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

3.1. Application of Z × Z2-graded Lie superalgebras to quadratic Lie superalgebras
Example 3.1.9. Let X ∈ Vx be a homogeneous element in V of degree x and define the endomorphism
ιX of E(V ) by
ιX(A)(X1,...,Xa−1) = (−1) xb A(X,X1,...,Xa−1), ∀ A ∈ E (a,b) (V ), X1,...,Xa−1 ∈ V.
Then ιX is a super-derivation of E(V ) of degree (-1,x) [BP89]. In particular
ιX(A ∧ A ′ ) = ιX(A) ∧ A ′ + (−1) −a+xb A ∧ ιX(A ′ ), ∀ A ∈ E (a,b) (V ), A ′ ∈ E(V ).
Lemma 3.1.10. Let X 0 ∈ V 0 and X 1 ∈ V 1 then for all Ω ⊗ F ∈ A ω ⊗ S f , one has:
(1) ιX 0 (Ω ⊗ F) = ιX 0 (Ω) ⊗ F,
(2) ιX 1 (Ω ⊗ F) = (−1) ω Ω ⊗ ιX 1 (F).
Proof. We have:
(1) ιX 0 (Ω ⊗ F) = ιX 0 ((Ω ⊗ 1) ∧ (1 ⊗ F)) = ιX 0 (Ω ⊗ 1) ∧ (1 ⊗ F) + (−1) −ω (Ω ⊗ 1) ∧ ιX 0 (1 ⊗
F) = ιX 0 (Ω) ⊗ F.
(2) ιX 1 (Ω ⊗ F) = ιX 1 ((Ω ⊗ 1) ∧ (1 ⊗ F)) = ιX 1 (Ω ⊗ 1) ∧ (1 ⊗ F) + (−1) −ω (Ω ⊗ 1) ∧ ιX 1 (1 ⊗
F) = (−1) ω Ω ⊗ ιX 1 (F).
Remark 3.1.11.
(1) If Ω ∈ A ω then ιX(Ω)(X1,...,Xω−1) = Ω(X,X1,...,Xω−1), for all X,X1,...,Xω−1 ∈ V 0.
(2) Let X be an element of the canonical basis B of V1 and p ∈ V ∗
1
Corollary II.1.52 in [Gié04] one has:
Moreover,
ιX(p n )(X n−1 ) = (−1) n p n (X n ) = (−1) n (−1) n(n−1)/2 n!.
be its dual form. By
∂ pn
∂ p (X n−1 ) = n(p n−1 )(X n−1 ) = (−1) (n−1)(n−2)/2 n!. It implies that
ιX(p n )(X n−1 ∂ pn
) = −
∂ p (X n−1 ).
Since each F ∈ S f is regarded as a polynomial in the variable p and by linearizing so
one has the following property: let X ∈ V1 and p ∈ V ∗ be its dual form then
1
{A,A ′ m
ω+ f +1
} = (−1) ∑
j=1
ι X j
0
ιX(F) = − ∂F
, ∀ F ∈ S .
∂ p
Proposition 3.1.12. Fix an orthonormal basis {X 1 0 ,...,X m 0 } of V0 and a canonical basis B =
{X 1 1 ,...,X n,Y 1,...,Y
n
1 1 1 } of V1. Then the super Z × Z2-Poisson bracket on E(V ) is given by:
ιX k(A)
∧ ιY k(A
1
1
′ ) − ιY k(A)
∧ ιX k(A
1
1
′
)
for all A ∈ A ω ⊗ S f and A ′ ∈ E(V ).
(A) ∧ ι j(A
X
0
′ n
ω
) + (−1) ∑
k=1
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