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 have no idea what you're talking about, andrien. He says the operator is unitary.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.
It's a unitary operator with a unitary matrix. Unitary means unitary. I see nothing subtle going on.Thus the operator O′= SOS−1 is unitary on S, and its matrix representations will also be unitary.
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.