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I SU(2) and su(2) have different dimensions?

  1. Jun 19, 2016 #1
    The Lie group SU(2) is the set of unitary 2x2 matrices with determinant 1.
    These matrices can be written

    a b
    -b* a*

    Thus, as a manifold, we can think of a coordinate chart consisting of the four real numbers making up the two complex numbers a and b. It is a manifold of dimension 4.

    The Lie algebra su(2) is the set of 2x2 matrices of the form

    ic d+if
    -d+if -ic

    where here c, d, f are real numbers. Thus the dimension of su(2) is 3.

    But su(2) is also supposed to be the tangent space of the identity element of SU(2). Shouldn't the tangent space of a manifold at any point have the same dimension as the manifold?

    Either my identification of the dimension of SU(2) or su(2) is wrong, or my claim that the tangent space dimension is always equal to the manifold dimension is wrong. Which is it?
  2. jcsd
  3. Jun 19, 2016 #2


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    Staff: Mentor

    No, the condition on the determinant reduces it by another dimension:
    ##|a|^2 + |b|^2 = 1##
    Manifold and tangent space are of the same dimension.
  4. Jun 19, 2016 #3
    D'oh! I knew that. Thanks!
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