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

Question about Lie Brackets in Group Theory

  1. Feb 6, 2009 #1
    What does it mean when a Lie Bracket has a subscript + or - directly after it?

    I found this notation in http://en.wikipedia.org/wiki/Special_unitary_group" [Broken] under the fundamental representation heading

    Those are Lie Brackets, right? I know Lie Brackets are being used elsewhere in the article.
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Feb 6, 2009 #2
    My knowledge of Lie groups/algebras are very limited, but in quantum physics this notation usually means commutator/anti-commutator (usually for bosons/fermions).
    So
    [tex][A,B]_{-} = AB-BA[/tex]
    [tex][A,B]_{+} = AB+BA[/tex]

    So i would guess, this is the same for the Lie Brackets.
     
  4. Feb 12, 2009 #3

    CompuChip

    User Avatar
    Science Advisor
    Homework Helper

    Note that Lie-brackets are abstract things, which take two elements x and y and produce a new one [x, y].
    In the section you are referring to, A and B are matrices and the square brackets are not Lie-brackets but simply commutators of matrices. For those it is common to use the notation explained by element4.

    Note that if we have a set of matrix commutation relations
    [tex][T_i, T_j]_- = T_i T_j - T_j T_i = c_{ij}^k T_k[/tex]
    then we can "promote" them into a Lie-group whose algebra is defined by
    [tex][t_i, t_j] = c_{ij}^k t_k[/tex]
    where the brackets in the former refers to the matrix commutator and in the latter expression to the abstract operation called "Lie-bracket". The matrices [itex]T_i[/itex] are one specific representation called the fundamental representation (if you remember that a representation is a function from the Lie-algebra to some subgroup of matrices, the fundamental representation would simply be the map [itex]t_i \to T_i[/itex] linearly extended).
    However, for a set of anti-commutation relations this is not generally possible, as the Lie-bracket must satisfy a requirement
    [tex][t_i, t_j] = -[t_j, t_i][/tex]
    which the matrix anti-commutator does not satisfy.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Question about Lie Brackets in Group Theory
  1. Question on Lie groups (Replies: 6)

Loading...