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

3.1. Application of Z × Z2-graded Lie superalgebras to quadratic Lie superalgebras
By straightforward calculating, it is easy to get simple properties of this bracket:
Proposition 3.1.5. E(V ) is a Lie superalgebra with the super Z × Z2-Poisson bracket. In other
words, for A ∈ E (a,b) (V ), A ′ ∈ E (a′ ,b ′ ) (V ) and A ′′ ∈ E (a ′′ ,b ′′ ) (V ):
(1) {A ′ ,A} = −(−1) aa′ +bb ′
{A,A ′ }.
(2) (−1) aa′′ +bb ′′
{A,{A ′ ,A ′′ }} + (−1) a′′ a ′ +b ′′ b ′
{A ′′ ,{A,A ′ }} + (−1) a′ a+b ′ b {A ′ ,{A ′′ ,A}} = 0.
Moreover, one has {A,A ′ ∧ A ′′ } = {A,A ′ } ∧ A ′′ + (−1) aa′ +bb ′
A ′ ∧ {A,A ′′ }.
Remark 3.1.6. The second formula in the previous proposition is equivalent to:
{{A,A ′ },A ′′ } = {A,{A ′ ,A ′′ }} − (−1) aa′ +bb ′
{A ′ ,{A,A ′′ }}.
Therefore, if we denote by adP(A) = {A,.}, A ∈ E (a,b) (V ) and by End(E(V )) the vector space
of endomorphisms of E(V ) then adP(A) ∈ End(E(V )), for all A ∈ E(V ) and:
for all A ′ ∈ E (a′ ,b ′ ) (V ).
adP({A,A ′ }) = adP(A) ◦ adP(A ′ ) − (−1) aa′ +bb ′
adP(A ′ ) ◦ adP(A)
We recall that End(E(V )) has a natural Z × Z2-gradation as follows:
deg(F) = (n,d), n ∈ Z, d ∈ Z2 if deg(F(A)) = (n + a,d + b), where A ∈ E (a,b) (V ).
Denote by Endn f (E(V )) the subspace of endomorphisms of degree (n, f ) of End(E(V )). It is
clear that if A ∈ E (a,b) (V ) then adP(A) has degree (a−2,b). Moreover, as we known, End(E(V ))
is also a Z×Z2-graded Lie algebra, frequently denoted by gl(E(V )), with the Lie super-bracket
defined by:
[F,G] = F ◦ G − (−1) np+ f g G ◦ F, ∀ F ∈ End n f (E(V )), G ∈ Endp g(E(V )).
Therefore, since Remark 3.1.6, one has the corollary:
Corollary 3.1.7.
adP({A,A ′ }) = [adP(A),adP(A ′ )], ∀ A,A ′ ∈ E(V ).
Definition 3.1.8. An endomorphism D ∈ gl(E(V )) of degree (n,d) is called a super-derivation
of E(V ) (for the super-exterior product) if
D(A ∧ A ′ ) = D(A) ∧ A ′ + (−1) na+db A ∧ D(A ′ ), ∀ A ∈ E (a,b) (V ), A ′ ∈ E(V ).
Denote by Dn d (E(V )) the space of super-derivations degree (n,d) of E(V ) then we obtain a
Z × Z2-gradation of the space of super-derivations D(E(V )) of E(V ) as follows:
D(E(V )) =
n∈Z
d∈Z2
D n d (E(V ))
and D(E(V )) becomes a subalgebra of gl(E(V )) [NR66]. Moreover, the last formula in Proposition
3.1.5 affirms that adP(A) ∈ D(E(V )), for all A ∈ E(V ).
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