|WikiProject Mathematics||(Rated B-class, High-importance)|
is this useful?
page where beside a precise echo of my impression I also learn that "IF you dont [find it useful] why not go read something else?". OK thanks. will do. So much for a "comprehensive written compendium" 220.127.116.11 (talk) 12:53, 23 October 2009 (UTC)anonymouscontributor
p-adic lie groups
so what's that? anybody up to writing a quick explanation?
Smooth (differentiable) or analytic?
Most sources and this article defines Lie groups as smooth but not necessarily analytic manifolds. Rossmann, in Lie groups: An introduction through linear groups, defines them with the analytic structure. He argues that it is the natural way to define them, and it is clear that Lie groups allow an analytic structure. One thing is clear to me; the theory of flows of an analytic vector field becomes much more elegant in this setting. See page 152 and on in the referenced book. The reason that flows can be called exp (in this context) is extremely transparent. The power series of the usual exponential occurs as something that actually characterizes flows on Lie groups completely. I don't think this is possible on smooth manifolds with "merely" smooth vector fields and functions. YohanN7 (talk) 21:50, 11 October 2014 (UTC)
The formula, due to Sophus Lie himself (1888), I'm thinking about is
Here X is an analytic vector field, φ an analytic function, and exp is the flow of X, p is a point on the manifold and τ is the "time" of the flow. Its proof depends on φ having a Taylor series. In a smooth manifold setting, the exponential mapping is simply defined to be the time one flow of a (left invariant) smooth vector field without the strong motivational point of the power series for its name. YohanN7 (talk) 06:09, 12 October 2014 (UTC)
On the left hand side,
is, for each τ ∈ R, a bi-analytic bijection (the analytic counterpart of diffeomorphism) from the analytic manifold (the Lie group) to itself. Thus
is a new point on the manifold. When τ varies (p fixed), an integral curve of X is obtained. With p = the identity,
can be considered as a one-parameter subgroup as τ varies over all R (provided X is in the Lie algebra). All one-parameter subgroups of the Lie group are expressible as such for some X in the Lie algebra (the left-invariant vector fields ≃ tangent space at the identity). On the right hand side, X is to be thought of as a first order differential operator operating on φ. Every X in the Lie algebra can canonically be associated with such a differential operator on the whole Lie group. This extends the concept of a Taylor series globally and in a coordinate-independent way.
I think this sort of stuff should go into the present (convoluted) section Lie group#The exponential map (with a renaming of the section). I also think the Baker-Campbell-Hausdorff formula generalizes almost verbatim too in the analytic manifold setting. That is, begin with analytic Lie groups, consider vector fields, and the whole thing falls out naturally. YohanN7 (talk) 07:51, 12 October 2014 (UTC)
- Here is my position:
- (1) We should have a section regarding Hilbert's fifth problem-type fact, like any smooth group or group homomorphism is analytic (category of smooth group is the same as the category of analytic group). This is very important if it is not treated in depth in elementary textbooks.
- (2) I'm not sure if we need to have too much stuff on exponential map; after all, the separate article is the best place for in-depth discussion. Perhaps more discussion exponential coordinate may make sense. In my opinion, there is too much emphasis on examples and structure theory and not enough abstract theory.
- -- Taku (talk) 21:35, 29 November 2014 (UTC)
- (2) Agreed, exponential coordinates should definitely be here, as well as the associated topology, the group topology. In relation to this a discussion on the equivalence of this topology with the subspace topology for Lie subgroups, relegating the bulk to Closed subgroup theorem. Also agreed that the exponential map need not be overly present here. But Lie's formula of above must get in somewhere. It's pretty and explains the name of exp in all generality without any a priory reference to the usual matrix exponential. But, it works only under the assumption of analyticity.
- Actually, Lie's formula works for all analytic manifolds, not only Lie groups. YohanN7 (talk) 11:22, 1 December 2014 (UTC)
- Yes, there should be a section on "topology and differentiable structure". As for Lie's formula, the obvious thing to do seems to create Lie's formula. It's also known as Taylor series, but it probably makes sense to have a separate article (and have obvious links between the two). As you said, the formula itself has nothing to do with Lie groups. -- Taku (talk) 22:19, 1 December 2014 (UTC)
- The first sentence of the definition section reads thus:
Biały 13:39, 27 February 2016 (UTC)
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