Connection between A(N-1) and SU(N)

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In summary, the book explains that the groups SU(N) and A(N-1) are considered equivalent because they both have N^2-1 traceless generators. However, SU(N) is over the real numbers and has N^2-1 dimensions, while A(N-1) is a subgroup of GL(N,C) and has N^2-1 real parameters. By requiring the generators to be anti-hermitian, the number of parameters for A(N-1) is reduced to (N^2-1)/2, making the groups identical. This is the correct way to view this, and they also share the same root system and are isomorphic algebras. The appropriate group morphism is not specified
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gouranja
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According to this book I'm reading these groups are said to be equivalent and I am trying to better understand how this is so.

The generators of SU(N) are N^2-1 traceless complex anti-hermitian matrices. The group is over the real numbers and is N^2-1 dimensional.

On the other hand we have the group A(N-1), a subgroup of GL(N,C). The generators of A(N-1) are N^2-1 traceless real matrices but this over the complex numbers. If your require that the generating element be anti-hermitian this reduces the number of complex parameters from N^2-1 to (N^2-1)/2. I.e. we have N^2-1 real parameters and hence the groups are identical.

Is this the correct way to view this?
 
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Well I can see no response then I shall rephrase the question.
Is it true, and if so why is it true, that the groups A(N-1) and SU(N) have the same root system? i.e. Isomorphic algebras? What is the appropriate group morphism? etc.
 

1. What is the connection between A(N-1) and SU(N)?

The connection between A(N-1) and SU(N) lies in the mathematical theory of Lie groups and Lie algebras. A(N-1) and SU(N) are both special types of Lie groups, with A(N-1) being a subgroup of SU(N). This means that A(N-1) is a smaller group contained within SU(N), and the elements of A(N-1) are a subset of the elements of SU(N).

2. What is the significance of A(N-1) and SU(N) in physics?

A(N-1) and SU(N) have significant applications in theoretical physics, particularly in the study of quantum mechanics and particle physics. They are used to describe the symmetries of physical systems and play a crucial role in understanding the fundamental interactions between particles.

3. How are A(N-1) and SU(N) related to each other?

A(N-1) and SU(N) are related through a process called "embedding". This means that A(N-1) is embedded within SU(N) as a subgroup, and the elements of A(N-1) can be represented as a subset of the elements of SU(N). In other words, A(N-1) is a smaller group contained within SU(N).

4. Can you give an example of A(N-1) and SU(N) in action?

One example of A(N-1) and SU(N) in action is the Standard Model of particle physics. In this model, the symmetries of particles are described by the SU(3) and A(2) groups, where A(2) is a subgroup of SU(3). This allows for the description of the strong nuclear force (described by SU(3)) and the electroweak force (described by A(2)).

5. How is the connection between A(N-1) and SU(N) used in real-world applications?

The connection between A(N-1) and SU(N) has various real-world applications, such as in quantum computing, cryptography, and signal processing. These fields utilize the principles of Lie groups and algebras, including A(N-1) and SU(N), to solve complex problems and develop new technologies.

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