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Math Help - why is Lie algebra better to work with than with Lie group

  1. #1
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    why is Lie algebra better to work with than with Lie group

    Can someone explain to me in a easy way why is it easier and better to work with Lie algebras than with Lie groups?

    I found somewhere that it is difficult with Lie groups because of their global properties which I must admitt do not understand....what does the author mean by saying ''because of the global properties''?

    Lie algebras are ''nicer'' because they are vector spaces....but how to interpretate this contribution, in which way the fact that they are vector spaces makes it easier to work with them than with Lie groups instead?

    Thank you very much for any help
    Last edited by CaptainBlack; May 3rd 2011 at 12:08 PM.
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  2. #2
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    In geometric terms, the Lie algebra is the tangent space at the identity element of the Lie group. Since a tangent space is flat, or linear, that obviously makes it easier to work with than the Lie group.

    The tangent line to a function at a point on its graph tells you how the function is behaving in the neighbourhood of that point, but it doesn't tell you anything about the global behaviour of the function. In the same way, the Lie algebra tells you what the Lie group looks like in the neighbourhood of the identity. By group multiplication you can translate the identity to any other point of the group. So the Lie algebra gives information about the behaviour of the group in the neighbourhood of any point. But it cannot tell you anything about global properties such as the topological structure of the group (for example, whether it is connected or simply-connected).
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    thank you very much! This explanaition really does make the difference
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