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

Appendix C
We will classify (up to isomorphisms) 3-forms on a vector space V with 1 ≤ dim(V ) ≤ 5
that can be applied in the classification of quadratic solvable or 2-step nilpotent Lie algebras of
low dimension in Chapter 2. The method is based only on changes of basis in the dual space
V ∗ .
Let I ∈ A 3 (V ) be a 3-form on V . It is obvious that I = 0 if dim(V ) = 1 or 2.
Case 1: dim(V ) = 3
If I = 0 then there exists a basis {α1,α2,α3} of V ∗ such that I = aα1 ∧α2 ∧α3. Replace α1
by 1 a α1, we get the result
I = α1 ∧ α2 ∧ α3.
Case 2: dim(V ) = 4
We will show that every 3-form on V is decomposable. Hence A 3 (V ) has only a non-zero
3-form (up to isomorphisms). Let {α1,α2,α3,α4} be a basis of V ∗ . Then I has the following
form:
I = aα1 ∧ α2 ∧ α3 + bα1 ∧ α2 ∧ α4 + cα1 ∧ α3 ∧ α4 + dα2 ∧ α3 ∧ α4,
where a,b,c,d ∈ C. We rewrite:
I = α1 ∧ α2 ∧ (aα3 + bα4) + (cα1 + dα2) ∧ α3 ∧ α4.
If aα3 +bα4 = 0 or cα1 +dα2 = 0 then I is decomposable. If aα3 +bα4 = 0 and cα1 +dα2 =
0, then we can assume that a = 0 and c = 0. We replace b a by b′ and d c by d′ to get:
We change the basis of V ∗ as follows:
I = aα1 ∧ α2 ∧ (α3 + b ′ α4) + c(α1 + d ′ α2) ∧ α3 ∧ α4.
β1 = α1 + d ′ α2, β2 = α2, β3 = α3 + b ′ α4, β4 = α4.
Then I = aβ1 ∧ β2 ∧ β3 + cβ1 ∧ β3 ∧ β4. It means that
Therefore, I is decomposable.
I = β1 ∧ (aβ2 − cβ4) ∧ β3.
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