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

3.4. Quadratic Lie superalgebras with 2-dimensional even part
(1) g s 4,1 is the double extension of the 2-dimensional symplectic vector space q = C2 by the
map having matrix:
C =
0 1
0 0
in a canonical basis {E1,E2} of q where B(E1,E2) = 1.
(2) g s 4,2 the double extension of the 2-dimensional symplectic vector space q = C2 by the
map having matrix:
C =
1 0
0 −1
in a canonical basis {E1,E2} of q where B(E1,E2) = 1.
(3) g s 6 is the double extension of the 4-dimensional symplectic vector space q = C4 by the
map having matrix:
⎛
0
⎜
C = ⎜0
⎝0
1
0
0
0
0
0
⎞
0
0 ⎟
0⎠
0 0 −1 0
in a canonical basis {E1,E2,E3,E4} of q where B(E1,E3) = B(E2,E4) = 1, the other are
zero.
Let (q,B) be a symplectic vector space. We recall that Sp(q) is the isometry group of the
symplectic form B and sp(q) is its Lie algebra, i.e. the Lie algebra of skew-symmetric maps
with respects to B. The adjoint action is the action of Sp(q) on sp(q) by conjugation (see
Chapter 1).
Proposition 3.4.10. Let (q,B) be a symplectic vector space. Let g = (CX0 ⊕ CY0 ) ⊥
⊕ q and
g ′ = (CX ′
⊥
⊕CY ′ ) ⊕ q be double extensions of q, by skew-symmetric maps C and C
0 0 ′ respectively.
Then:
(1) there exists a Lie superalgebra isomorphism between g and g ′ if and only if there exist an
invertible map P ∈ L (q) and a non-zero λ ∈ C such that C ′ = λ PCP −1 and P ∗ PC = C
where P ∗ is the adjoint map of P with respect to B.
(2) there exists an i-isomorphism between g and g ′ if and only if C ′ is in the Sp(q)-adjoint
orbit through λC for some non-zero λ ∈ C.
Proof. The assertions are obvious if C = 0. We assume C = 0.
(1) Let A : g → g ′ be a Lie superalgebra isomorphism then A(CX0 ⊕ CY0) = CX ′ 0
A(q) = q. It is obvious that C ′ = 0. It is easy to see that CX0 = Z(g) ∩ g0 and CX ′ 0 =
Z(g ′ ) ∩ g ′ 0 then one has A(CX0) = CX ′ 0 . It means A(X0) = µX ′ for some non-zero µ ∈ C.
0
Let A|q = Q and assume A(Y0) = βY ′
0 + γX ′ . For all X, Y ∈ q, we have A([X,Y ]) =
0
. Also,
µB(C(X),Y )X ′
0
A([X,Y ]) = [Q(X),Q(Y )] ′ = B(C ′ Q(X),Q(Y ))X ′
0 .
80
⊕ CY ′
0 and