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Linear combinations of structure constants

  1. Apr 12, 2008 #1
    I have two Lie algebras with structure constants [tex]f^{a}_{bc}[/tex] and [tex]g^{a}_{bc}[/tex], the number of generators being the same (as will become clear).

    Due to a particular symmetry/construct, I have that the system needs to be valid under [tex]g \to af + bg[/tex] (and a similar transform for g), which leads through to the Jacobi constraint [tex]f^{d}_{[ab}g_{c]d}^{e} + g^{d}_{[ab}f_{c]d}^{e} = 0[/tex], once you take into account that f.f = g.g = 0.

    Now this new equation is trivially satisfied if f=Ag for some number A. However, I want to know if there's some way of working out what systems are or aren't possible to exist under this symmetry. For instance, if f is from a bunch of U(1)'s then it's trivially true for any g. But what if I have an SU(2) group? To be more specific, I'm working over 6 generators so the list of nice Lie algebras is pretty short but the list of other algebras such as combinations of 2 or 3 generator algebras and the Nilpotent algebras aren't quite so short.

    Rather than computing an explicit form of f and g for every pair of possible algebras, can I say "The algebra [something] with structure constant f cannot pair with the algebra [something] because ..." (well I cannot think of any such examples). Does it matter if one of them is rotated away from the canonical form? The model which this relates to doesn't allow for independent rotations of the structure constants.

    I find it hard to believe that there isn't some kind of restriction on what two algebras can be combined. But I've no idea how to go about finding such methods.

    Even a few pointers, not a complete walk through, would be nice, so I can go through the method on my own to make sure I end up grasping it. Thanks :)
     
  2. jcsd
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