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