Prove Perfect Lie Algebra of R^3 Euclidean Motions Isn't Semisimple

[arz2000]

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

 

[matt grime]

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)

 

[arz2000]

does the lie group of Euclidean motions of R^3 means O(3) i.e. rotations?

 

[matt grime]

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)).

 

[arz2000]

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!

 

[matt grime]

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).

 

[arz2000]

Ok!

many thanks