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Math Help - Tangent Space in Lie Group

  1. #1
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    Tangent Space in Lie Group

    Hello,

    i try to understand how one can compute the tangent space on a Lie Group G at a given point, say e \ in G (That is to compute the Lie Algebra of G).

    Is there a general method or trick?

    My second question is about the special tangent space of SO(3) at e. Is there any simple way to compute the tangent space at e? I think it is more easy than in the general case, but i don't know how to do it.

    Thanks in advance.
    Last edited by Sogan; January 15th 2011 at 03:02 PM.
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  2. #2
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    I'll confine G as a matrix lie group.

    Think of tangent vectors as "velocity vectors" of points moving smoothly through the point 1.

    If M \in O(n), then MM^T=1. Let M=M(t) be a smooth path originating at 1. Apply d/dt to M(t)M(t)^T=1. Then, we have

    M^\prime(t)M(t)^T + M(t)M^\prime(t)^T=0.

    Since M(0)=M(0)^T=1, if we let X=M^\prime(0), then we have

    X+X^T=0 for t=0. Further, for any vector X with X + X^T=0, the matrix e^X is in SO(n) (verify this). Thus, the tangent space of SO(n) consists of  n \times n real vectors X such that X+X^T=0.

    You can apply this method for U(n), SP(n) as well.
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