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

3.1. Application of Z × Z2-graded Lie superalgebras to quadratic Lie superalgebras
= −
n
∑
k=1
ι X k
1
(I) ∧ ιY k(γl)
1
(X,Y ) = −
ι X l
1
(I) ∧ ιY l(γl)
(X,Y )
1
= ιX l(I)(X,Y
) = B(X
1
l 1 ,[X,Y ]) = γl([X,Y ]) = −d(γl)(X,Y ).
Therefore, d = −adP(I).
Moreover, adP({I,I}) = [adP(I),adP(I)] = [d,d] = 2d2 = 0. Therefore, for all 1 ≤ i ≤ m,
1 ≤ j,k ≤ n one has {αi,{I,I}} = {β j,{I,I}} = {γk,{I,I}} = 0. Those imply ιX ({I,I}) = 0 for
all X ∈ g and hence, we obtain {I,I} = 0.
Conversely, let g be a quadratic Z2-graded vector space equipped with a bilinear form B
and I be an element in E (3,0) (g). Define d = −adP(I) then d ∈ D 1 0 (E(g)). Therefore, d2 = 0 if
and only if {I,I} = 0. Let F be the struture in g corresponding to d by the isomorphism D in
Lemma 3.1.15, one has
Proposition 3.1.18. F becomes a Lie superalgebra structure if and only if {I,I} = 0. In this
case, with the notation [X,Y ] := F(X,Y ) one has:
Moreover, the bilinear form B is invariant.
I(X,Y,Z) = B([X,Y ],Z), ∀ X,Y,Z ∈ g.
Proof. We need to prove that if F is a Lie superalgebra structure then I(X,Y,Z) = B([X,Y ],Z),
for all X,Y,Z ∈ g. Indeed, let {X 1 0 ,...,X m 0 } be an orthonormal basis of V0 and {X 1 1 ,...,X n 1 ,
Y 1,...,Y
n
1 1 } be a canonical basis of V1 then one has
It implies that
d = −adP(I) = −
F =
m
∑
j=1
Therefore, for all i we obtain
m
∑
j=1
ι X j
0
ι j(I)
⊗ X
X
0
j
(I) ∧ ι X j
0
0 +
n
∑
k=1
+
n
∑
k=1
ι X k
1
(I) ∧ ι Y k
1
ιX k(I)
⊗Y
1
k
1 −
n
∑
k=1
−
n
∑
k=1
ι Y k
1
ιY k(I)
⊗ X
1
k 1 .
B([X,Y ],X i 0 ) = ιX i(I)(X,Y
) = I(X
0
i 0 ,X,Y ) = I(X,Y,X i 0 ),
B([X,Y ],X i 1 ) = −ι Xi (I)(X,Y ) = −I(X
1
i 1 ,X,Y ) = I(X,Y,X i 1 ),
B([X,Y ],Y i
1 ) = −ι Y i(I)(X,Y
) = −I(Y i
i
1 ,X,Y ) = I(X,Y,Y 1 ).
These show that I(X,Y,Z) = B([X,Y ],Z), for all X,Y,Z ∈ g.
1
(I) ∧ ιX k .
1
Remark 3.1.19. The element I defined as above is also an invariant of g since LX(I) = 0, for all
X ∈ g where LX = D(ad(X)) the Lie super-derivation of g. Therefore, I is called the associated
invariant of g.
Lemma 3.1.20. Let (g,B) be a quadratic Lie superalgebra and I be its associated invariant.
Then ιX(I) = 0 if and only if X ∈ Z(g).
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