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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3.6. Quasi-singular quadratic Lie superalgebras<br />
3.6 Quasi-singular quadratic Lie superalgebras<br />
By Definition 3.5.3, it is natural to question: let (q = q0 ⊕ q1,Bq) be a quadratic Z2-graded<br />
vector space and C be an endomorphism of q. Let (t = span{X1,Y1},Bt) be a 2-dimensional<br />
symplectic vector space with Bt(X1,Y 1) = 1. Is there an extension g = q ⊥<br />
⊕ t such that g equipped<br />
with the bilinear form B = Bq + Bt becomes a quadratic Lie superalgebra such that g0 = q0, g1 = q1 ⊕ t and the Lie super-bracket is represented by C ? In this last section of Chapter 3, we<br />
will give an affirmative answer to this question.<br />
The dup-number and the form of the associated invariant I in the previous sections suggest<br />
that it would be also interesting to study a quadratic Lie superalgebra g whose associated<br />
invariant I has the form<br />
I = J ∧ p<br />
where p ∈ g ∗ 1 is non-zero, J ∈ A 1 (g 0) ⊗ S 1 (g 1) is indecomposable. We obtain the first result<br />
as follows:<br />
Proposition 3.6.1. {J,J} = {p,J} = 0.<br />
Proof. Apply Proposition 3.1.5 (1) and (2) to obtain<br />
{I,I} = {J ∧ p,J ∧ p} = {J ∧ p,J} ∧ p + J ∧ {J ∧ p, p}<br />
= −{J,J} ∧ p ∧ p + 2J ∧ {p,J} ∧ p − J ∧ J ∧ {p, p}.<br />
Since the super-exterior product is commutative then one has J ∧ J = 0. Moreover, {I,I} = 0<br />
implies that:<br />
{J,J} ∧ p ∧ p = 2J ∧ {p,J} ∧ p.<br />
That means {J,J} ∧ p = 2J ∧ {p,J}.<br />
If {J,J} = 0 then {J,J} ∧ p = 0, so J ∧ {p,J} = 0. Note that {p,J} ∈ A 1 (g0) so J must<br />
contain the factor p, i.e. J = α ⊗ p where α ∈ g∗. But {p,J} = {p,α ⊗ p} = −α ⊗ {p, p} = 0<br />
0<br />
since {p, p} = 0. This is a contradiction and therefore {J,J} = 0.<br />
As a consequence, J ∧ {p,J} = 0. Set α = {p,J} ∈ A 1 (g0) then we have J ∧ α = 0. If<br />
α = 0 then J must have the form J = α ⊗ q where q ∈ S 1 (g1). That is a contradition since J is<br />
indecomposable.<br />
Definition 3.6.2. We continue to keep the condition I = J ∧ p with p ∈ g∗ non-zero and J ∈<br />
1<br />
A 1 (g0) ⊗ S 1 (g1) indecomposable. We can assume that<br />
J =<br />
n<br />
∑ αi ⊗ pi<br />
i=1<br />
where αi ∈ A 1 (g 0), i = 1,...,n are linearly independent and pi ∈ S 1 (g 1). A quadratic Lie<br />
superalgebra having such associated invariant I is called a quasi-singular quadratic Lie superalgebra.<br />
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