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

Introduction
The thesis has been divided into four parts. The classifications of O(m)-adjoint orbits of
o(m) and Sp(2n)-adjoint orbits of sp(2n) are presented in Chapter 1. Singular quadratic Lie
algebras and 2-step nilpotent quadratic Lie algebras are studied in Chapter 2. We will prove the
equality {I,I} = 0 and introduce the class of singular quadratic Lie superalgebras in Chapter
3. However, since the classifying method is not new, we only focus on two cases: elementary
and 2-dimensional even part. The classification of singular quadratic Lie algebras and singular
quadratic Lie superalgebras having 2-dimensional even part can be regarded as an application
of the problem of orbits classification in Chapter 1. We present quasi-singular quadratic Lie
superalgebras without classification in Chapter 3. Such algebras can be found in [BBB] where
the generalized double extension notion is reduced into the one-dimensional extension of an
Abelian superalgebra. Pseudo-Euclidean Jordan algebras that are the one-dimensional double
extension of an Abelian algebra and 2-step nilpotent pseudo-Euclidean Jordan algebras are
given in Chapter 4 with a classifying characterization. The structure of symmetric Novikov
algebras is studied in the last section of Chapter 4 with a more detail than in [AB10].
There are four appendices containing rather obvious and lengthy results but yet useful for
our problems. Appendix A supplies a source about skew-symmetric maps for Chapter 1 and
Chapter 2. Appendix B gives a non trivial proof of a fact that every non-Abelian 5-dimensional
quadratic Lie algebra is singular. Another proof can be found in Appendix C where we classify
(up to isomorphisms) 3-forms on a vector space V with 1 ≤ dim(V ) ≤ 5. Appendix D is a small
result used in Chapter 4 for pseudo-Euclidean Jordan algebras.
xviii