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

2.4. 2-step nilpotent quadratic Lie algebras
2.4.2 2-step nilpotent quadratic Lie algebras
Let (g,B) be a quadratic Lie algebra and g = h ⊕ g ⊕ h ∗ be the double extension of g by
h by means of π as in Definition 2.1.9. If g is a 2SN-Lie algebra then g and h should be also
2-step nilpotent.
Proposition 2.4.10. Let (g,B) be a 2-step nilpotent quadratic Lie algebra (or 2SNQ-Lie algebra
for short), h be another 2SN-Lie algebra and π : h → Dera(g) be a representation of h by
means of skew-symmetric derivations of g. Then the double extension of g by h by means of π
is 2-step nilpotent if and only if π is a 2SN-admissible representation of h in g.
Proof. We can prove directly by checking the conditions of Definition 2.4.1 for the Lie algebra
g. However, it is easy to see that g is the semi-direct product of h by g⊕h ∗ by means of π ⊕ad ∗
where g⊕h ∗ is the central extension of g by h ∗ by means of φ. Therefore, the result follows.
Combined with Proposition 2.4.7 and Definition 2.4.8 we obtain the following result:
Corollary 2.4.11. Let (g,B) be a 2SNQ-Lie algebra and D ∈ Dera(g) be a skew-symmetric
derivation of g. Then the double extension of g by means of D is a 2SNQ-Lie algebra if and only
if D 2 = 0 and [D(x),y] = 0, for all x,y ∈ g.
Proposition 2.4.12. Let (g,B) be a 2SNQ-Lie algebra of dimension n + 2, n ≥ 0. Then g is
the double extension of a 2SNQ-Lie algebra of dimension n. Consequently, every 2SNQ-Lie
algebra can be obtained from an Abelian algebra by a sequence of double extensions by onedimensional
algebra.
Proof. If g is Abelian then g is the double extension of an Abelian algebra by means of the
zero map. If g is non-Abelian. By Corollary 2.1.6, there exists a central element x such that
x is isotropic. Then there is an isotropic element y such that B(x,y) = 1 and g is the double
extension of (h = (Cx ⊕ Cy) ⊥ ,B ′ ) where B ′ = B|h×h. Certainly, h is still 2-step nilpotent.
Proposition 2.4.13. Let g be a Lie algebra, θ be a cyclic 2-cocycle of g with value in g∗ and
T ∗
θ (g) be the T ∗-extension of g by means of θ (see Definition 2.1.12). Then T ∗
θ (g) is a 2SNQ-Lie
algebra if and only if g is 2-step nilpotent and θ satisfies
Proof. For all x,y,z ∈ g, f ,g,h ∈ g ∗ one has
θ(x,y) ◦ adg(z) + θ([x,y]g,z) = 0, ∀ x,y,z ∈ g.
[[x + f ,y + g],z + h] = [[x,y],z]g + ( f ◦ adg(y) − g ◦ adg(x) + θ(x,y)) ◦ adg(z)
−h ◦ adg([x,y]g) + θ([x,y]g,z).
Therefore, T ∗
θ (g) is 2-step nilpotent if and only if g is 2-step nilpotent and θ satisfies
θ(x,y) ◦ adg(z) + θ([x,y],z) = 0, ∀ x,y,z ∈ g.
By the above proposition, if T ∗
θ (g) is a 2SNQ-Lie algebra then g should be 2-step nilpotent.
However, we can only consider T ∗
θ -extensions of an Abelian algebra by the following
proposition.
56