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

2.4. 2-step nilpotent quadratic Lie algebras
Proposition 2.4.14. Let (g,B) be a reduced quadratic Lie algebra. Then g is 2-step nilpotent
if and only if it is i-isomorphic to a T ∗
θ -extension of an Abelian algebra by means of a nondegenerate
cyclic 2-cocycle θ.
Proof. Assume that g is 2-nilpotent then [g,g] ⊂ Z(g). Since g is reduced one has [g,g] = Z(g)
and dim(g) even. By Proposition 2.1.13 (given in [Bor97]) and Z(g) a totally isotropic ideal,
we write g = V ⊕ Z(g) with V totally isotropic. We can identify V with the quotient algebra
h g/Z(g) and Z(g) with h∗ . Then g is i-isomorphic to the T ∗
θ -extension of h by θ defined by
θ(p0(x), p0(y)) = φ(p1([x,y])),
where p0, p1 are respectively the projections from g into V and Z(g). Certainly, h is Abelian
since [g,g] = Z(g). We write g = h ⊕ h ∗ and the bracket on g becomes
[x + f ,y + g] = θ(x,y), ∀ x,y ∈ h, f ,g ∈ h ∗ .
Since Z(g) = h∗ then θ is non-degenerate on h × h.
Conversely, if g is i-isomorphic to the T ∗
θ -extension T ∗
θ (h) of Abelian algebra h by means
of a non-degenerate cyclic 2-cocycle θ, it is obvious that g is 2-step nilpotent and Z(g)
Z(T ∗
θ (h)) = h∗ . Since h∗ is totally isotropic then Z(g) is also totally isotropic. Therefore g is
reduced.
Consequently, we have a restricted definition for the reduced 2-step nilpotent case as follows:
Definition 2.4.15. Let h be a complex vector space and θ : h × h → h ∗ be a non-degenerate
cyclic skew-symmetric bilinear map. Let g = h ⊕ h ∗ be the vector space equipped with the
bracket
[x + f ,y + g] = θ(x,y)
and the bilinear form
B(x + f ,y + g) = f (y) + g(x),
for all x,y ∈ h, f ,g ∈ h ∗ . Then (g,B) is a 2SNQ-Lie algebra. We say that g is the T ∗ -extension
of h by θ.
Theorem 2.4.16. Let g and g ′ be T ∗ -extensions of h by θ1 and θ2 respectively. Then:
(1) there exists a Lie algebra isomorphism between g and g ′ if and only 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)), ∀ x,y ∈ h.
(2) there exists an i-isomorphism between g and g ′ if and only if there exists an isomorphism
A1 of h such that
θ1(x,y) = θ2(A1(x),A1(y)) ◦ A1, ∀ x,y ∈ h.
Proof.
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