Understanding Simple Lie Groups: Definition and Common Misconceptions

[Matterwave]

Hello,

I am reading Naive Lie Theory by John Stillwell, and he gives the definition of a simple Lie group as a Lie group which has no non-trivial normal subgroups.

Wikipedia, on the other hand, defines it as a Lie group which has no connected normal subgroups.

I was wondering, which definition is more common in the literature, and which definition did Lie use, etc? There seems to be quite a bit of difference between the two definitions since under the first definition, for example, SU(2) is not simple, [STRIKE]while under the second definition, it is (since it's non-trivial normal subgroups 1 and -1 aren't connected).[/STRIKE]

The first definition is certainly more restrictive, and would eliminate all SU(n) and Sp(n).

EDIT: Wait a second...SU(2) is a connected group so...even under the second definition, it's not simple right? ... So why do I see it labeled as simple all the time? I'm confused.

 

[morphism]

The Wikipedia definition is the more common (and more useful) one. Some authors make a point of saying "almost simple" to make sure there is no misunderstanding.

I don't think Lie was concerned with such notions as simplicity (but I could be wrong). It probably wasn't until the work of E. Cartan and Killing that simplicity was defined; I would guess it first popped up in the context of the Cartan-Killing classification of complex semisimple Lie algebras. Incidentally, the reason for the utility of the "almost simple" concept for Lie groups is that it's really the appropriate definition to make if you want to port the classification over from the Lie algebra to the Lie group side of things.

EDIT: Wait a second...SU(2) is a connected group so...even under the second definition, it's not simple right? ... So why do I see it labeled as simple all the time? I'm confused.

What does the connectedness of SU(2) have to do with its simplicity? You're looking for proper connected normal subgroups.

 

[Matterwave]

I thought that SU(2) was "simple" because I incorrectly thought that it's 2 normal subgroups weren't connected. But they are, so SU(2) is not simple by either definition of "simple".

I guess it's simply "almost simple"? And then classifying it as simple, like I've seen many people do, is just a slight sloppiness?

 

[morphism]

I thought that SU(2) was "simple" because I incorrectly thought that it's 2 normal subgroups weren't connected. But they are, so SU(2) is not simple by either definition of "simple".

I should have read your post more carefully. The wikipedia definiton you quoted is no good either: you want to exclude closed normal connected subgroups. This is because you don't want 'interesting' normal Lie subgroups. (Another technical provisio: you don't want your group G to be abelian.)

I guess it's simply "almost simple"? And then classifying it as simple, like I've seen many people do, is just a slight sloppiness?

In some sense, yes. But really it's more of an issue of overloading the word "simple". A simple Lie group isn't a Lie group that's simple as a group. It really ought to be a Lie group that has no 'interesting' Lie groups as quotients. That's one point of view. Another (but essentially equivalent) point of view is that a simple Lie group should be one whose Lie algebra is simple (=is nonabelian and has no nontrivial ideals). In some sense, this is really where the problem with excluding all normal subgroups comes from: the Lie algebra won't detect discrete normal Lie subgroups, in the sense that they correspond to the zero subalgebra, and so the simple Lie algebra <-> simple Lie group correspondence doesn't hold for the naive definition of a "simple Lie group".

 

[Matterwave]

ok, thanks for the info. =]