# Talk:Lie group

WikiProject Mathematics (Rated B-class, High-importance)
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
 B Class
 High Importance
Field: Geometry

## 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" 130.246.132.26 (talk) 12:53, 23 October 2009 (UTC)anonymouscontributor

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

${\displaystyle \varphi (\mathrm {exp} (\tau X)p)=\sum _{k=0}^{\infty }{\frac {\tau ^{k}}{k!}}X^{k}\varphi (p).}$

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,

${\displaystyle \mathrm {exp} (\tau X)}$

is, for each τR, a bi-analytic bijection (the analytic counterpart of diffeomorphism) from the analytic manifold (the Lie group) to itself. Thus

${\displaystyle \mathrm {exp} (\tau X)p}$

is a new point on the manifold. When τ varies (p fixed), an integral curve of X is obtained. With p = the identity,

${\displaystyle \mathrm {exp} (\tau X)}$

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)

## Definition?!

Where is the definition? There is no definition in the definition section! — Preceding unsigned comment added by 78.128.194.120 (talk) 02:47, 27 February 2016 (UTC)

The first sentence of the definition section reads thus:
"A real Lie group is a group that is also a finite-dimensional real smooth manifold, in which the group operations of multiplication and inversion are smooth maps."
-- 13:39, 27 February 2016 (UTC)

Hello fellow Wikipedians,

I have just modified one external link on Lie group. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

You may set the |checked=, on this template, to true or failed to let other editors know you reviewed the change. If you find any errors, please use the tools below to fix them or call an editor by setting |needhelp= to your help request.

• If you have discovered URLs which were erroneously considered dead by the bot, you can report them with this tool.
• If you found an error with any archives or the URLs themselves, you can fix them with this tool.

If you are unable to use these tools, you may set |needhelp=<your help request> on this template to request help from an experienced user. Please include details about your problem, to help other editors.

Cheers.—InternetArchiveBot 11:56, 15 May 2017 (UTC)