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

3.1. Application of Z × Z2-graded Lie superalgebras to quadratic Lie superalgebras
Proof. Let A = Ω ⊗ F ∈ A ω ⊗ S f and A ′ = Ω ′ ⊗ F ′ ∈ A ω′ ⊗ S
f ′
, then
{A,A ′ } = (−1) f ω′ {Ω,Ω ′ } ⊗ FF ′ + (Ω ∧ Ω ′ ) ⊗ {F,F ′ } .
By the formula (I) in Chapter 2, one has
{Ω,Ω ′ m
ω+1
} = (−1) ∑
j=1
Combined with Lemma 3.1.10 (1), we obtain
{Ω,Ω ′ } ⊗ FF ′ m
ω+1
= (−1) ∑
j=1
= (−1) f (ω′ −1)+ω+1 m
∑
j=1
ι X j
0
ι X j
0
ι j(Ω)
⊗ F
X
0
= (−1) f ω′ +ω+ f +1 m
∑
j=1
n
∑
k=1
ι X j
0
(Ω) ∧ ι j(Ω
X
0
′ ).
(Ω) ∧ ι j(Ω
X
0
′
) ⊗ FF ′
∧
(A) ∧ ι j(A
X
0
′ ).
Let {p1,..., pn,q1,...,qn} be the dual basis of B then
{F,F ′
∂F ∂F
} =
∂ pk
′
−
∂qk
∂F ∂F
∂qk
′
.
∂ pk
By Remark 3.1.11, one has
{F,F ′ } =
n
∑
k=1
ι X k
1
(F)ι Y k
1
Combined with Lemma 3.1.10 (2), we obtain
(Ω ∧ Ω ′ ) ⊗ {F,F ′ } = (Ω ∧ Ω ′ ) ⊗
n
( f −1)ω′
= (−1) ∑
k=1
Ω ⊗ ι X k
1
= (−1) f ω′ +ω n
∑
k=1
Therefore, the result follows.
(F) ∧
ι X k
1
n
∑
k=1
(F ′ ) − ι Y k
1
Ω ′ ⊗ ι Y k
1
(A) ∧ ι Y k
1
ι X k
1
(F)ι Y k
1
(F ′
) −
(A ′ ) − ι Y k
1
ι j(Ω
X
0
′ ) ⊗ F ′
(F)ι Xk (F
1
′
) .
(F ′ ) − ι Y k
1
Ω ⊗ ι Y k
1
(F)ι Xk (F
1
′
)
(F) ∧
(A) ∧ ιX k(A
1
′
) .
Ω ′ ⊗ ι X k
1
(F ′
)
Corollary 3.1.13. Define the (even) isomorphism φ : V → V ∗ by φ(X) = B(X,.), for all X ∈ g
then one has
(1) {α,A} = ι φ −1 (α) (A),
(2) {α,α ′ } = B(φ −1 (α),φ −1 (α ′ )),
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