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Prerequisites for Lie Groups

  1. Sep 6, 2014 #1
    I don't know if this is the correct section for this thread. Anyway, I'm taking a graduate course in General Relavity using Straumann's textbook. I skimmed through the pages to see his derivation of the Schwarzschild metric and it assumes knowledge of Lie groups. I've never had an abstract algebra course, but my background in linear algebra is pretty strong. So, I was wondering what are the algebra prerequisites for Lie groups?

    Thank you. :)
     
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  3. Sep 6, 2014 #2
  4. Sep 6, 2014 #3

    D H

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    As Incnis mentioned, you don't need to know full-blown Lie theory to be able to use that theory in physics. Physicists and engineers use "Lie theory lite".

    You don't really need abstract algebra (but knowing a bit of it will help). You do need a strong grip on linear algebra.

    Basic concepts needed:

    • Matrix multiplication is in general non-commutative and therefore has a non-zero commutator ##[A,B]=AB-BA##
    • Matrix to some non-negative integral power: ##A^0\equiv I, A^n=A^{n-1}A = A A^{n-1}##
    • Matrix exponential ##\exp(A) = \sum_{n=0}^{\infty} A^n/n!##
    • For every invertible matrix A, there exists a matrix ##B## such that ##A = \exp(B)##
    • The set of ##N\times N## invertible matrices form a non-abelian group (as I wrote earlier, knowing a bit of abstract algebra does help).
    • There's a mapping from the set of ##N\times N## matrices to the set of ##N\times N## invertible matrices via the matrix exponential. This, along with a few other things makes the set of ##N\times N## invertible matrices a Lie group and makes the set of ##N\times N## invertible matrices the Lie algebra for that group.


    One last thing: I've found wikipedia to be an absolutely terrible source for learning this subject. There are lots of online references that are much, much better than wikipedia, and of course there are those old fashioned things called "books".
     
    Last edited: Sep 6, 2014
  5. Sep 6, 2014 #4

    Fredrik

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    The book by Brian Hall requires only basic linear algebra, and a little bit about limits and continuity.

    The most general definition of Lie group involves terms from differential geometry. Hall avoids this by noticing that almost all the interesting groups are matrix groups (i.e. groups whose elements are matrices) and that a matrix group is a Lie group if and only if it satisfies some simple conditions. (He doesn't prove this as a theorem, because that would require differential geometry). He then takes those conditions as the definition of a matrix Lie group.

    Lie algebras are treated in a similar way.
     
  6. Sep 6, 2014 #5
    But if the OP is interested in GR he shouldn't skip the connections of groups with differential geometry wich he should be at least slightly acquainted with if he is dealing with graduate GR. Furthermore, given GR nonlinearity it might be beneficial for him to get some notions outside matrix groups, with groups that don't have faithful finite dimensional matrix representation in certain cases(i.e. : SL(2,R), SL(2,C)...) and their action in manifolds.
     
    Last edited: Sep 7, 2014
  7. Sep 6, 2014 #6

    Matterwave

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    This sounds like the approach in "Naive Lie Theory" by John Stillwell, which I like very much. Of course, doing things this way you miss out on all the exceptional simple Lie groups! But those are much more prominent in string theory and quantum gravity and less prominent in GR or QFT.
     
  8. Sep 6, 2014 #7
    Less prominent? SU(2), SU(3), SL(2,C) are all simple groups and fundamental to QFT and the latter also to GR.
     
  9. Sep 6, 2014 #8

    Matterwave

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    I am talking about the exceptional simple lie groups, i.e. G2, F4, E6, E7, and E8. Those are not matrix groups, and I have never seen them in QFT or GR. But of course my knowledge of either QFT or GR is not complete.

    SU(2), SU(3), SL(2,C) (or really any subgroup in GL(n,C)) are all matrix Lie groups, and they can be analyzed without much differential geometry.
     
  10. Sep 6, 2014 #9
    Sorry, I misinterpreted the word exceptional(I gave it literal-like in outstanding:blushing:- instead of mathematical meaning), I listed the classical simple groups. You are right that the exceptional simple groups are not used in QFT/GR.
    For SL(2,C), being a non-compact group, you are not guaranteed to always have faithful matrix rep.
     
  11. Sep 6, 2014 #10

    Fredrik

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    Isn't the identity map on SL(2,ℂ) a faithful representation of SL(2,ℂ)?
     
  12. Sep 6, 2014 #11

    Matterwave

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    Since the special linear group is defined as the group of (real/complex) matrices of dimension n, and determinant 1... I'm not sure...how it might not have a faithful matrix representation? I don't understand how your statement could possibly be true since it it defined in terms of a matrix representation! Perhaps, do you mean it might not have a faithful, finite dimensional, UNITARY representation?

    EDIT: Whoop, got sniped by Fredrik.
     
  13. Sep 6, 2014 #12
    Sure, I was referring to finite dimensional rep.
     
  14. Sep 6, 2014 #13

    Matterwave

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    Actually the keyword in my statement was the term "unitary", which is why I capitalized it. ;) SL(2,C) is a legitimate matrix group. It has itself as a faithful, finite dimensional, representation as Fredrik mentioned. This representation just happens not to be unitary, since SL(2,C) matrices are not necessarily unitary matrices. But from a mathematical point of view, whether we have a unitary representation or not is not so important. From a QFT point of view, then obviously we want unitary representations of groups due to Wigner's theorem.
     
  15. Sep 6, 2014 #14
    That's why I wrote you are not always(in all cases) guaranteed a faithfull matrix rep. Since we were talking about QFT unitarity is assumed.
    Edit: I was actually referring to the universal covering case as pond Dragon pointed out.
    And of course in QFT the unitary non-trivial matrix representations of SL(2,C) must be infinite dimensional.
     
    Last edited: Sep 7, 2014
  16. Sep 6, 2014 #15
    I think you are referring to their universal covering groups...?
     
  17. Sep 7, 2014 #16
    Yes, I should have clarified that.
     
    Last edited: Sep 7, 2014
  18. Sep 7, 2014 #17
  19. Sep 7, 2014 #18
    I guess thinking about groups in purely mathematical or formal terms is different than in a more physical way. SL(2,C) is of course a matrix group formally, but when thinking in more physical terms, here meaning requiring unitarity, one sees it under a different light.
     
  20. Sep 7, 2014 #19

    Fredrik

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    Regardless of how we think of it, it is covered by the book. I don't see many advantages of studying a book based on differential geometry, if we're interested in representations. But I must confess that I'm not familiar with applications of Lie group theory to GR, so maybe I'm missing something. The OP mentioned that he encountered Lie groups in a presentation of how to find the Schwarzschild solution. I don't know how Lie group theory enters that picture, so I can't guarantee that Hall's book contains everything that the OP needs. But if it doesn't, I suspect that all he has to do is to read a few pages in a book like "introduction to smooth manifolds" by Lee...but I must also confess that I haven't read that chapter.
     
  21. Sep 8, 2014 #20
    I had a look at Straumann chapter on Schwarzschild and it actually just mentions in a couple of paragraphs 3D rotational symmetry group in Lorentz manifolds to justify the 2-spheres foliation of Schwarzschild spacetime. I don't think any knowledge about groups besides a very basic familiarity with groups is really necessary to follow it.
    Towards the end of the book group theory is used in GR's spinor analysis in a short appendix where a working knowledge of SL(2,C) is probably required.
    Barring this I'm not familiar either with other applications of group theory in GR outside the obvious fact that isometry groups are Lie groups.
    In any case I've always found that learning even if just a bit about Lie groups(the basic ones: rotations, SU(2) and the Lorentz group), really pays off in terms of my understanding of differential geometry.
     
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