Lie Groups and Lie Algebras

Abstract

AbstractIn this chapter, we recall some well-known results on Lie groups and Lie algebras. In particular, we discuss the third Lie theorem, the Ado theorem, and the Cartan semisimplicity criterion. Some important types of Lie algebras and Lie groups together with their important ideals and normal subgroups are discussed. Much attention is paid to compact Lie groups and compact Lie algebras. We consider deep structure results on compact Lie algebras and Cartan subalgebras, discuss important properties of maximal tori in connected compact Lie groups, as well as their roots, root systems, and the Weyl groups. Moreover, we provide useful formulas for calculation of the Ricci curvature and the scalar curvature of Lie groups, supplied with left-invariant Riemannian metrics. Some important geometric properties of nilpotent and compact Lie groups with left-invariant Riemannian metrics are discussed.