2
$\begingroup$

I've been struggling with a paragraph appearing in this book:

http://www.math.sunysb.edu/~kirillov/liegroups/liegroups.pdf

( Example 3.10 bottom half of page 31 ).

It says that "by theorem 3.7, elements of the form $ exp(tJ_{x}), exp(tJ_{y}), exp(tJ_{z}) $ generate SO(3). Where $ J_{x}, J_{y}, J_{z} $ are the usual basis vectors for the Lie Algebra so(3)

What I am struggling with is this:

First of all, I know that these elements do generate SO(3), this is sometimes called the Tait-Bryan angle parametrization of SO(3). What I don't understand, is how elements of the above form generate a neighbourhood of the identity, and how it follows from anything that the book has said.

I understand that he is trying to say, if they do generate a neighbourhood of the identity, then they will in turn generate the entire group, as SO(3) is connected. However, from the results of the book up to this section, I can only justify as much as saying every element in SO(3) is of the form $ exp(tJ) $ for some J in the Lie algebra (not necessarily one of the basis elements) This is because the exponential map for a compact and connected Lie group is surjective. This is result is also known as Euler's axis-angle parametrization of SO(3).

So, how is the book supposed to justify the statement:

"by Theorem 3.7, elements of the form $ exp(tJ_{x}), exp(tJ_{y}), exp(tJ_{z}) $ generate a neighbourhood of the identity"?

EDIT: Sorry, Theorem 3.7 is on the page before ( page 30 ). It gives properties of the exponential map ( it is a local diffeomorphism, commutes with lie group homomorphisms..et c )

I even tried to work out the BCH formula for the product of two exponentials, but still, it is not been clear to me.

Thank you

$\endgroup$
2
  • $\begingroup$ Could you tell us what theorem 3.7 is? $\endgroup$ Mar 10, 2013 at 19:53
  • $\begingroup$ I made an edit. It's on the page directly before ( on page 30 ). I just lists the well known basic properties of the exp map. Thanks $\endgroup$
    – Elliot
    Mar 10, 2013 at 19:56

1 Answer 1

2
$\begingroup$

I found this post here:

Is a basis for the Lie algebra of a Lie group also a set of infinitesimal generators for the Lie group?

This is actually an answer to my question. Whether or not it would follow from anything the book has already said I guess is open to interpretation.. Except Rob Sliversmith's proof does not contain any information except for basic facts about the exp map and generating an open neighbourhood of the identity. So I like the explanation.

Thanks!

$\endgroup$

Not the answer you're looking for? Browse other questions tagged .