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

Introduction
for all x,y ∈ h, f ,g ∈ h ∗ . Then (g,B) is a 2-step nilpotent quadratic Lie algebra and called the
T ∗ -extension of h by θ (or T ∗ -extension, simply).
The set of T ∗ -extensions is enough to represent all 2-step nilpotent quadratic Lie algebras
by a result in Proposition 2.4.14 that every reduced 2-step nilpotent quadratic Lie algebra is
i-isomorphic to a T ∗ -extension. Thus, we focus on the isomorphic classes and i-isomorphic
classes of T ∗ -extensions. We prove that (Theorem 2.4.16):
THEOREM 9:
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.
Now, if we define I(x,y,z) = θ(x,y)z, for all x,y,z ∈ h then the element I is the 3-form
associated to the T ∗ -extension of h by θ. Moreover, there is a one-to-one correspondence
between the set of T ∗ -extensions of h and the set of 3-forms {I ∈ A 3 (h) | ιx(I) = 0, ∀ x ∈
h \ {0}}. Remark that by Theorem 9 the i-isomorphic classification of T ∗ -extensions of h can
be reduced to the isomorphic classification of such 3-forms on h. As a consequence, we obtain
the same result as in [Ova07] and further that there exists only one reduced 2-step nilpotent
quadratic Lie algebra of dimension 10 (Appendix C and Remark 2.4.21).
In Chapter 3, we give a graded version of the main results in Chapter 2: singular quadratic
Lie superalgebras. We begin with a quadratic Z2-graded vector space V = V 0 ⊕V 1 with a nondegenerate
bilinear form B. Recall that the bilinear form B is symmetric on V 0, skew-symmetric
on V 1 and B(V 0,V 1) = 0.
Consider the super-exterior algebra of V ∗ defined by a Z × Z2-gradation
with the natural super-exterior product
E(V ) = A (V 0) ⊗
Z×Z2
S (V 1)
(Ω ⊗ F) ∧ (Ω ′ ⊗ F ′ f ω′
) = (−1) (Ω ∧ Ω ′ ) ⊗ FF ′ ,
for all Ω,Ω ′ in the algebra A (V 0) of alternating multilinear forms on V 0 and F,F ′ in the algebra
S (V 1) of symmetric multilinear forms on V 1. It is clear that this algebra is commutative and
associative. In [MPU09], I. A. Musson, G. Pinczon and R. Ushirobira presented the super
Z × Z2-Poisson bracket on E(V ) as follows:
{Ω ⊗ F,Ω ′ ⊗ F ′ } = (−1) f ω′ {Ω,Ω ′ } ⊗ FF ′ + (Ω ∧ Ω ′ ) ⊗ {F,F ′ } ,
for all Ω ∈ A (V0), Ω ′ ∈ A ω′ (V0), F ∈ S f (V1), F ′ ∈ S (V1). xii