TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...
TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...
TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...
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2.4. 2-step nilpotent quadratic Lie algebras<br />
Proposition 2.4.14. Let (g,B) be a reduced quadratic Lie algebra. Then g is 2-step nilpotent<br />
if and only if it is i-isomorphic to a T ∗<br />
θ -extension of an Abelian algebra by means of a nondegenerate<br />
cyclic 2-cocycle θ.<br />
Proof. Assume that g is 2-nilpotent then [g,g] ⊂ Z(g). Since g is reduced one has [g,g] = Z(g)<br />
and dim(g) even. By Proposition 2.1.13 (given in [Bor97]) and Z(g) a totally isotropic ideal,<br />
we write g = V ⊕ Z(g) with V totally isotropic. We can identify V with the quotient algebra<br />
h g/Z(g) and Z(g) with h∗ . Then g is i-isomorphic to the T ∗<br />
θ -extension of h by θ defined by<br />
θ(p0(x), p0(y)) = φ(p1([x,y])),<br />
where p0, p1 are respectively the projections from g into V and Z(g). Certainly, h is Abelian<br />
since [g,g] = Z(g). We write g = h ⊕ h ∗ and the bracket on g becomes<br />
[x + f ,y + g] = θ(x,y), ∀ x,y ∈ h, f ,g ∈ h ∗ .<br />
Since Z(g) = h∗ then θ is non-degenerate on h × h.<br />
Conversely, if g is i-isomorphic to the T ∗<br />
θ -extension T ∗<br />
θ (h) of Abelian algebra h by means<br />
of a non-degenerate cyclic 2-cocycle θ, it is obvious that g is 2-step nilpotent and Z(g) <br />
Z(T ∗<br />
θ (h)) = h∗ . Since h∗ is totally isotropic then Z(g) is also totally isotropic. Therefore g is<br />
reduced.<br />
Consequently, we have a restricted definition for the reduced 2-step nilpotent case as follows:<br />
Definition 2.4.15. Let h be a complex vector space and θ : h × h → h ∗ be a non-degenerate<br />
cyclic skew-symmetric bilinear map. Let g = h ⊕ h ∗ be the vector space equipped with the<br />
bracket<br />
[x + f ,y + g] = θ(x,y)<br />
and the bilinear form<br />
B(x + f ,y + g) = f (y) + g(x),<br />
for all x,y ∈ h, f ,g ∈ h ∗ . Then (g,B) is a 2SNQ-Lie algebra. We say that g is the T ∗ -extension<br />
of h by θ.<br />
Theorem 2.4.16. Let g and g ′ be T ∗ -extensions of h by θ1 and θ2 respectively. Then:<br />
(1) there exists a Lie algebra isomorphism between g and g ′ if and only if there exist an<br />
isomorphism A1 of h and an isomorphism A2 of h ∗ such that<br />
A2(θ1(x,y)) = θ2(A1(x),A1(y)), ∀ x,y ∈ h.<br />
(2) there exists an i-isomorphism between g and g ′ if and only if there exists an isomorphism<br />
A1 of h such that<br />
θ1(x,y) = θ2(A1(x),A1(y)) ◦ A1, ∀ x,y ∈ h.<br />
Proof.<br />
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