Question about SO(N) group generators

[Einj]

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

 

[homeomorphic]

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]

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:


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

 

[Einj]

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!

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 $\lambda \phi^4$ 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.