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Adjoint representation vs the adjoint of a matrix

  1. Jul 25, 2014 #1
    In my limited study of abstract Lie groups, I have come across the adjoint representation ##Ad: G \to GL(\mathfrak{g})## on the lie algebra ##\matfrak{g}##. It is defined through the conjugation map ##C_g(h) = ghg^{-1}## as the pushforward ##C_{g*}|_{g=e}: \mathfrak{g} \to \mathfrak{g}##.

    Does this bear any relation to that one would call the adjoint of a matrix? I.e. the operation where one transposes and takes the complex conjugate?
  2. jcsd
  3. Jul 29, 2014 #2


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    I a not sure, but I would think the answer is no, at least not in the case of groups since when A a V (finite-dim. since we are using matrices)in the target space , i.e., A is in GL(V) , then A^T is a map in End(V*) , not in GL(V).
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