Lie groups that are not matrix groups

From: Zig (ziggurism_at_gmail.com)
Date: 03/29/05

  • Next message: David Madore: "f split => f' split" 
    Date: Tue, 29 Mar 2005 02:00:07 +0000 (UTC)

    So in the book Representation Theory by Fulton and Harris, they're
    classifying Lie algebras of dimension 3 in one of their beginning
    chapters. It made me want to ask some questions about Lie groups which
    are not matrix Lie groups, i.e. have no faithful finite dimensional
    representations.

    Let me briefly describe the stuff in the text. They get to the real
    Lie algebra [X,Y]=[X,Z]=0, [Y,Z]=X, and goes on to exponentiate some
    representation of the algebra to get the group G of upper triangular
    3x3 matrices with 1s along the diagonal. This group has center R,
    matrices with the only nonzero entry in the upper corner. To get other
    Lie groups with some algebra, they take the quotient of G/N with N any
    discrete normal subgroup, which must therefore be central. Z being a
    discrete normal subgroup, G/Z is the only other group with this
    algebra, presumably G/nZ will be isomorphic.

    Then he proves that this group has no faithful irreps: if it did, the
    center, S^1, would be proportional to the identity in that irrep, and
    we would have [Y,Z]~1, but tr[Y,Z]=0.

    My first question is about this proof. I can't understand why this
    proof doesn't apply to the covering group G (which clearly does have
    finite dimensional faithful reps). I think its center R should also be
    proportional to the identity in an irrep. The difference between the
    two cases is that S^1 is compact, while R is not. But so what? Did I
    miss the theorem in the book where he showed that a compact central
    subgroup must be proportional to the identity? Does not Schur's lemma
    apply to anything that commutes with the whole group?

    He also talks about how the tower of covers of SL(2) are not matrix
    groups. And I seem to recall once hearing that the universal cover of
    GL(n) is not a matrix group. These Lie groups that are not matrix
    groups seem a bit mysterious. Is there a general way to find groups
    who are not matrix groups? Or turn one that is not into one that is?
    What are some others? I think I've heard that the universal cover of
    GL(n) is not. Is that right? What about Spin(n)? The Spin(n)s that
    I've met have been, but I've never seen a general case.

    thanks
    zig

  • Next message: David Madore: "f split => f' split"

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