Quadratic Lie algebras. Algorithms and (de)constructions stars

Résumé

In this dissertation we are going to study quadratic Lie algebras, with special interest in the ones which are 2-step nilpotent, and to give algorithmic procedures to build a wide range of examples. After an introduction and a overview of the known results in this matter, we start with a deconstruction process to reduce the study of general quadratic Lie algebras to the nilpotent ones. This is obtained undoing successive double extensions on quotients given from the location of some important ideals. The variety of nilpotent quadratic Lie algebras can be established from free nilpotent Lie algebras and their invariant bilinear forms. But this is a tough problem, so we focus ourselves in the 2-step case. We start by introducing a new method to obtain them using multilinear algebra. Later we prove this new method is equivalent to the two main classical techniques: double and T∗ -extensions. In combination with trivectors, we end up giving a classification of these algebras up to dimension 17. Once covered the 2-step nilpotent case, we start building larger and more general quadratic Lie algebras. This is achieved via double extensions using their skew-derivations, which can be described through the Universal Mapping Property of free nilpotent Lie algebras. After, we study the family of quadratic Lie algebras with only one maximal ideal: the local ones. These algebras have strong structural properties and include the well-known family of real oscillator algebras, which are the quadratic algebras attached to metric Lorentzian forms. The next part is devoted to the ideal structure of quadratic Lie algebras, specially those whose ideals form a chain by their inclusions. Finally, we introduce and explain how to use a computational package we have developed. This software is supported on the thesis results and includes many tools used along this work.