# Question about SO(N) group generators

• Einj
In summary, the conversation discusses the properties of the generators of the SO(N) group and the commutation relations they satisfy. It is mentioned that the Lie algebra of SO(n) is the skew-symmetric matrices and that this comes from differentiating a path of orthogonal matrices at the identity. The n-dimensionality of SO(n) is also referenced. The conversation ends with the question of why the knowledge of the Lie commutators in SO(n) is needed, to which the response is related to a specific theory in quantum field theory.
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

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.

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.

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

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homeomorphic said:
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!

Tenshou said:
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.

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.

## 1. What is the SO(N) group?

The SO(N) group, also known as the special orthogonal group, is a mathematical group that consists of all the orthogonal matrices of size N x N with determinant 1. It is a Lie group, which means it is a continuous group that can be described by a finite number of parameters.

## 2. What are the generators of the SO(N) group?

The generators of the SO(N) group are the set of N(N-1)/2 antisymmetric matrices, which can be represented as skew-symmetric matrices with a single non-zero element equal to 1 or -1. These matrices are also known as the basis elements of the Lie algebra of the SO(N) group.

## 3. How are the generators of the SO(N) group related to rotations?

The generators of the SO(N) group are related to rotations through the exponential map. This means that any element of the SO(N) group can be expressed as an exponential of a linear combination of the generators. In other words, the generators represent the infinitesimal rotations of the group.

## 4. Can the generators of the SO(N) group be diagonalized?

Yes, the generators of the SO(N) group can be diagonalized, as they are real, antisymmetric matrices. This means that they can be written as a product of a diagonal matrix and an orthogonal matrix. The diagonal elements of this diagonal matrix are known as the eigenvalues of the generators.

## 5. What is the significance of the SO(N) group in physics?

The SO(N) group plays a crucial role in physics, particularly in the study of rotations and angular momentum. It is used to describe symmetries in physical systems, such as in quantum mechanics and classical mechanics. It also has applications in fields such as crystallography, where symmetries play an important role.

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