Hello. I have a question that has been on my mind for some time. I always see in mathematical physics books that they identify elements of the Lie algebra with group elements "sufficiently close" to the identity. I have never seen a real good proof of this so went on an gave a proof. Let X(adsbygoogle = window.adsbygoogle || []).push({}); _{i}be the generators of the Lie algebra and Y_{i}are their images in coordinates. Let β be the coordinate map around identity of e. (identifying that of R^{n}with its tangent space at 0) I have shown for analytic group operation mapping and for some small nbd around e,

[tY_{i},tY_{j}] = β[exp(tY_{i})exp(tY_{j})] - β[exp(tY_{j})exp(tY_{i})] + o(t^{3})

The last order of correction basically comes from the exponential coordinates and other higher corrections. In writing the exponential coordinates, I think the correct expansion is:

β(exp(tY_{i})) = tY_{i}+ o(t^{2})

Now that is where most other books say that for small enough t we have

β(exp(tY_{i})) = tY_{i}

In fact, Dubrovin, Novikov, Fomenko's book they directly say that in canonical coordinates of the first kind we have β(exp(tY_{i})) = tY_{i}. And by throwing out that last order of t^{2}, they show that if the lie algebra of a connected lie group is abelian then lie group is also abelian (which also follows from my estimate above if you disregard o(t^{2}), but can you?).

So my questions are

1- What happens to that order of correction to the expansion :) Why do people treat it like "lets take the limit zero and it is no more important". I could not find a way to gauge away the lesser t orders out of the equation by taking derivatives so I really don't understand the methodology here.

2- In many books, when they build this kind of correspondance between Lie groups and Lie algebras, they assume the group operation is analytic and carry out the proof with Taylor expansions. But then they say that actually analyticity is not required but is harder to give proof without that assumption. I also used that assumption to have a taylor series expansion. But that is of course bothersome because generally theory of differentiable manifolds is built assuming the manifold is not analytic so that one can use hat functions. So my second question is what is the idea of passage to non-analytic coordinates. For instance can one build a non-analytic charts for a Lie group such that at least in some nbd of identity coordinates are analytic? When we give up real analyticity of the manifold, do we lose anything other than Taylor expansions?

Thanks alot in advance :)

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Lie Groups and Canonical Coordinates

**Physics Forums | Science Articles, Homework Help, Discussion**