Re: Lie groups and Lie alegbras -- some simple questions
From: Suresh Venkat (sureshv_at_gmail.com)
Date: 09/24/04
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Date: 24 Sep 2004 14:12:04 -0700
Johan Kullstam <kullstj-nn@comcast.net> wrote in message news:<87y8j2m4mf.fsf@sophia.axel.nom>...
> I am trying to learn about Lie groups/algebras in order to model rigid
> body rotations in a dynamic system. I have about a half dozen books
> but none seem to be at my (elementary) level and assume that a person
> has a pretty good knowlege of working with Lie groups and such. (As
> an example, I had to search the web to discover that the "Ad" means
> "ajoint" since none of my books would deign to explain such simple
> things).
>
> Anyhow, take SO(3) as the Lie group. Represent this group with 3x3
> orthogonal matrices. The Lie algebra so(3) is represented by 3x3
> anti-symmetric matrices.
>
> Ad[R] maps so(3) to so(3) and is linear
>
> Let R be in SO(3), let X be in so(3). Then
>
> Ad[R] X = R X R^{-1}
>
> In the representation, I can see how to multiply matrices. I am
> disturbed by the mixing of group elements with algebra elements. Do
> not these things belong to different classes? How does this work on
> the abstract group and algebra (as in without the representation as
> embedding space for the group)?
>
> Can the adjoint be broken apart? Does it make any sense to consider
> RX alone? What space would that live in?
>
> Any suggestions as to books on *introduction* to Lie groups and Lie
> algebra would be helpful.
If it's any consolation, I am in the same boat: trying to learn about
lie groups and differential geometry in order to do stuff with SO(3).
Chevalley's book is excellent; I am going thru that myself. However, I
have found that books from physics tend to be easier to deal with,
especially for someone like myself approaching this from a more
computer science (as opposed to math) background.
A small book by Bernard Schutz
(http://www.amazon.com/exec/obidos/tg/detail/-/0521298873/102-1095917-1426506?v=glance)
deals with some of the basics of manipulating manifolds in a way that
I found very accessible. it was the best way for me to understand the
idea of geodesics and covariant derivatives. Also, because SO(3) has
been studied a lot, there are many papers/texts that specialize in
SO(3) related work on lie groups that might be more efficient (even if
they don't give you the right grounding in the fundamentals).
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