Lie algebras over non-algebraically closed fields

Question

I am independently studying Lie algebras (in preparation for grad school) from James Humphrey's text "Introduction to Lie Algebras and Representation Theory. Fairly early on in the development (Chapter 2), the assumption that the underlying field is algebraically closed sneaks in and remains in place all the way through to the classification of semi-simple Lie algebras in Chapter 3. I understand where we need the closure in Lie's theorem etc., but I couldn't help wondering

How far can you get in the development of Lie algebras without assuming algebraic closure of the underlying field?

Are there classifications, or partial results etc.? Where can I find them in a format suitable for a beginner? Thanks for your help!

Answer 1

Hmmm... I guess I should try to say something in self-defense, beyond the reference I made to Jacobson's 1962 book Lie Algebras (since republished in paperback by Dover). It's conventional in many algebraic classifications to start with a splitting field, then descend to smaller fields by something like Galois cohomology. In the case of semisimple Lie algebras, which emerged from Elie Cartan's work a century ago on Lie groups and differential geometry, most people simplify by working first over an algebraically closed field of characteristic 0 (often specified arbitrarily as $\mathbb{C}$). Classically the objective was to understand simple Lie algebras (and Lie groups) over $\mathbb{R}$, but this gets quite complicated even though the field extension involved is quadratic.

The notion of splitting field here is fairly straightforward: you need all eigenvalues of ad-operators to be in your field in order to get a clean structure and classification theory. This mostly comes down to looking at Cartan subalegras (or "maximal toral subalgebras") in semisimple Lie algebras. But it's easier just to think about a familiar large field. Here the classification is relatively elementary, thanks to Killing and Cartan and their successors. But working over an arbitrary ground field gets quite intricate. Here some motivation comes from the parallel but more difficult study of semisimple algebraic groups.

Books and lecture notes abound, but usually at a fairly high level, and some books like those in the defunct Dekker series are hard to find as well as being photocopied from typescript. Reviews I wrote for the AMS Bulletin long ago of a couple of these are free online from AMS at emath.org, written by Goto-Grosshans and by my graduate advisor George Seligman from a different viewpoint. I didn't mention in my various reference lists the standard old book by Helgason now republished by AMS: Differential Geometry, Lie Groups, and Symmetric Spaces. These and more recent books are pretty formidable, as are the much older research papers on structure and classification. It's all there, but are there readable and accessible modern references? Good luck.

Answer 2

Well, certainly things get more complicated when the field is not algebraically closed, as you can see from the classification of finite-dimensional simple Lie algebras over $\mathbb{R}$. But there are many cases where one just needs to be more careful with hypotheses.

1. In the proof of Lie's Theorem: for a solvable Lie algebra $\mathfrak{g}$ and a finite-dimensional representation $\pi : \mathfrak{g} \to \mathfrak{gl}(V)$, you can get the theorem as long as you assume that all of the eigenvalues of all of the endomorphisms $\pi(X)$ lie in the field you are working over. I don't know how often this happens in practice when the field is not algebraically closed, but the proof does go through.
2. Engel's theorem goes through without a problem over any field, I believe.
3. The Poincare-Birkhoff-Witt Theorem works over any field (and in fact, for Lie algebras over any commutative ring $R$ where the underlying $R$-module of the Lie algebra is free).
4. Cartan's Criterion for Semisimplicity works over any subfield of $\mathbb{C}$, essentially because $\mathfrak{g}$ is semisimple if and only if $\mathfrak{g} \otimes \mathbb{K}$ is semisimple for any extension field $\mathbb{K}$ (note that this is not true for simplicity; if $\mathfrak{g}$ is a simple real Lie algebra which is not obtained from a simple complex Lie algebra by restriction of scalars, then the complexification $\mathfrak{g} \otimes \mathbb{C}$ is not simple.)

Hmmm... I'm sure there's a lot more to say. Let's just wait until Professor Humphreys shows up.