Left-invariant Pseudo-Riemannian Metrics On Lie Groups And Some Related Algebraic Problems

Lie groups are beautiful, important, and useful because they play crucial roles in each of the two great divisions of mathematics-algebra and geometry. Their algebraic properties derive from the group axioms. Their geometric properties derive from the identification of group operations with points in a topological space.Since nineteen seventies, the systematical studies of Lie groups with left-invariant Riemannian metrics have begun. In nineteen nineties, Riemannian metrics were gen-eralized to pseudo-Riemannian cases. From then on, the studies of Lie groups with left-invariant pseudo-Riemannian metrics never stop.In this paper, we study the Lie groups with left-invariant pseudo-Riemannian met-rics and some related algebraic problems.In chapter 2, we follow the ideals of [1], and give parallel results on Lie groups with left-invariant pseudo-Riemannian metrics. Our main result is the uniqueness of the orthogonal strong decomposition of Lie groups with left-invariant pseudo-Riemannian metrics.In chapter 3, we consider pseudo-Riemannian Lie algebras. We show first that if g is a pseudo-Riemannian Lie algebra with C(g)≠0 then gg(?)g. On the other hand, we show that gg(?)g if g is a 4-dimensional pseudo-Riemannian Lie algebra. The proof is based on a result of [2] and the classification of solvable 4-dimensional Lie algebras [3]. Finally, the classification of 4-dimensional pseudo-Riemannian Lie algebras is deduced by a direct computation.In chapter 4, we study another kind of invariant metric structures, which are on the n-Lie algebras. We give a classification of metric n-Lie algebras of dimension n+2 and n+3. In some senses, the method we used here is similar to those used on chapter 3.