Description

This work gives a detailed structure theory for semisimple Lie algebras over arbitrary fields of characteristic 0. Starting from the well-known classification over algebraically closed fields via root systems, we mimic the language of reductive groups, so that part of our work can be seen as an introduction to a simpler version of Borel-Tits theory ; but we also express our results in the language of classical (matrix) algebra as well as Galois cohomology. In the second part of the work, we focus on p-adic fields and achieve a complete classification of semisimple Lie algebras over them. This classification consists essentially of a list of so-called Satake-Tits diagrams, which extend the Dynkin diagrams from the split case. Several instructive examples and historical notes supplement the text. The work can be used as the basis for a lecture on semisimple Lie algebras beyond the beginner's level, and as a reference for researchers.