Lie Groups and Representation theory?

[pivoxa15]

What is the connection between the two if any?

What kind of algebra would Lie groups be best labeled under?

 

[quasar_4]

I am just learning this, but from what I have heard (not sure this is 100% it), it's the following.

Lie Groups are infinite groups.

Representations are group homomorphisms from G-->GL(V).

We are usually interested in finding out if we can decompose a representation into irreducibles. If there is an invariant subspace for the representation, then it is not irreducible. If there is no invariant subspace, then it is irreducible. Careful: A representation that is not irreducible is NOT necessarily reducible! In order to be reducible, there must be an invariant subspace AND an invariant complement. Then we have a reducible representation.

Conveniently, for every FINITE dimensional group if there exists an invariant subspace, then there exists an invariant complement. We can then decompose the representation into irreducible bits, namely trivial, alternating and standard representations.

Now I don't know much about Lie groups, but it seems to be that they are of the infinite dimensional case, so we cannot necessarily always find an invariant complement to an invariant subspace. A semisimple representation can always be decomposed like the finite dimensional case, so you might want to look for Lie groups that will help you do this...

Then again, I'm just in a class on the basics of the prerequisite material to all this, so I don't know much. But our text gets to this whole topic, with the first part on finite groups and the latter part on the Lie groups. It's called "Representation Theory: A First Course", by Fulton and Harris. You might check it out, since it's bound to be much more accurate then I am...

 

[HallsofIvy]

More specifically, Lie Groups are topological groups. That is, they have a topology (that of a manifold) as well as group operations and the group operation, (a, b)->a*b is a continuous function from GxG to G. Also, the function G->G defined by a-> a^-1 is a continuous function.

There certainly is a connection between Lie Groups and Representation Theory! Historically, the first Lie Groups were "groups of transformations"- that is, they were exactly the kind of structures used to "represent" groups.

 

[quasar_4]

oh, is that why they are regarded as infinite groups? I suppose if we associate them with Manifolds then there are assumptions regarding continuity...

 

[HallsofIvy]

Yes, as I said above, "a, b)->a*b is a continuous function from GxG to G. Also, the function G->G defined by a-> a-1 is a continuous function."

 

[DeadWolfe]

Everywhere halls said continuous he should have said smooth. Also, we can study finite dimensional representations of infinite groups, but the infinite dimensional representations are more interesting. Also, it is the COMPACT groups whose reps are always completely irreducible, and one perhaps should think of finite groups as 0 dimensional compact lie groups.

 

[Chris Hillman]

The OP is in my Ignore List, but skimming the responses I noted two of you talking past one another due to a simple misunderstanding: HallsofIvy remarked (correctly) that Lie groups are examples of topological groups, and then he discussed some properties of topological groups. Deadwolfe complained (correctly) that the multiplication and inverse maps are actually not just continuous (required for topological grouphood) but also smooth (required for Lie grouphood). So there is no disagreement here!

quasar_4 thought that the definition requires a Lie group to be an infinite group, but in fact as Deadwolfe said, zero dimensional Lie groups include finite Lie groups since we can use an appropriate "discrete topology" (this is one of those places in mathematics where a solid course in "general topology" is helpful to students!)

quasar_4 mentioned representations and cited the very book I was about to recommend. I'd also recommend P. M. Cohn, Lie Groups, Cambridge University Press, 1957 (which has a fine concise introduction to topological groups), and Daniel Bump, Lie Groups, GTM 225, Springer, 2004, which is IMO the best all around modern introduction, and includes some good background for the finite-dimensional versus infinite-dimensional representation issues which Deadwolfe mentioned.