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Thread: Deriving the Lie algebra

  1. #1 Deriving the Lie algebra 
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    Well, I had thought I was OK with that, but it seems not.

    Specifically, on a friend's recommendation, I went to our library and looked in Fulton & Harris: Representation Theory.

    There I find that, for the generic Lie group , the map is a nice mapping that fixes the the identity.

    This is glaringly obvious, right?

    But they, F&H, that is, go on to say this defines a mapping . Now this step I don't quite see.

    Clearly is the group of all automorphisms , and, by their (F&H's) construction, then is a Lie group homomorphism.

    What I DON'T get is the the leap from to .

    Any help out there?


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  3. #2 Re: Deriving the Lie algebra 
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    Quote Originally Posted by Guitarist
    Well, I had thought I was OK with that, but it seems not.

    Specifically, on a friend's recommendation, I went to our library and looked in Fulton & Harris: Representation Theory.

    There I find that, for the generic Lie group , the map is a nice mapping that fixes the the identity.

    This is glaringly obvious, right?

    But they, F&H, that is, go on to say this defines a mapping . Now this step I don't quite see.

    Clearly is the group of all automorphisms , and, by their (F&H's) construction, then is a Lie group homomorphism.

    What I DON'T get is the the leap from to .

    Any help out there?
    It is


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  4. #3  
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    Rocket I thank you for your response.

    Either you have read the book I referenced, or you are more clever than you appear (own up!) since I made a confusing typo.

    The mapping should be

    So may I say this? Since, by the group axioms , and for the same reason has inverse, then so does .

    The operation is, as you well know, called "conjugation".

    So since each conjugates all thusly, I may have that .

    Hmm. Still feels like I am missing something, but I can't quite see what.
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  5. #4  
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    Quote Originally Posted by Guitarist
    Rocket I thank you for your response.

    Either you have read the book I referenced, or you are more clever than you appear (own up!) since I made a confusing typo.
    I had not even heard of that book before your post.

    But I understand the basic theory so I knew what the author was trying to do.
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