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

3.1. Application of Z × Z2-graded Lie superalgebras to quadratic Lie superalgebras
for all α,α ′ ∈ V ∗ , A ∈ E(V ).
Proof.
(1) We apply Proposition 3.1.12, respectively for α = (X i 0 )∗ = φ(X i), for all i = 1,...,m,
0
α = (Y l
1 )∗ = φ(X l 1 ) and α = (−X l 1 )∗ = φ(Y l),
for all l = 1,...,n to obtain the result.
1
(2) Let α ∈ g ∗ x, α ′ ∈ g ∗ x ′ be homogeneous forms in g ∗ , one has
{α,α ′ } = ιφ −1 (α) (α ′ ) = (−1) xx′
α ′ (φ −1 (α)) = (−1) xx′
B(φ −1 (α ′ ),φ −1 (α))
= B(φ −1 (α),φ −1 (α ′ )).
Proposition 3.1.12 and Corollary 3.1.13 are enough for our purpose. But as a consequence
of Lemma 6.9 in [PU07], one has a more general result as follows:
Proposition 3.1.14. Let {X 1 0 ,...,X m 0 } be a basis of V0 and {α1,...,αm} its dual basis. Let
{Y 1,...,Y
m
0 0 } be the basis of V0 defined by Y i
0 = φ −1 (αi). Set B = {X 1 1 ,...,X n,Y 1,...,Y
n}
be a
1 1 1
canonical basis of V1. Then the super Z × Z2-Poisson bracket on E(V ) is given by:
{A,A ′ m
ω+ f +1
} = (−1) ∑
n
ω
+(−1) ∑
k=1
for all A ∈ A ω ⊗ S f and A ′ ∈ E(V ).
Now, we consider the vector space
i, j=1
ι X k
1
B(Y i j
0 ,Y
0 )ι Xi 0
(A) ∧ ι Y k
1
E =
E n ,
n∈Z
(A) ∧ ι j(A
X
0
′ )
(A ′ ) − ι Y k
1
(A) ∧ ιX k(A
1
′ )
where E n = {0} if n ≤ −2, E −1 = V and E n is the space of super-antisymmetric n + 1-linear
mappings from V to V . Each of the subspaces E n is Z2-graded then the space E is Z × Z2-
graded by
E =
n∈Z
f ∈Z2
Moreover, E is a Z×Z2-graded Lie algebra and called the graded Lie algebra of the Z2-graded
vector space V [BP89]. Recall that there exists a Z × Z2-graded Lie algebra isomorphism D
between E and D(E(V )) (see a precise construction in [Gié04]) satisfying if F = Ω⊗X ∈ E n ω+x
then DF = −(−1) xωΩ ∧ ιX ∈ Dn f (E(V )).
Lemma 3.1.15. ([BP89], [Gié04])
Fix F ∈ E 1
0 , denote by d = DF and define the product [X,Y ] = F(X,Y ), for all X,Y ∈ V .
Then one has
66
E n f .