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.

 

[blue_raver22]

for your question. The generators of the SO(N) group, also known as the special orthogonal group, are a set of matrices that generate the group when exponentiated. These generators are defined as antisymmetric matrices, meaning that they satisfy the relation λ^T = -λ. This property is important because it guarantees that the group elements will have a determinant of 1, which is a necessary condition for any special orthogonal group.

In addition, the generators of the SO(N) group also satisfy the commutation relation [λ_i, λ_j] = iε_ijk λ_k, where ε_ijk is the Levi-Civita symbol. This means that the generators do not commute with each other, but rather form a Lie algebra, which is a fundamental concept in group theory.

I hope this helps clarify the properties of the generators of the SO(N) group. If you have any further questions, please feel free to ask. Thank you.