TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...
TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...
TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...
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3.1. Application of Z × Z2-graded Lie superalgebras to quadratic Lie superalgebras<br />
= −<br />
n<br />
∑<br />
k=1<br />
ι X k<br />
1<br />
(I) ∧ ιY k(γl)<br />
1<br />
<br />
<br />
(X,Y ) = −<br />
ι X l<br />
1<br />
<br />
(I) ∧ ιY l(γl)<br />
(X,Y )<br />
1<br />
= ιX l(I)(X,Y<br />
) = B(X<br />
1<br />
l 1 ,[X,Y ]) = γl([X,Y ]) = −d(γl)(X,Y ).<br />
Therefore, d = −adP(I).<br />
Moreover, adP({I,I}) = [adP(I),adP(I)] = [d,d] = 2d2 = 0. Therefore, for all 1 ≤ i ≤ m,<br />
1 ≤ j,k ≤ n one has {αi,{I,I}} = {β j,{I,I}} = {γk,{I,I}} = 0. Those imply ιX ({I,I}) = 0 for<br />
all X ∈ g and hence, we obtain {I,I} = 0.<br />
Conversely, let g be a quadratic Z2-graded vector space equipped with a bilinear form B<br />
and I be an element in E (3,0) (g). Define d = −adP(I) then d ∈ D 1 0 (E(g)). Therefore, d2 = 0 if<br />
and only if {I,I} = 0. Let F be the struture in g corresponding to d by the isomorphism D in<br />
Lemma 3.1.15, one has<br />
Proposition 3.1.18. F becomes a Lie superalgebra structure if and only if {I,I} = 0. In this<br />
case, with the notation [X,Y ] := F(X,Y ) one has:<br />
Moreover, the bilinear form B is invariant.<br />
I(X,Y,Z) = B([X,Y ],Z), ∀ X,Y,Z ∈ g.<br />
Proof. We need to prove that if F is a Lie superalgebra structure then I(X,Y,Z) = B([X,Y ],Z),<br />
for all X,Y,Z ∈ g. Indeed, let {X 1 0 ,...,X m 0 } be an orthonormal basis of V0 and {X 1 1 ,...,X n 1 ,<br />
Y 1,...,Y<br />
n<br />
1 1 } be a canonical basis of V1 then one has<br />
It implies that<br />
d = −adP(I) = −<br />
F =<br />
m<br />
∑<br />
j=1<br />
Therefore, for all i we obtain<br />
m<br />
∑<br />
j=1<br />
ι X j<br />
0<br />
ι j(I)<br />
⊗ X<br />
X<br />
0<br />
j<br />
(I) ∧ ι X j<br />
0<br />
0 +<br />
n<br />
∑<br />
k=1<br />
+<br />
n<br />
∑<br />
k=1<br />
ι X k<br />
1<br />
(I) ∧ ι Y k<br />
1<br />
ιX k(I)<br />
⊗Y<br />
1<br />
k<br />
1 −<br />
n<br />
∑<br />
k=1<br />
−<br />
n<br />
∑<br />
k=1<br />
ι Y k<br />
1<br />
ιY k(I)<br />
⊗ X<br />
1<br />
k 1 .<br />
B([X,Y ],X i 0 ) = ιX i(I)(X,Y<br />
) = I(X<br />
0<br />
i 0 ,X,Y ) = I(X,Y,X i 0 ),<br />
B([X,Y ],X i 1 ) = −ι Xi (I)(X,Y ) = −I(X<br />
1<br />
i 1 ,X,Y ) = I(X,Y,X i 1 ),<br />
B([X,Y ],Y i<br />
1 ) = −ι Y i(I)(X,Y<br />
) = −I(Y i<br />
i<br />
1 ,X,Y ) = I(X,Y,Y 1 ).<br />
These show that I(X,Y,Z) = B([X,Y ],Z), for all X,Y,Z ∈ g.<br />
1<br />
(I) ∧ ιX k .<br />
1<br />
Remark 3.1.19. The element I defined as above is also an invariant of g since LX(I) = 0, for all<br />
X ∈ g where LX = D(ad(X)) the Lie super-derivation of g. Therefore, I is called the associated<br />
invariant of g.<br />
Lemma 3.1.20. Let (g,B) be a quadratic Lie superalgebra and I be its associated invariant.<br />
Then ιX(I) = 0 if and only if X ∈ Z(g).<br />
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