What Is a Lie-Algebra Valued Form in Connection One-Forms?

[camel_jockey]

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!

 

[NateD]

A Lie algebra valued form is a type of one-form that takes a vector and returns an element of the Lie algebra associated with the vector space. This means that instead of producing a number, as in the case of a vector-valued form, it produces an element of the Lie algebra, which is itself a vector space. The Lie algebra element can be thought of as a linear combination of elements of the Lie algebra, which gives us a way to represent the Lie algebra element as a vector. This then allows us to use the connection one-form matrix to take a vector and return an element of the Lie algebra associated with the vector space.