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Question about SO(N) group generators

by Einj
Tags: generators
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Einj
#1
Jan20-13, 05:19 AM
P: 324
Hi all. I have a question about the properties of the generators of the SO(N) group.
What kind of commutation relation they satisfy? Is it true that the generators λ are such that:

$$\lambda^T=-\lambda$$ ??

Thank you very much
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homeomorphic
#2
Jan21-13, 10:23 PM
P: 1,273
The commutators are complicated, in general--or too complicated for me.

Yes, the Lie algebra of SO(n) is the skew-symmetric matrices, which is the condition you wrote. That comes from differentiating a path of orthogonal matrices at the identity, or rather differentiating the equation that defines an orthogonal matrix.
Tenshou
#3
Jan21-13, 10:53 PM
P: 150
Notice, that the n-dimensionality of SO(n) are triangle numbers in ℝn hopefully this can help you figure out a reason why, also I set a link to a video I think that might be able to help.

Link:
http://www.youtube.com/watch?v=-W6JWck4__Y

Edit: Also may I ask why do you need to know this thing about the lie commutators in SO(n)?

Einj
#4
Jan22-13, 01:44 AM
P: 324
Question about SO(N) group generators

Quote Quote by homeomorphic View Post
Yes, the Lie algebra of SO(n) is the skew-symmetric matrices, which is the condition you wrote. That comes from differentiating a path of orthogonal matrices at the identity, or rather differentiating the equation that defines an orthogonal matrix.
Thank you very much! That solves some problems!

Quote Quote by Tenshou View Post
Edit: Also may I ask why do you need to know this thing about the lie commutators in SO(n)?
I am working on the SO(N) symmetry of a [itex]\lambda \phi^4[/itex] theory in QFT and I need the exact expression of the commutator of two conserved charges, so I need to know the commutator of the generators.


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