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

3.6. Quasi-singular quadratic Lie superalgebras
Corollary 3.6.7. Keep the notations as in Corollary 3.6.5 and replace 2X 0 by X 0 then for all
X,Y ∈ q, one has:
(1) [Y 1,Y 1] = X 0,
(2) [Y 1,X] = C(X) − B(X,X 0)X 1,
(3) [X,Y ] = −B(C(X),Y )X 1.
Hence, we have a more general result of Proposition 3.6.6:
Theorem 3.6.8. Let (q = q 0 ⊕ q 1,Bq) be a quadratic Z2-graded vector space and C an odd
endomorphism of q such that C is skew-supersymmetric and C 2 = 0. Let X0 be an isotropic
element q0, X0 ∈ ker(C) and (t = span{X1,Y1},Bt) a 2-dimensional symplectic vector space
with Bt(X1,Y 1) = 1. Consider the space g = q ⊥
⊕ t and define the product on g by:
[Y 1,Y 1] = X 0, [Y 1,X] = C(X) − Bq(X,X 0)X 1 and [X,Y ] = −Bq(C(X),Y )X 1
for all X ∈ q. Then g becomes a 2-nilpotent quadratic Lie superalgebra with the bilinear form
B = Bq + Bt. Moreover, one has g 0 = q 0, g 1 = q 1 ⊕ t.
A quadratic Lie superalgebra obtained in the above theorem is a special case of the generalized
double extensions given in [BBB] where we consider the generalized double extension of a
quadratic Z2-graded vector space (regarded as an Abelian superalgebra) by a one-dimensional
Lie superalgebra.
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