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Question on Lie groups

  1. Mar 16, 2007 #1
    Does anybody know the answer of the following problem?

    Show that the Lie group of Euclidean motions of R^3 has a Lie algebra g which is perfect i.e., Dg=g but g is not semisimple.
    By Dg I mean the commutator [g,g] and a semisimple lie algebra is one has no nonzero solvable ideals.

    Regards
     
  2. jcsd
  3. Mar 16, 2007 #2

    matt grime

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    Is the following helpful?
    1. Write down the lie algebra
    2. show that Dg=g
    3. find a solvable non-zero ideal (there will be an obvious candidate - this must be the easiest part)
     
  4. Mar 16, 2007 #3
    does the lie group of Euclidean motions of R^3 means O(3) i.e. rotations?
     
  5. Mar 16, 2007 #4

    matt grime

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    No. I don't think so. I think it means distance preserving maps (not just linear maps) - so rotations and translations, and reflections. (SO(3) is the group of rotations, not O(3)).
     
  6. Mar 16, 2007 #5
    you're right. So what is the Lie algebra of transformation preserving distance?
    Since I'm learning Lie theory by myself, I have lots of questions even in definitions!!
     
  7. Mar 16, 2007 #6

    matt grime

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    Write down the lie group, then work out its lie algebra. I'll give you the lie group.

    G:= O(3)\sdp R^3, that is the semi direct product. It is the set of pairs (X,V) with X in O(3) and v in R^3, and the composition:

    (X,v)(Y,w)=(XY,Xw+ v).

    You have been told how to find lie algebras (or someone else has in another thread currently active in this forum).
     
  8. Mar 16, 2007 #7
    Ok!

    many thanks
     
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