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

Introduction
In this thesis, we study Lie algebras, Lie superalgebras, Jordan algebras and Novikov algebras
equipped with a non-degenerate associative bilinear form. Such algebras are considered
over the field of complex numbers and finite-dimensional. We add the condition that the bilinear
form is symmetric or even, supersymmetric in the graded case. We call them respectively
quadratic Lie algebras, quadratic Lie superalgebras, pseudo-Euclidean Jordan algebras
and symmetric Novikov algebras.
Let g be a finite-dimensional algebra over C and (X,Y ) ↦→ XY be its product. A bilinear
form B : g × g → C is called associative (or invariant) if it satisfies:
B(XY,Z) = B(X,Y Z)
for all X,Y,Z in g and non-degenerate if B(X,g) = 0 implies X = 0. Such a bilinear form has
arisen in several areas of Mathematics and Physics. It can be seen as a generalization of the
Killing form on a semisimple Lie algebra, the inner product of an Euclidean Jordan algebra or
simply, as the Frobenius form of a Frobenious algebra. The associativity of a bilinear form also
can be found in the conditions of an admissible trace function defined on a power-associative
algebra. For details, the reader can refer to a paper by M. Bordemann [Bor97].
We begin with a quadratic Lie algebra g and its product, the bracket [ , ]. A result in the
work of G. Pinczon and R. Ushirobira [PU07] leads to our first problem: define the 3-form I on
g by I(X,Y,Z) = B([X,Y ],Z) for all X,Y,Z in g. Then I satisfies {I,I} = 0 where { , } is the
super-Poisson bracket defined on A (g), the Grassmann algebra of skew-symmetric multilinear
forms on g by:
{Ω,Ω ′ n
k+1
} = (−1) ∑ ιXj
j=1
(Ω) ∧ ιXj (Ω′ ), ∀ Ω ∈ A k (g),Ω ′ ∈ A (g)
in a fixed orthonormal basis {X1, ..., Xn} of g.
In this case, the element I is called the 3-form associated to g. Conversely, given a quadratic
vector space (g,B) and a non-zero 3-form I on g such that {I,I} = 0, then there is a non-Abelian
quadratic Lie algebra structure on g such that I is the 3-form associated to g. By a classical result
in a N. Bourbaki’s book [Bou58] that is also recalled in Proposition 2.2.3, we set the following
vector space:
VI = {α ∈ g ∗ | α ∧ I = 0}
The dup-number dup(g) of a non-Abelian quadratic Lie algebra g is defined by dup(g) =
dim(VI). It measures the decomposability of the 3-form I and its range is {0,1,3}. For instance,
I is decomposable if and only if dup(g) = 3 and then the corresponding quadratic Lie algebra
vi