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

2.4. 2-step nilpotent quadratic Lie algebras
2.4 2-step nilpotent quadratic Lie algebras
Conveniently, we redefine a 2-step nilpotent Lie algebra in another way as follows:
Definition 2.4.1. An algebra g over C with a bilinear product g × g → g,(x,y) ↦→ [x,y] is called
a 2-step nilpotent Lie algebra if it satisfies [x,y] = −[y,x] and [[x,y],z] = 0 for all x,y,z ∈ g.
Sometimes, we use the notion 2SN-Lie algebra as an abbreviation.
According to this definition, a commutative Lie algebra is a trivial case of 2SN-Lie algebras.
2.4.1 Some extensions of 2-step nilpotent Lie algebras
Definition 2.4.2. Let g be a 2SN-Lie algebra, V be a vector space and ϕ : g × g → V be a
bilinear map. On the space g = g ⊕V we define the following product:
[x + u,y + v] = [x,y] + ϕ(x,y), ∀ x,y ∈ g,u,v ∈ V.
Then it is easy to see that g is a 2SN-Lie algebra if and only if ϕ is skew-symmetric and
ϕ([x,y],z) = 0, for all x,y,z ∈ g. In this case V is contained in the center Z(g) of g so the Lie
algebra g is called the 2SN-central extension of g by V by means of ϕ.
Proposition 2.4.3. Let g be a 2SN-Lie algebra then g is the 2SN-central extension of an Abelian
algebra h by some vector space V .
Proof. Denote by V = [g,g] and let h = g/[g,g]. Then h is Abelian. Set the map ϕ : h × h → V
by
ϕ(p(x), p(y)) = [x,y], ∀ x,y ∈ g,
where p : g → h is the canonical projection. This map is well defined since g is 2-step nilpotent.
So g is the 2SN-central extension of h by V by means of ϕ.
Let g be a 2SN-Lie algebra, V be a vector space and π : g → End(V ) be a linear map. On
the space g = g ⊕V we define the following product:
[x + u,y + v] = [x,y] + π(x)v − π(y)u, ∀ x,y ∈ g,u,v ∈ V.
Proposition 2.4.4. The vector space g is a 2SN-Lie algebra if and only if π satisfies the condition:
π([x,y]) = π(x)π(y) = 0, ∀ x,y ∈ g.
In this case, π is called a 2SN-representation of g in V .
Proof. For all x,y,z ∈ g, u,v,w ∈ V the condition [[x + u,y + v],z + w] = 0 is equivalent to
π([x,y])w−π(z)π(x)v+π(z)π(y)u = 0 for all u,v,w ∈ V . This happens if and only if π([x,y]) =
π(x)π(y) = 0, for all x,y ∈ g.
Remark 2.4.5. The adjoint representation and the coadjoint representation are 2SN-representations
of a 2SN-Lie algebra g. Therefore, the extensions of g by itself or its dual space with respect to
these representations are 2-step nilpotent.
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