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

2.4. 2-step nilpotent quadratic Lie algebras
(1) Let A : g → g ′ be a Lie algebra isomorphism. Since h ∗ = Z(g) = Z(g ′ ) is stable by A then
there exist linear maps A1 : h → h, A ′ 1 : h → h∗ and A2 : h ∗ → h ∗ such that
A(x + f ) = A1(x) + A ′ 1(x) + A2( f ), ∀ x ∈ h, f ∈ h ∗ .
It is obvious that A2 is an isomorphism of h ∗ . We show that A1 is also an isomorphism
of h. Indeed, if there exists x0 ∈ h such that A1(x0) = 0 then 0 = [A(x0),g ′ ] ′ =
A([x0,A −1 (g ′ )]) = A([x0,g]). It means that [x0,g] = 0. Since h ∗ = Z(g) then x0 = 0.
Therefore A1 is an isomorphism of h.
For all x,y ∈ h, f ,g ∈ h ∗ , one has
A([x + f ,y + g]) = A(θ1(x,y)) = A2(θ1(x,y)).
and [A(x + f ),A(y + g)] ′ = θ2(A1(x),A1(y)).
Therefore, A2(θ1(x,y)) = θ2(A1(x),A1(y)) for all x,y ∈ h.
Conversely, if there exist an isomorphism A1 of h and an isomorphism A2 of h ∗ such that
A2(θ1(x,y)) = θ2(A1(x),A1(y)) for all x,y ∈ h, we define A : g → g ′ by A(x+ f ) = A1(x)+
A2( f ) for all x + f ∈ g. Then it is easy to check that A is a Lie algebra isomorphism.
(2) Asumme A : g → g ′ is an i-isomorphism then there exist A1 and A2 defined as in (1). Let
x ∈ h, f ∈ h ∗ , one has
B ′ (A(x),A( f )) = B(x, f ) ⇒ A2( f )(A1(x)) = f (x).
Therefore, A2( f ) = f ◦ A −1
1 , for all f ∈ h∗ .
On the other hand, since A2(θ1(x,y)) = θ2(A1(x),A1(y)) we obtain
θ1(x,y) = θ2(A1(x),A1(y)) ◦ A1, ∀ x,y ∈ h.
Conversely, define A(x + f ) = A1(x) + f ◦ A −1
1 , for all x ∈ h, f ∈ h∗ then A is an iisomorphism.
Example 2.4.17. We keep the notations as above. Let g ′ be the T ∗ -extension of h by θ ′ = λθ
where λ = 0. Then g and g ′ are i-isomorphic by A : g → g ′ defined by A(x + f ) = 1
3√ λ x + 3√ λ f ,
for all x + f ∈ g.
For a non-degenerate cyclic skew-symmetric bilinear map θ of h, define the 3-form
I(x,y,z) = θ(x,y)z, ∀ x,y,z ∈ h.
Then I ∈ A 3 (h). The non-degenerate condition of θ is equivalent to ιx(I) = 0 for all x ∈ h\{0}.
Conversely, let h be a vector space and I ∈ A 3 (h) such that ιx(I) = 0 for every non-zero vector
x ∈ h. Define θ : h × h → h∗ by θ(x,y) = I(x,y,.), for all x,y ∈ h then θ is skew-symmetric
and non-degenerate. Moreover, since I is alternating one has θ is cyclic and then we obtain a
reduced 2SNQ-Lie algebra T ∗
θ (h) defined by θ. Therefore, there is a one-to-one map from the
set of all T ∗-extension of h onto the subset {I ∈ A 3 (h) | ιx(I) = 0, ∀ x ∈ h \ {0}}. We has a
corollary of the above theorem as follows:
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