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

3.1. Application of Z × Z2-graded Lie superalgebras to quadratic Lie superalgebras
and in terms of the Z2-gradations
E0(V ) = A ⊗ ⊕ S
j≥0
2 j
and E1(V ) = A ⊗
2 j+1
⊕ S
j≥0
.
In other words, if A = Ω ⊗ F ∈ A ω ⊗ S f then A ∈ E (ω+ f , f ) (V ) where f denotes the residue
class modulo 2 of f .
Next, we define the super-exterior product on E(V ) as follows:
for all Ω ∈ A , Ω ′ ∈ A ω′ , F ∈ S f , F ′ ∈ S .
(Ω ⊗ F) ∧ (Ω ′ ⊗ F ′ f ω′
) = (−1) (Ω ∧ Ω ′ ) ⊗ FF ′ ,
Proposition 3.1.1. The vector space E(V ) with this product becomes a commutative and associative
Z × Z2-graded algebra. We call E(V ) the super-exterior algebra of V ∗ .
Proof. It is easy to see that the algebra E(V ) with this product is a Z × Z2-graded algebra. Let
A = Ω ⊗ F ∈ A ω ⊗ S f , A ′ = Ω ′ ⊗ F ′ ∈ A ω′ ⊗ S f ′
and A ′′ = Ω ′′ ⊗ F ′′ ∈ A ω′′ ⊗ S f ′′
then
A ′ ∧ A = (Ω ′ ⊗ F ′ ) ∧ (Ω ⊗ F) = (−1) f ′ ω (Ω ′ ∧ Ω) ⊗ F ′ F = (−1) f ′ ω+ωω ′
(Ω ∧ Ω ′ ) ⊗ FF ′
= (−1) f ′ ω+ωω ′ + f ω ′
(Ω ⊗ F) ∧ (Ω ′ ⊗ F ′ ) = (−1) (ω+ f )(ω′ + f ′ )+ f f ′
A ∧ A ′ .
Similarly, (A∧A ′ )∧A ′′ = A∧(A ′ ∧A ′′ ) = (−1) f (ω′ +ω ′′ )+ f ′ ω ′′
(Ω∧Ω ′ ∧Ω ′′ )⊗FF ′ F ′′ . Therefore,
we get the result.
Remark 3.1.2. There is another equivalent construction in [BP89], that is, E(V ) is the space of
super-antisymmetric multilinear mappings from V into C. The algebras A and S are regarded
as subalgebras of E(V ) by identifying Ω := Ω ⊗ 1, F := 1 ⊗ F, and the tensor product Ω ⊗ F =
(Ω ⊗ 1) ∧ (1 ⊗ F) for all Ω ∈ A , F ∈ S .
Now, we assume that the vector space V is equipped with a non-degenerate even supersymmetric
bilinear form B. That means B is symmetric on V 0, skew-symmetric on V 1, B(V 0,V 1) = 0
and B|V 0 ×V 0 , B|V 1 ×V 1 are non-degenerate. In this case, dim(V 1) = 2n must be even and V is also
called a quadratic Z2-graded vector space. We recall the definition of the Poisson bracket on
S as follows.
Definition 3.1.3. Let B = {X1,...,Xn,Y1,...,Yn} be a canonical basis of V 1 such that B(Xi,Xj) =
B(Yi,Yj) = 0, B(Xi,Yj) = δi j and {p1,..., pn,q1,...,qn} its dual basis. Then the Poisson bracket
on the algebra S regarded as the polynomial algebra C[p1,..., pn,q1,...,qn] is defined by:
{F,G} =
n
∂F ∂G
−
∂ pi ∂qi
∂F
∂G
,
∂qi ∂ pi
∀ F,G ∈ S .
∑
i=1
Combined with the notion of super-Poisson bracket on A in Chapter 2, we have a new
bracket as follows [MPU09].
Definition 3.1.4. The super Z × Z2-Poisson bracket on E(V ) is defined by:
{Ω ⊗ F,Ω ′ ⊗ F ′ } = (−1) f ω′ {Ω,Ω ′ } ⊗ FF ′ + (Ω ∧ Ω ′ ) ⊗ {F,F ′ } ,
for all Ω ∈ A , Ω ′ ∈ A ω′ , F ∈ S f , F ′ ∈ S .
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