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

Appendix C
Corollary C.4. Let V be a vector space, V ∗ its dual space and I be a 3-form on V . Define
Then dup(V ) = 0 if 1 ≤ dim(V ) ≤ 5.
VI = {α ∈ V ∗ | α ∧ I = 0} and dup(V ) = dim(VI).
Proof. It is easy to see that if I = 0 then VI = V ∗ . If I is a non-zero decomposable 3-form then
dup(V ) = 3. Assume that I is indecomposable. Since dim(V ) ≤ 5 then it happens only in the
case dim(V ) = 5. In this case I has the form as in Proposition C.1 with a,b = 0. Therefore one
has dup(V ) = 1.
Consequently, we obtain the result given in Appendix B as follows:
Corollary C.5. Let g be a non-Abelian quadratic Lie algebra such that dim[g,g] ≤ 5. Then g
is singular.
Proof. Define the element I as in Proposition 2.2.1 then I ∈ 3 (WI) where WI = φ([g,g])
(Corollary 2.2.6). Since dim[g,g] ≤ 5 then dup(g) = 0 and therefore g is singular.
Corollary C.6. Let g be a non-Abelian quadratic solvable Lie algebra such that dim(g) ≤ 6.
Then g is singular.
Proof. Since g is solvable then [g,g] = g and therfore dim[g,g] ≤ 5. Apply the above corollary,
we get the result.
In the case of higher dimensions, we have the following proposition:
Proposition C.7. Let V be a vector space such that dim(V ) ≥ 6. Then there exists an element
I ∈ A 3 (V ) satisfying ιx(I) = 0 for all non-zero x in V .
Proof. We denote by n = dim(V ) and fix a basis {α1,...,αn} of V ∗ . Then the element I is
defined in the following cases:
• If n = 3k then we set
• If n = 3k + 1 = 3(k − 2) + 7 then we set
I = α1 ∧ α2 ∧ α2 + ··· + αn−2 ∧ αn−1 ∧ αn.
I = α1 ∧ α2 ∧ α2 + ··· + αn−9 ∧ αn−8 ∧ αn−7
+αn−6 ∧ (αn−5 ∧ αn−4 + αn−3 ∧ αn−2 + αn−1 ∧ αn).
• If n = 3k + 2 = 3(k − 1) + 5 then we set
I = α1 ∧ α2 ∧ α2 + ··· + αn−7 ∧ αn−6 ∧ αn−5
+αn−4 ∧ (αn−3 ∧ αn−2 + αn−1 ∧ αn).
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