Lie groups and Lie alegbras -- some simple questions

From: Johan Kullstam (kullstj-nn_at_comcast.net)
Date: 09/22/04 

Date: Wed, 22 Sep 2004 20:27:05 GMT

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. 

-- 
Johan KULLSTAM

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