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

3.5. Singular quadratic Lie superalgebras of type S1 with non-Abelian even part
Theorem 3.5.7. Let g = (CX0 ⊕ CY0) ⊥
⊕ (q0 ⊕ q1) and g ′ = (CX ′ 0
double extensions of q = q0 ⊕ q1 by C = C0 +C1 and C ′ = C ′
0 +C ′
C1 is non-zero. Then
⊥
⊕ CY ′ ) ⊕ (q 0 0 ⊕ q1) be two
1, respectively. Assume that
(1) there exists a Lie superalgebra isomorphism between g and g ′ if and only if there exist an
invertible maps P ∈ L (q 0), Q ∈ L (q 1) and a non-zero λ ∈ C such that
(i) C ′
0 = λPC0P −1 and P ∗ PC0 = C0.
(ii) C ′
1 = λQC1Q −1 and Q ∗ QC1 = C1.
where P ∗ and Q ∗ are the adjoint maps of P and Q with respect to B|q 0 ×q 0 and B|q 1 ×q 1 .
(2) there exists an i-isomorphism between g and g ′ if and only if there is a non-zero λ ∈ C
such that C ′
0 is in the O(q0)-adjoint orbit through λC0 and C ′
1 is in the Sp(q1)-adjoint orbit through λC1.
Proof.
(1) Assume that there is a Lie superalgebra isomorphism A : g → g ′ . Obviously, A|g : g 0
0 → g0 is a Lie algebra isomorphism. Moreover, g0 and g ′ 0 are the double extensions of q0 by C0
and C ′
0, respectively. By Theorem 2.2.30 (1), there exist an invertible map P ∈ L (q0) and
a non-zero λ ∈ C such that
C ′
0 = λPC0P −1 , P ∗ PC0 = C0 and A(Y0) = 1
λ
Y ′
0 +Y,
where Y ∈ g 0 ∩ (X 0) ⊥ . Let Q = A|g 1 . Since C1 is non-zero one has [g 1,g 1] = 0 and then
[g ′ 1 ,g′ 1 ]′ = 0, where [ , ] ′ denotes the Lie super-bracket on g ′ . We have:
Therefore, A(X 0) = µX ′ 0
Let X,Y be elements in g 1 then
A([g 1,g 1]) = A(CX 0) = [A(g 1),A(g 1)] ′ = [g ′ 1 ,g′ 1 ]′ = CX ′ 0 .
for some non-zero µ ∈ C.
A([X,Y ]) = µB(C1(X,Y ))X ′ 0
It implies that Q ∗ C ′
1Q = µC1. Similarly, one has
= [A(X),A(Y )] = B(C′ 1Q(X),Q(Y ))X ′ 0 .
QC1(X) = A([Y0,X]) = [A(Y0),A(X)] ′ = [ 1 ′
Y 0
λ +Y,Q(X)]′ = 1
λ C′ 1Q(X).
So we obtain C ′
1 = λQC1Q −1 and then Q∗QC1 = µ
λ C1.
1
λµ 2
Replace with Q := Q then
Q∗QC1 = C1 and C ′
1 = λQC1Q −1 .
Conversely, if there is a triple (P,Q,λ) satisfying (i) and (ii) then we set A(X0) = λX0, A(Y 0) = 1
λ Y 0 and A(X +Y ) = P(X) + Q(Y ), for all X ∈ q 0, Y ∈ q 1. It is easy to check that
A is a Lie superalgebra isomorphism.
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