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Math Help - the adjoint representation of a Lie group

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    the adjoint representation of a Lie group

    Hello! Can someone please explain to me the concept behind the adjoint representation of a Lie group? I have run through the theory but I ma not quite getting it.

    In my book the author gives such definition

    ''Take any  a\in G and define homomorphism  ad_{a}:G\rightarrow G by conjugation
     ad_{a}:g\mapsto aga^{-1}
    This homomorphism is called the adjoint representation of G''

    What does it do?

    I have also noticed that there is also an adjoint representation of a Lie algebra? what is the difference?

    Thank you
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by rayman View Post
    Hello! Can someone please explain to me the concept behind the adjoint representation of a Lie group? I have run through the theory but I ma not quite getting it.

    In my book the author gives such definition

    ''Take any  a\in G and define homomorphism  ad_{a}:G\rightarrow G by conjugation
     ad_{a}:g\mapsto aga^{-1}
    This homomorphism is called the adjoint representation of G''

    What does it do?

    I have also noticed that there is also an adjoint representation of a Lie algebra? what is the difference?

    Thank you
    Hello friend, you have been asking quite a few questions about these (pardon my language, I don't mean to be offensive) basic topics. It sound's though that the problem is not in yourself but in the book you are using. Do you mind if I ask which one you are?
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    Quote Originally Posted by Drexel28 View Post
    Hello friend, you have been asking quite a few questions about these (pardon my language, I don't mean to be offensive) basic topics. It sound's though that the problem is not in yourself but in the book you are using. Do you mind if I ask which one you are?
    Thank you for your reply.

    Yes I am an undergraduate student so I am mostly asking about basic things
    We use quite advanced book ''Topology, geometry and physics'' of Nakahara. His book is quite difficult because it is directed to graduated students and requires very much mathematical preliminaries (which undergraduate students quite ofthe dont have)

    I have an idea what the adjoint representation might be but I have problems with some physical interpretation of it which is most important in this case.

    Writting just a definition and some formulae does not help with understanding the whole concept behind it at all. It just makes it boring for the reader. It is always very helpful to try to interpret physically and give examples.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by rayman View Post
    Thank you for your reply.

    Yes I am an undergraduate student so I am mostly asking about basic things
    We use quite advanced book ''Topology, geometry and physics'' of Nakahara. His book is quite difficult because it is directed to graduated students and requires very much mathematical preliminaries (which undergraduate students quite ofthe dont have)

    I have an idea what the adjoint representation might be but I have problems with some physical interpretation of it which is most important in this case.

    Writting just a definition and some formulae does not help with understanding the whole concept behind it at all. It just makes it boring for the reader. It is always very helpful to try to interpret physically and give examples.
    I suppose the difficulty I am having in trying to answer your question is that I don't know what you want.I could spout off the definition of the adj. representation of a Lie group and now it determines the adjoint representation of the associated Lie algebra, but I don't think that's what you want. Also, when you say 'what does it do' I'm not quite sure what you mean. Adjoints provide a classic way of producing representations on Lie Algebras say since the map :\mathfrak{g}\to\mathfrak{gl(g)}:g\mapsto \text{ad}_g is a representation. It is so fundamental it is one of the reasons one insists that a Lie bracket satisfy the Jacobi Identity
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    Quote Originally Posted by Drexel28 View Post
    I suppose the difficulty I am having in trying to answer your question is that I don't know what you want.I could spout off the definition of the adj. representation of a Lie group and now it determines the adjoint representation of the associated Lie algebra, but I don't think that's what you want. Also, when you say 'what does it do' I'm not quite sure what you mean. Adjoints provide a classic way of producing representations on Lie Algebras say since the map :\mathfrak{g}\to\mathfrak{gl(g)}:g\mapsto \text{ad}_g is a representation. It is so fundamental it is one of the reasons one insists that a Lie bracket satisfy the Jacobi Identity

    You are right, I do not need the definition. I need rather some help to interpret the definition. By saying what does it do I wonder what can this be good for, where we can use it, how this representations apply to Lie groups and so on so when I present it to the other students I can give them something better than just 'dry' boring definition which not everyone might understand it.

    Your last sentence is precisely what I am after, but I think you are describing the adjoint representation of Lie algebras- not Lie groups???
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by rayman View Post
    You are right, I do not need the definition. I need rather some help to interpret the definition. By saying what does it do I wonder what can this be good for, where we can use it, how this representations apply to Lie groups and so on so when I present it to the other students I can give them something better than just 'dry' boring definition which not everyone might understand it.

    Your last sentence is precisely what I am after, but I think you are describing the adjoint representation of Lie algebras- not Lie groups???
    Ok, so is the purpose of your course representation theory of lie groups/lie algebras?
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