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Exponential map of a lie group

  1. Jul 10, 2009 #1
    I am trying to read through this paper on the standard model. The ideas seem straightforward enough, but as always, I'm tripping over the "physicist's math" it uses. I was wondering if I can get some clarification or general guidance.

    http://aps.arxiv.org/PS_cache/arxiv/pdf/0810/0810.3328v1.pdf

    I understand everything in chapter 2 up and through section 2.2.1. Groups are groups. Lie groups are smooth manifold groups. A representation of a group is a group isomorphism between it and a matrix group. Important groups include GR, O, U, SO, SU, blah blah blah, all easy stuff.

    Now I'm onto generators. The motivation seems clear enough. Since rotations are continuous things, we need to be able to talk about some sort of derivatives. If an object's state is a rotation-valued function of time, for instance, we can take the time derivative to create an mathematical object to represent its angular velocity. Informally then, a generator is an infinitesimal rotation of an object over an infinitesimal amount of time.

    I hate infinitesimals. I'm fairly solid on my understanding of smooth manifolds, so I am trying to pose the problem in terms of tangent spaces. I have most of the argument used in section 2.2.2 mapped to my understanding of tangent spaces, but the argument breaks down at a certain point where the exponential map is (anonymously) introduced.

    Here is as much as I've been able to reason out myself.

    The article talks about [tex]\alpha_i[/tex] as being the parameters for the group with [tex]g(\alpha_i)[/tex] being the corresponding element of the group. The way I see it is that g : R^n -> G is a chart on the manifold G. [tex]D_n : G -> R^{nxn}[/tex]is the representation. The point [tex]\delta\alpha[/tex], which is infinitesimally close to the group identity, is actually a tangent vector at the identity.

    The partial derivatives [tex]-i \frac{\partial D_n(g(\alpha_i))}{\partial \alpha_i} = X_i[/tex] are called the generators of the group. They are very similar to a transition map, acting on the tangent vector in the coordinate space and mapping it to a tangent vector in the representation manifold.

    The problem with this interpretation is there is no analogy to the formula [tex]\lim_{N \to \infty} (1 + i\delta\alpha_iX_i)^N[/tex], which leads up to the definition of the exponential map. The meaning of that formula in the paper itself is confusing on its own. The author seems to have implicitly mapped the identity matrix ([tex]\mathbb{I}[/tex]) in the previous equation to 1, the multiplicative identity in R. The meaning seems to be that the tangent vector is anchored at the identity of [tex]R^{nxn}[/tex], but in my formulation, this is implicit, and there is no analogous way of coming up with the limit formula for the exponential function.

    If it's too much of a bother to follow along with my reasoning, I guess I can ask the question more simply as this.

    How does the exponential map come about for a manifold?

    On Wikipedia, the article talks about Lie algebras. However, those are discussed in the following chapter of this paper, so there must be an explanation of exponential maps which do not depend on Lie algebras.
     
  2. jcsd
  3. Jul 10, 2009 #2

    George Jones

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    The exponential map gives a diffeomorphism between a neighbourhood of the identity of a Lie group and a neighbourhood of zero in the tangent space (Lie algebra) at the identity. I think the isomorphism between the space of left-invariant vector fields and the tangent space of the identity, and flows of these vector fields are used to establish this diffeomorphism, but I only have one relevant book home with me, and I only have had time to take a brief glance at things.
     
  4. Jul 10, 2009 #3
    So, tell me if I'm off base here. You start at the origin and travel in the direction of some tangent vector for one unit of time. The exponential map is the thing that tells you what point of the manifold you ended up at? (If that's correct), can the exponential map also be used when starting from an arbitrary point in the manifold? And again, if my understanding is correct, on simple R^n space, wouldn't the "exponential map" simply be the function f(x) = x?

    What does "left-invariant" mean in this context? I believe I have come across it before. Is it related to the idea of a connection on a manifold?
     
  5. Jul 10, 2009 #4
    After thinking about this for a while, I had a thought.

    The exponential map is a device which allows us to pretend we have a run-of-the-mill basis for a curvy space. The set of generators forms the basis. Then any point in the manifold can be written as some sort of linear combination of the generators, under the exponential map. So that for any g in G, [tex]g = \Sigma_i e^{a_i X_i}[/tex] for some coefficients [tex]a_i[/tex]. Or something close to that. I'm working at an intuitive level here, but it seems like there should be something close to this. I know that commutativity plays a role in here too, but I'm not sure how.
     
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