = [X,D(Y 0)], we find
D(C(X)) = B(D(X 0),Y 0)C(X), ∀ X ∈ g.
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