# Real Lie groups versus real linear algebraic groups: differences in connexity and fundamental group

There are many introductory texts on real Lie groups, and many on linear algebraic groups in general, but fewer on the specific case of linear algebraic groups over the reals, and even fewer that try to adequately explain the exact relationship between real linear algebraic groups and real Lie groups. Worse, it's easy to get very confused because words like "connected" and "simply connected" mean different things for algebraic varieties or for the set of real points of such an algebraic variety. Even the notation is confusing (what is meant by $\mathit{PSL}_n$, for example, is always fairly mysterious).

My main question is where can I find a book or introductory/survey article that will build the bridge between real linear algebraic groups and real Lie groups and dispel confusion instead of adding to it.

Here's the sort of thing I'd like to see discussed. First, if $G$ is a semisimple linear algebraic group over $\mathbb{R}$ that is connected as an algebraic variety, it need not remain connected when we pass to the differentiable manifold defined by its real points: so $\mathit{SO}_{p,q}$ or $\mathit{PGL}_{2n,\mathbb{R}}$ is connected as an algebraic variety but has two real connected components (the real identity component, $\mathit{SO}^0_{p,q}$ or $\mathit{PSL}_{2n,\mathbb{R}}$, does not come from a real algebraic group). And the same problem occurs with $\pi_1$ as with $\pi_0$: if $G$ is simply connected as an algebraic variety, it need not remain simply connected when we pass to the real points: so $\mathit{SL}_{2n,\mathbb{R}}$ is simply connected as an algebraic variety but has a non-trivial fundamental group as a real manifold. I'm not entirely sure how one can compute the $\pi_0$ and $\pi_1$ of the real manifold of points of a linear algebraic group over $\mathbb{R}$ that is algebraically connected and simply connected (I know one can get some information from the maximal compact subgroup which can itself be related to its complexification, but I'm not sure I didn't miss some fine print and I'd like a clear exposition of this). I'm even more confused as to what happens without semisimplicity.

If no clear reference like I ask exists, is there at least a table somewhere containing the number of connected components and fundamental groups of the real points of the algebraically connected and simply connected simple real algebraic groups?

PS: I should mention that I'm aware of §14 in Borel and Tits's 1965 IHÉS paper "Groupes réductifs", which at least sheds some light on the matter (specifically corollary 14.5: the group of connected components of the real points of an algebraically connected reductive real algebraic group $G$ is a 2-group whose rank is at most the real rank of $G$).

• There can be some material in Borel-Serre's CMH paper and in Margulis' book. Concerning connectedness, if $G$ (linear alg. group) is semisimple simply connected then $G(\mathbf{R})$ is connected. Hence if $Z$ is central discrete in $G$ then the non-connectedness of $(G/Z)(\mathbf{R})$ can be thought as a lack of right-exactness of taking real points (with respect to the exact sequence $1\to Z\to G\to G/Z\to 1$). A useful keyword here can be "Whitehead group". – YCor May 22 '14 at 18:30
• @YvesCornulier: Where can I find that statement (that the real points of a semisimple s.c. group are connected) written down and proven? That's exactly the sort of things I was wondering about, and what I was getting confused upon (e.g., the indefinite Spin groups: it doesn't help that no two authors can agree on how they are defined, let alone how connected they are); so this really explains a lot. – Gro-Tsen May 22 '14 at 18:43
• I guess it's in Margulis' book. In the compact case it's due to the fact that compact Lie groups are linear and compact subgroups of real matrices are Zariski closed, yielding that real points of connected anisotropic reductive real groups are connected; in the purely isotropic case (no compact factor) the result is that a simply connected $G$ is such that $G(K)$ is generated by $U(K)$ for $U$ ranging over unipotents $K$-subgroups, for any $K$ field of char. zero. – YCor May 22 '14 at 19:33
• @Yves: I've added a reference to my answer, but I don't know if it's the most direct way to deal with the connectedness of real points in a simply connected complex algebraic group. In the background is the fact that complex simply connected semisimple Lie groups are the same as complex simply connected semisimple algebraic groups while "simply connected" means the same thing in both cases, which comes essentially from Chevalley's work. – Jim Humphreys May 22 '14 at 23:02
• May be looking up Onishchik-Vinberg's book "Lie groups and algebraic groups" might give some useful information. – Claudio Gorodski May 23 '14 at 0:34

In the book Lie groups and algebraic groups by Onishchik and Vinberg, Theorem 3 in Section 5.2.1 on page 240 says: Let $S$ be a real structure on a simply connected complex semisimple Lie group $G$. Then the real form $G^S$ is algebraic and connected.

This should mean that if $G$ is a simply connected semisimple algebraic $\mathbf{R}$-group, then the group of real points $G(\mathbf{R})$ is connected.

EDIT: In the book Algebraic Groups and Number Theory by Platonov and Rapinchuk, in Section 7.2 we find:

Proposition 7.6 (E. Cartan). Let $G$ be a simply connected simple algebraic group over $\mathbf R$. Then $G(\mathbf R)$ does not have any nontrivial noncentral normal subgroups. In particular, $G(\mathbf R)$ is connected, and if in addition $G$ is $\mathbf R$-isotropic, then $G(\mathbf R)^+=G(\mathbf R)$.

Here $G(\mathbf R)^+$ is the subgroup of $G(\mathbf R)$ generated by the unipotent elements (this subgroup is clearly normal).

Probably there is no single book which covers these things, especially because some of your questions involve the detailed classification of (real) semisimple Lie groups along with that of (complex) semisimple algebraic groups. The transition from "semisimple" to "reductive" in not entirely trivial here, but most of what you look for involves mainly the semisimple case.

On the other hand, Armand Borel made deep contributions to both Lie groups and linear algebraic groups, and planned to give lectures in Hong Kong (but was unable to). Leslie Saper prepared the lecture notes for publication in a proceedings volume: Lie groups and linear algebraic groups. I. Complex and real groups. Lie groups and automorphic forms, 1–49, AMS/IP Stud. Adv. Math., 37, Amer. Math. Soc., Providence, RI, 2006. These notes give a good overview of the whole subject along with selected examples.

Many previous questions on Math Overflow have also dealt with parts of your question, so it's worth searching for "linear algebraic group" or "fundamental group" (etc.) with the tag lie-groups.

Complete lists involving connectedness properties and such are probably not so easily found but might be worth compiling as a supplement to Helgason's book. In any case, the topology of a connected semisimple Lie group reduces (by the Iwasawa decomposition) to the topology of a maximal compact subgroup.

ADDED: Concerning the discussion in the comments to the original question, the result wanted goes back in some sense to the work of E. Cartan. But a more modern proof is given in a broader study of fundamental groups by Borel-Tits here: see 4.7-4.8.

In Algebraic Groups and Number Theory by Platonov and Rapinchuk, section 3.2 deals with algebraic groups over $\mathbb{R}$ and starts with some connectedness results. Besides, I would also have betted on the Onishchik-Vinberg book mentioned in the comments, although with a quick look I could not find something specifically about connectedness.