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

3.1. Application of Z × Z2-graded Lie superalgebras to quadratic Lie superalgebras
(1) d(φ)(X,Y ) = −φ([X,Y ]), for all X,Y ∈ V, φ ∈ V ∗ .
(2) The product [ , ] becomes a Lie super-bracket if and only if d 2 = 0. In this case, d is
called a differential super-exterior of E(V ).
Next, we will apply the above results for quadratic Lie superalgebras defined as follows:
Definition 3.1.16. A quadratic Lie superalgebra (g,B) is a Z2-graded vector space g equipped
with a non-degenerate even supersymmetric bilinear form B and a Lie superalgebra structure
such that B is invariant, i.e. B([X,Y ],Z) = B(X,[Y,Z]), for all X,Y,Z ∈ g.
Theorem 3.1.17. Let (g,B) be a quadratic Lie superalgebra. Define a trilinear form I on g by
Then one has
I(X,Y,Z) = B([X,Y ],Z), ∀ X,Y,Z ∈ g.
(1) I ∈ E (3,0) (g) = A 3 (g 0) ⊕ A 1 (g 0) ⊗ S 2 (g 1) .
(2) d = −adP(I).
(3) {I,I} = 0.
Proof. The assertion (1) follows the properties of B, note that B([g0,g 0],g 1) = B([g1,g 1],g 1) = 0.
For (2), fix an orthonormal basis {X 1 0 ,...,X m 0 } of V0 and a canonical basis {X 1 1 ,...,X n 1 ,
Y 1,...,Y
n
1 1 } of V1. Let {α1,...,αm} and {β1,...,βn,γ1,...,γn} be their dual basis respectively.
Then for all X,Y ∈ g, i = 1,...,m, l = 1,...,n one has:
adP(I)(αi)(X,Y ) =
(αi) − ιX k(I)
∧ ιY k(αi)
− ιY k(I)
∧ ιX k(αi)
1
1
1
1
(X,Y )
=
adP(I)(βl)(X,Y ) =
=
adP(I)(γl)(X,Y ) =
m
∑
j=1
m
∑
j=1
m
∑
j=1
n
∑
k=1
ι X j
0
ι X j
0
(I) ∧ ι X j
0
(I) ∧ ι j(αi)
X
0
n
∑
k=1
(X,Y ) =
ι X i
0
(I) ∧ ιX i(αi)
(X,Y )
0
= B(X i 0 ,[X,Y ]) = αi([X,Y ]) = −d(αi)(X,Y ),
ι X j
0
ι Y k
1
(I) ∧ ι X j
0
(βl) −
(I) ∧ ιX k(βl)
1
n
∑
k=1
ι X k
1
(X,Y ) =
(I) ∧ ι Y k
1
ι Y l
1
(βl) − ι Y k
1
(I) ∧ ιX l(βl)
(X,Y )
1
= −ι Y l(I)(X,Y
) = −B(Y
1
l
1 ,[X,Y ]) = βl([X,Y ]) = −d(βl)(X,Y ),
m
∑
j=1
ι X j
0
(I) ∧ ι X j
0
(γl) −
n
∑
k=1
67
ι X k
1
(I) ∧ ι Y k
1
(γl) − ι Y k
1
(I) ∧ ιX k(βl)
1
(X,Y )
(I) ∧ ιX k(γl)
1
(X,Y )