If one generator commutes with all other,why must it be generator of U(1)?

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In summary, the conversation discusses the relationship between a generator of a Lie algebra and U(1) transformation, and mentions the three families of compact simple Lie algebras, with a focus on the five exceptional groups.
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ndung200790
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Please teach me this:

If one generator of a Lie algebra commutes with all other generator,then why must this generator be the generator of U(1) trasformation?(I know that generator of U(1) commutes with all other generators).

By the way,compact simple Lie algebras belong to 3 families called classical groups,with only 5 exceptions.What are the five exceptions?(QFT of Peskin and Schroeder)

Thank you very much in advance
 
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ndung200790 said:
If one generator of a Lie algebra commutes with all other generator,then why must this generator be the generator of U(1) transformation?(I know that generator of U(1) commutes with all other generators).
This question needs more context to be answered properly, but I guess you're dealing with a case where the representation of the other generators is irreducible. If so, I suspect Schur's lemma applies.

http://en.wikipedia.org/wiki/Schur's_lemma

This leads to the generator in question being a multiple of the identity, which generates U(1) when represented on a Hilbert space (i.e., when exponentiated to become a unitary operator).

By the way, compact simple Lie algebras belong to 3 families called classical groups,with only 5 exceptions.What are the five exceptions?

Well (surprise!) they are the "exceptional groups".

See the section "Exceptional cases" in this Wiki page:
http://en.wikipedia.org/wiki/Exceptional_group
which also has links to details of these groups.
 

FAQ: If one generator commutes with all other,why must it be generator of U(1)?

Why is it important for a generator to commute with all others in U(1)?

In U(1), the generators represent symmetry transformations. If a generator commutes with all others, it means that it preserves the symmetry of the system under all transformations. This is crucial in understanding the behavior and properties of the system.

What does it mean for a generator to commute with others in U(1)?

Two generators commuting with each other means that the order in which they are applied does not matter. In other words, their operations can be interchanged without affecting the outcome, which is a key characteristic of U(1).

3. Can a generator that does not commute with others be a generator of U(1)?

No, if a generator does not commute with all others, it cannot be a generator of U(1). This is because U(1) is defined as a group of transformations that commute with each other.

4. How does commutation relate to the structure of U(1)?

The commutation between generators is a fundamental aspect of the structure of U(1). It reflects the group's abelian nature, where the order of operations does not change the result. This allows for simpler and more efficient calculations in U(1).

5. What are some real-world applications of understanding the commutation of generators in U(1)?

Understanding the commutation of generators in U(1) has various applications in different fields. In physics, it is essential in studying fundamental particles and their interactions. In mathematics, it is used in the theory of Lie groups and algebras. It also has applications in signal processing, quantum mechanics, and computer science.

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