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Group Theory

  1. Jan 30, 2010 #1
    Hi there,

    I 'm currently reading topics relating to type IIB superstring theory. One of the things I am always confused with is Groups. I looked on various websites including Wikipedia but I still haven't quite got it. Could anyone please give me a nice introduction about Groups?

    What are GL(n), SL(n), O(n), SO(n), U(n), SU(n), Sp(n) ?? I know what they 're called, but I don't understand their operations and significance.

    For example:
    What are the differences between SO(24) and SO(25)?
    What are the differences between SO(3) and SO (2, 1)?
    Why is the two sphere S^{2} = SO(3) / SO(2)?
    Why is the Anti-de Sitter space AdS^{2} = SO(2, 1) / SO(1, 1)?

    Many thanks!
  2. jcsd
  3. Jan 30, 2010 #2


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    SO(N) is the rotation group acting on an N dimensional real vector space
    SU(N) is the rotation group acting on an N dimensional complex vector space

    Dropping the "S" it means dropping the constraint det = 1
  4. Jan 30, 2010 #3


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    When I was in secondary school, and then as a sophomore or whatever you call the first year as undergraduate, it was one of my problems, to try to read the advanced books and get muddled in the nomenclature. After all it was "Matrix Groups", and I already know about matrix, so it can not be so hard, can it?

    One of the problems is that sometimes they use the same nomenclature for close objects, such as the pin and spin groups. Worse, Sp() and Spin() is not the same, but in some books it seems to be!?! Worse, sometimes people use uppercase for the algebras instead of the groups, while the standarda is that uppercase is a matrix group, lowercase its Lie algebra. And what the **** is a Lie algebra? It is the "infinitesimal generator", a funny but accurate way to abstract the relationship "exp(t X)=A", for A in the group and X in the Lie algebra.

    Now I think we should ask to the public for a good introductory book or article...
  5. Jan 30, 2010 #4
    I don't have time to answer everything but I'll take a stab at this one. I'll assume that you know the meaning of the division in this expression.

    SO(3) is the group of symmetries of S^2, the unit sphere in R^3. The manifold of SO(3), ie. the set of all differentpossible transformation parameters, it is bigger than S^2. The following reason: Consider the point (1,0,0) on the unit sphere. The transformations in SO(3) are able to transport this point to any other point in the sphere. Thus we have made a "correspondence" between the set of possible transformations in SO(3), i.e. the manifold of SO(3), and the set of points on S^2.

    So this manifold contains at least as many elements as the sphere itself since any target point on S^2 can be reached from (1,0,0) with a transformation in SO(3). But in addition to this, there are elements in SO(3) that act as rotations, while keeping the target point fixed. These are rotations around the axis defined by the target point on S^2 and the origin of R^3. This subgrup corresponds to SO(2) since this is rotation about an axis in R^3. So we need to divide them out from the correspondence we had defined. Therefore S^2 = SO(3)/SO(2).

    It can be made more rigorous of course...

    I like the book the book "Lectures on Lie groups" by Hsiang. It is concise, and suitable for those who are mathematically inclined.

    EDIT: Maybe the argument becomes nicer by instead saying that SO(3) can take any point on S^2 to the point (1,0,0), and then say that there is a redundant SO(2) that rotates around the x-axis. In that way, it is always the same subgroup SO(2), which may make it easier to understand. Something also needs to be said about the fact that it is SO(3) instead of O(3) and so on, but a proper proof can surely be found elsewhere.

  6. Jan 31, 2010 #5


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    Heh, I've been trying to figure this out for at least a year :wink:

    Here's my take on it, starting with the basics: A group is a set of objects together with an operation that can be performed on those objects. (Example: real numbers and multiplication, but that's only an example) The operation, denoted '*', needs to satisfy four properties,

    -closure: for all pairs of elements A and B in the group, [itex]A*B[/itex] is in the group
    -identity: the group must contain an element I which satisfies [itex]A*I=A[/itex] for all A in the group
    -invertibility: for all elements A in the group, there must be an element [itex]A^{-1}[/itex] which satisfies [itex]A*A^{-1}=I[/itex]
    -associativity: for all triples of group elements A, B, C, [itex](A*B)*C = A*(B*C)[/itex]

    A group can be completely defined by a "multiplication table" - that is, if you know enough to determine the result of A*B for any given A and B in the group, you know everything there is to know about the group. It turns out that for any such multiplication rule, or at least any of the ones we care about in physics, you can find one or more sets of matrices that which obey the rule. These matrices constitute a representation of the group. The groups get their names from their simplest matrix representations. So for example, SO(3) is the group of all special (that is, determinant=1) orthogonal 3x3 matrices. SO(4) is the group of all special orthogonal 4x4 matrices. (There is also e.g. a group of 4x4 matrices that follows the SO(3) multiplication rule; that would be the quadruplet representation of SO(3), which is not the same thing as SO(4).) And so on. So SO(24) needs a 24x24 matrix representation, SO(25) needs a 25x25 matrix representation, etc.

    There's also SU(N), which is the group of special unitary NxN matrices. And then there are O(N) and U(N), which are just like SO(N) and SU(N), except without the requirement that they be "special" (so the matrices can have determinant not equal to 1). Obviously, O(N) includes SO(N) as a subgroup, U(N) includes SU(N) as a subgroup, etc. I don't know all the abbreviations but you could certainly look them up.

    The commas, e.g. SO(3) vs. SO(2,1), I have only seen in the context of special relativity. So for example, the matrices of SO(3) represent rotations in 3D space. But this is 3D Euclidean space, which has a metric that has +1,+1,+1 on the diagonal. If you were to switch to Minkowski space, which would have a metric of -1,+1,+1, rotations in that space would no longer be represented by SO(3) matrices, but instead by SO(1,2) matrices, since the metric has 1 negative and then 2 positives (or 1 positive and 2 negatives, but still, it's 1 then 2) on the diagonal. But I'm no expert on group theory so I can't really give a proper explanation of that.
  7. Jan 31, 2010 #6


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    Most groups can be thought of symmetry groups of a scalar product.

    For SO(N) you define a scalar product
    (r, s) = r s = rn sn; n=1..N
    Then you can rotate the vectors like
    r' = M r
    using the matrix M which leaves the scalar product invariant
    (Mr, Ms) = r Mt M s = r 1 s = r s

    For SU(N) the scalar product has to be defined as follows
    (r, s) = rn* sn
    instead of transposing the matrix Mtone has to use cc as well: Mt*

    Now if you look at SO(2) the scalar product can be written explizitly as
    (r,s) = r1 s1 + r2 s2

    In SO(1,1) this changes to
    (r,s) = r1 s1 - r2 s2

    If you look at the rotation matrices M using rotation angles appearing via sin and cos in M, the minus sign causes the rotation angle to become imaginary or (equivalently) transforming th sin into sinh and cos into cosh. Therefore the four-dimensional Lorentz group is something like SO(1,3) directly reflecting the metric diag(+1, -1, -1, -1).

    If you want to understand the Lie algebra in more detail you should start with SU(2). The generators ta are (half of the) Pauli matrices. Therefore M = exp(ifata) can be expressed as a Taylor series involving products of Pauli matrices. That means that products like tatb can be calculated easily and the matrix M an be derived explicitly.

    One should add one difference between O(N) and U(N). O(N) is SO(N) times the discrete group of reflections which simply means +1, -1. Instead U(N) is SU(N) times the continuous group U(1) with group elements exp(ia).

    You should have a look at http://en.wikipedia.org/wiki/Table_of_Lie_groups
  8. Jan 31, 2010 #7


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    Good table. Points to note:

    I am not sure if the http://en.wikipedia.org/wiki/Pin_group]Pin(n) [Broken] and Spin(n) groups are Lie groups for some or all n. They are coverings, so in any case they have the same Lie algebra that the groups they cover.

    For groups of the kind (n,m) a lot of properties happen to depend on the "signature" of the metric, which is the difference n-m.

    WHat is really critical, for particle physics, is representation theory of lie algebras. Regretly the wikipedia pages are poor on this topic, perhaps some OpenCourseWare should do better. To read across a book, in a first contact, it is perhaps enough to be aware of the words singlet, triplet, octet, decuplet :smile: Still, and specially when GUT and SUSY enter game, some essential vocabulary is "fundamental representation", "adjoint representation" and so on; math books tell what it is, but only the american teachers in payed university sessions will really delve into its meaning in a Lagrangian.
    Last edited by a moderator: May 4, 2017
  9. Feb 4, 2010 #8


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    That was fantastic! Thank you for sharing that, torquil!
  10. Feb 10, 2010 #9
    Can anybody please solve this task : Prove that all groups with row not bigger then 5 are comutative
  11. Feb 10, 2010 #10
    Groups don't have rows, or at least that's not standard nomenclature. Do you mean order (the number of elements in a group)? If so, you can prove that yourself given the definition of a group. Write the multiplication table for the one and only group of order one. Now, write the multiplication table for the one and only group of order two. Keep going! :)
  12. Feb 10, 2010 #11
    the point is that i understand what are you talking about, but i don't know how to write that, can you write the solution please :((
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