Why the generator operators of a compact Lie algebra are Hermitian?

[ndung200790]

Why generator matrices of a compact Lie algebra are Hermitian?I know that generators of adjoint representation are Hermitian,but how about the general representaion of Lie groups?

 

[DrDu]

Besides thinking that this question is probably more apt for the algebra forum, I think that it is basically the content of the Peter Weyl theorem:

http://en.wikipedia.org/wiki/Peter–Weyl_theorem

 

[Bill_K]

Why generator matrices of a compact Lie algebra are Hermitian?I know that generators of adjoint representation are Hermitian,but how about the general representaion of Lie groups?

Perhaps Theorem 3.11 in this paper.

 

[andrien]

Why generator matrices of a compact Lie algebra are Hermitian?I know that generators of adjoint representation are Hermitian,but how about the general representaion of Lie groups?

I think it is true only in case of su(n),where it follows follows from U⁺U=1.

 

[Bill_K]

I think it is true only in case of su(n),where it follows follows from U⁺U=1.

Are you just guessing, or did you find an error in the paper that I cited.

 

[andrien]

Well,the theorem given is talking about unitary representation of some operator representation of some compact group G.This unitary representation is different from the unitarity of operator.A unitary representaion simply concerns about finding an orthogonal set of basis in the group space,so I don't see a direct connection of that theorem with the unitarity which leads to hermitian generator.Probably you know how to make a connection.

 

[Bill_K]

Well,the theorem given is talking about unitary representation of some operator representation of some compact group G.This unitary representation is different from the unitarity of operator.A unitary representaion simply concerns about finding an orthogonal set of basis in the group space,so I don't see a direct connection of that theorem with the unitarity which leads to hermitian generator.Probably you know how to make a connection.

I have no idea what you're talking about, andrien. He says the operator is unitary.

Thus the operator O′= SOS⁻¹ is unitary on S, and its matrix representations will also be unitary.

It's a unitary operator with a unitary matrix. Unitary means unitary. I see nothing subtle going on.

 

[dextercioby]

Why generator matrices of a compact Lie algebra are Hermitian?I know that generators of adjoint representation are Hermitian,but how about the general representaion of Lie groups?

You must first distinguish between the matrices forming a group, forming a Lie algebra, forming the matrix elements of a linear operator representing a group/Lie algebra on some (topological) vector space. With this being said, let's try to see what can be understood from your first question: so you've got a compact Lie algebra of matrices and I assume the 'generator matrices' of it are the basis vectors of the linear space underlying the algebra. Why would they necessarily be Hermitean ? I see no reason for it.

 

[Bill_K]

I see no reason for it.

You might, if you took a look at the paper I cited in #3 above. The proof is trivial.

 

[dextercioby]

His 1st question was not related to representations. Surely, the irreds of a compact Lie group are equivalent to unitary ones as a result of Peter-Weyl's theorem. We use this result in Quantum Mechanics. That's why I wrote the first 2 sentences. Because I was suspecting a wrong terminology.