# Classification of real semisimple lie algebras

We know that each complex semisimple lie algebra $$L$$ is a direct sum of a chosen Cartan subalgebra $$H$$ and finitely many weight spaces, each of which is associated with an element in $$H^*=\operatorname{Hom}(H,\mathbb{C})$$, also known as a root. The set of roots of $$L$$ forms a root system which can be classified by Dynkin diagrams.

According to wikipedia, one classifies simple Lie algebras over the algebraic closure, then for each of these, one classifies simple Lie algebras over the original field which have this form (over the closure). For example, to classify simple real Lie algebras, one classifies real Lie algebras with a given complexification, which are known as real forms of the complex Lie algebra"

Can someone point me to a good source of reference for classification over the reals?

A good reference are the course notes Lie algebras by Alberto Elduque. Pages $$89-104$$ gives the classification of simple real Lie algebras in detail.

• This link is now broken, unfortunately. – the_lar Oct 27 '20 at 3:09
• @the_lar I fixed the link, thank you. Alberto has a new homepage, and he is still active and searchable. – Dietrich Burde Oct 27 '20 at 10:12
• Thank you for addressing this! It is very helpful. – the_lar Oct 27 '20 at 13:07

Here's a diploma thesis on the topic: https://www.mat.univie.ac.at/~cap/files/wisser.pdf

You should not expect much literature which focuses exclusively on Lie algebras, as the classification of semisimple algebraic / Lie groups is naturally connected to this.

In my thesis, I did almost exclusively deal with Lie algebras. Although in later chapters I focus on $$p$$-adic fields rather than $$\Bbb R$$, I think chapter 3 gives a good introduction to Satake-Tits diagrams which you will find used in all sources.

Satake's original monograph on the issue is https://books.google.ca/books/about/Classification_theory_of_semi_simple_alg.html?id=HQbvAAAAMAAJ&redir_esc=y. It has an appendix by M. Sugiura which applies the Satake diagram machinery to $$\Bbb R$$. An earlier paper doing the same thing is

Araki, Shôrô. On root systems and an infinitesimal classification of irreducible symmetric spaces. J. Math. Osaka City Univ. 13 (1962), no. 1, 1--34. https://projecteuclid.org/euclid.ojm/1353055009

Besides, I have found the treatise by Onishchik and Vinberg quite helpful, even though (or because) it is written in a very shorthand style; it demands that you fill in many steps as exercises along the way. Chapter 5 covers the case of your interest: http://people.maths.ox.ac.uk/drutu/tcc2/onishchik-vinberg.pdf

Added: In a comment, user YCor mentions the book Differential Geometry, Lie Groups, and Symmetric Spaces, doi:10.1090/gsm/034 by S. Helgason, which I also recommend and unfortunately forgot to include earlier. I have also read good things about A. Knapp's Lie Groups Beyond an Introduction.