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A 3dim Poincare Algebra - isl(2,R)

  1. Sep 29, 2016 #1
    The Poincare algebra is given by isl(2, R) ~ sl(2,R) + R^3. What exactly does the i stand for?
    Thanks a lot in advance!
     
    Last edited: Sep 29, 2016
  2. jcsd
  3. Sep 29, 2016 #2

    fresh_42

    Staff: Mentor

    I have never seen a notation ##isl(2,\mathbb{R})##. Can you give some more background? And what does ##sl_2 + \mathbb{R}^3## mean? And as far as I know, the Poincaré algebra is ten dimensional.
     
  4. Sep 29, 2016 #3
    I'm sorry, my statement was incorrect. You are right about the Poincare algebra. I meant the Poincare algebra in 3dim (not 4dim) space and not the dimension of the Lie algebra. by sl(2,R) + R^3 the direct sum of the special linear algebra sl(2, R) and R^3 is denoted
     
  5. Sep 29, 2016 #4

    fresh_42

    Staff: Mentor

    In this case my question is: ##sl(2,\mathbb{R}) \oplus \mathbb{R}^3## as a Lie algebra or simply the vector space? I assume ##isl(2,\mathbb{R})## is simply an abbreviation the author of your source uses for this (presumably Lie algebra) sum. And I further assume that it is not a direct product, but an indirect, i.e. I suppose ##sl(2,\mathbb{R})## to act non-trivially on ##\mathbb{R}^3##.
    If you can write down this action, you will get the multiplications in it.
    And by the way, this one is six dimensional.
     
  6. Sep 29, 2016 #5

    fresh_42

    Staff: Mentor

    Let me make another guess. ##sl(2,\mathbb{R}) ## might formerly have been ##o(2,1)## which (I don't know without proving it) is isomorphic to ##sl_2##. So if you consider Lorentz transformations plus translations on ##\mathbb{R}^3 \cong \mathbb{R}^{2,1}## and call this algebra ##isl_2##, then it makes sense to write ##isl_2 \cong sl_2 \ltimes \mathbb{R}^3## or sloppy ##isl_2 \text{ ~ } sl_2 + \mathbb{R}^3##
     
    Last edited: Sep 29, 2016
  7. Oct 18, 2016 #6
    sl(2,R) as I remember is isomorphic to so(2,1). Thanks so much for your reply- this way of interpreting it makes sense. Sorry, that I am replying so late- I thought I had already replied
     
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