Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

  1. Feb 3, 2014 #1
    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?
     
  2. jcsd
  3. Feb 3, 2014 #2

    DrDu

    User Avatar
    Science Advisor

  4. Feb 3, 2014 #3

    Bill_K

    User Avatar
    Science Advisor

    Perhaps Theorem 3.11 in this paper.
     
  5. Feb 3, 2014 #4
    I think it is true only in case of su(n),where it follows follows from U+U=1.
     
  6. Feb 3, 2014 #5

    Bill_K

    User Avatar
    Science Advisor

    Are you just guessing, or did you find an error in the paper that I cited.
     
  7. Feb 3, 2014 #6
    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.
     
  8. Feb 3, 2014 #7

    Bill_K

    User Avatar
    Science Advisor

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

    It's a unitary operator with a unitary matrix. Unitary means unitary. I see nothing subtle going on.
     
  9. Feb 3, 2014 #8

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
  10. Feb 3, 2014 #9

    Bill_K

    User Avatar
    Science Advisor

    You might, if you took a look at the paper I cited in #3 above. The proof is trivial.
     
  11. Feb 3, 2014 #10

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Why the generator operators of a compact Lie algebra are Hermitian?
  1. Hermitian Operator (Replies: 3)

  2. Hermitian Operators? (Replies: 34)

Loading...