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Lie 3-Algebra

  1. Sep 16, 2009 #1
    Hi!

    Im messing around with a Lie 3-Algebra at the moment(Im not sure how widespread these are. They obey similar rules to a Lie Algebra).

    I have a bracket that looks like this:
    [x1,x2,x3]=c*x0+a*x1+g*x2+h*x3, with c,a,g,h, some structure constants.

    I also have the relation a=-g.
    Am I allowed to then say [x1,x2,x3]=c*x0+a*x2-g*x2+h*x3
    or [x1,x2,x3]=c*x0=h*x3?

    I suppose my question amounts to does a=-g also mean a*x^1=-g*x^2?

    thanks so much
     
  2. jcsd
  3. Sep 16, 2009 #2

    HallsofIvy

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    Why would you ask? you have replaced x1 by x2 and g by -g. What right do you have to do that?

    No, of course not. Knowing that a=-g tells you nothing about ax1. Knowing that a= -g tells you that ax1= -gx1, not -gx2.

     
  4. Sep 16, 2009 #3
    in hindsight that was a silly question
    thanks for your help though
     
  5. Sep 20, 2009 #4
    Bumping this for some more structure constant help.
    I'm attempting to figure out what a particular Lie 3-algebra is by classifying it's structure constants, which are constrained by what's called the Fundamental Identity(Which is the 3-analog of the Jacobi Identity for normal Lie algebras).

    I'm convinced i'm not doing something right as i've actually never computed an algebra this way before(I've only previously studied matrix lie algebras): could someone take a look at my equations?
    I've unwound the identity into a system of equations that looks like

    1. a=b
    2. cd=ea
    3. cf=eg
    4.cb=eh
    5. -g=j
    6.ck=la
    7.cm=lg
    8.cj=lh
    9.k=-f
    10.ld=ek
    11.lf=em
    12.lb=ej

    where {a,b,c,d,e,f,g,h,j,k,l,m} are all 4 indexed structure constants. I was proceeding before trying to divide by various things, but now I've realized thats probably not a legal move as these are indexed objects. Any advice would be wonderful!
     
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