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Homework Help: Lie-algebra valued forms

  1. Sep 4, 2010 #1
    In trying to understand connection one-forms, I have to learn what a Lie-algebra valued form is.

    I already understand what a vector-valued form is. I also understand why

    \nabla (e) = e \otimes \omega

    where \omega is a one-form, \nabla is an affine connection /covariant differentiation and e is some basis vector. Here \omega (vector) = number

    But in the case of a connection one-form matrix, I am trying to understand why, when supplied with a "vector", it produces an element of the Lie-algebra. So all of a sudden, it would appear, \omega (vector) = lie-algebra element = element of a vector space!

    Can someone explain this to me?

    Many thanks!
  2. jcsd
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