SO(N) adjoint rep. under SO(3) subgroup

In summary, the SO(N) adjoint representation is a mathematical concept used to describe the transformation properties of a group element under the special orthogonal group SO(N). It is closely related to the Lie algebra of the group and has significant applications in physics, particularly in the study of symmetries and properties of physical systems. The SO(3) subgroup is of particular importance as it represents the rotation group in three-dimensional space and the matrix representations of the adjoint representation under this subgroup can be used to construct representations of the group.
  • #1
mkgsec
5
0
Hi.

I'm having trouble figuring out how SO(N) adjoint rep. transforms

under a SO(3) subgroup.

Unlike SU(N), SO(N) fundamental N gives

\begin{equation} N \otimes N = 1 \oplus A \oplus S \end{equation}

So the \begin{equation} S \end{equation} part really bothers.

Can you give a help?
 
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  • #2
What about using indexes for the vector spaces associated to the tensor products? Sometimes it helps to visualize the situation, specially now that you want to separate the symmetric and antisymmetric parts.
 

1. What is the SO(N) adjoint representation?

The SO(N) adjoint representation is a mathematical concept used to describe the transformation properties of a group element under the special orthogonal group SO(N). It is a fundamental concept in the study of Lie algebras and plays an important role in various areas of physics and mathematics.

2. What is the significance of the SO(3) subgroup in the SO(N) adjoint representation?

The SO(3) subgroup is significant because it represents the rotation group in three-dimensional space. The SO(N) adjoint representation under this subgroup describes the transformation properties of a group element under rotations in three dimensions, making it relevant in many physical systems such as rigid body rotations and quantum mechanics.

3. How is the SO(N) adjoint representation related to the Lie algebra of SO(N)?

The adjoint representation of SO(N) is closely related to the Lie algebra of the group. In fact, the adjoint representation can be seen as a linear map between the Lie algebra and the group itself, providing a way to study the structure and properties of the group through its algebraic properties.

4. What are the matrix representations of the SO(N) adjoint representation under the SO(3) subgroup?

The matrix representations of the SO(N) adjoint representation under the SO(3) subgroup are given by the generators of the Lie algebra of SO(N). These generators are anti-symmetric matrices for N > 3, and for N = 2, the generators are given by the Pauli matrices. These matrices satisfy the commutation relations of the Lie algebra and can be used to construct representations of the group.

5. How is the SO(N) adjoint representation useful in physics?

The SO(N) adjoint representation is useful in physics because it provides a powerful tool for understanding the symmetries and properties of physical systems. It is particularly useful in quantum mechanics, where the representation plays a crucial role in the study of spin and angular momentum. It is also used in classical mechanics to analyze the motion of rigid bodies and other systems with rotational symmetry.

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