- #1

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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

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- Thread starter arz2000
- Start date

- #1

- 15

- 0

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

matt grime

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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)

- #3

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

- #4

matt grime

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- #5

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Since I'm learning Lie theory by myself, I have lots of questions even in definitions!!

- #6

matt grime

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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).

- #7

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Ok!

many thanks

many thanks

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