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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3.3. Elementary quadratic Lie superalgebras<br />
Similarly and by invariance of B, we also obtain:<br />
and (2) follows.<br />
[X,Y ] = −2B(X 0 ,X)B(X 1,Y )X 1, ∀ X ∈ g 0 , Y ∈ g 1<br />
(3) If dim(U) = 2 and U is non-degenerate then B(X 1,Y 1) = a = 0. Replace with<br />
X1 := X1 p<br />
, p :=<br />
a a , X0 := aX0 and Y0 := Y0 a<br />
then g = (CX 0 ⊕ CY 0 ) ⊕ (CX 1 ⊕ CY 1), B(X 0 ,Y 0 ) = B(X 1,Y 1) = 1 and I = α ⊗ pq.<br />
Let X ∈ g 0, Y,Z ∈ g 1, one has:<br />
B(X,[Y,Z]) = I(X,Y,Z) = −α(X)(p(Y )q(Z) + p(Z)q(Y )).<br />
Therefore, the Lie super-brackets are defined:<br />
and (3) follows.<br />
[Y,Z] = −(B(X 1,Y )B(Y 1,Z) + B(X 1,Z)B(Y 1,Y ))X 0 , ∀ Y,Z ∈ g 1,<br />
[X,Y ] = −B(X 0 ,X)(B(X 1,Y )Y 1 + B(Y 1,Y )X 1), ∀ X ∈ g 0 , Y ∈ g 1<br />
(4) If dim(U) = 2 and U is totally isotropic. Let V = CZ 1 ⊕CT 1 be a totally isotropic subspace<br />
of g 1 such that g 1 = CX 1 ⊕ CY 1 ⊕ CZ 1 ⊕ CT 1, B(X 1,Z 1) = B(Y 1,T 1) = 1.<br />
If X ∈ g 0,Y,Z ∈ g 1 then:<br />
B(X,[Y,Z]) = I(X,Y,Z) = −α(X)(p(Y )q(Z) + p(Z)q(Y )).<br />
We obtain the Lie super-brackets as follows:<br />
[Y,Z] = −(B(X 1,Y )B(Y 1,Z) + B(X 1,Z)B(Y 1,Y ))X 0 , ∀ Y,Z ∈ g 1,<br />
[X,Y ] = −B(X 0 ,X)(B(X 1,Y )Y 1 + B(Y 1,Y )X 1), ∀ X ∈ g 0 , Y ∈ g 1.<br />
Thus, [Z 1,T 1] = −X 0 , [Y 0 ,Z 1] = −Y 1, [Y 0 ,T 1] = −X 1.<br />
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