Connections on principal bundles

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In summary, the conversation discusses the confusion surrounding the definition of the connection one form in principal bundles. The speaker initially thought it would be a product of two elements from different groups, but the given example does not match this assumption. The suggestion is to learn more about linear algebra, specifically multilinear algebra, to better understand the concept. The connection one form is explained as a ##\mathfrak g##-valued one form.
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Silviu
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Hello! I am reading about connections on principal bundles and the book I read introduces the connection one form as ##\omega \in \mathfrak{g} \otimes T^*P##, where ## \mathfrak{g}## is the Lie Algebra associated with the principle bundle P. I am a bit confused about what does this mean. Initially I thought ##\omega## would be something of the form ##(a,b)## with a and b belonging to the 2 groups ## \mathfrak{g}## and ##T^*P##, respectively. This confused me a bit, but I was willing to accept. However they give a basic example ##P(M,\mathbb{R})=M \times \mathbb{R}##, with ##M=\mathbb{R}^2-{0}## and then define the connection one form as ##\omega = \frac{ydx-xdy}{x^2+y^2}+df##, with ##(x,y) \in \mathbb{R}^2-{0}## and ##f \in \mathbb{R}##. This doesn't look at all as a tensor product to me and I am not sure where ##\mathfrak{g}## comes into play, so can someone explain to me how should I think of this connection one form? Thank you!
 
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Which book? It seems that the problem is with tensor products not principal bundles. You could try learning more linear algebra first (multilinear algebra in fact). If it helps think of ##\omega## as a ##\mathfrak g## valued one form.
 

1. What is a principal bundle?

A principal bundle is a fiber bundle that is locally isomorphic to a product space of a base space and a group. It is used to describe the symmetries of a space or system, with the group acting as the structure group.

2. What are connections on principal bundles?

Connections on principal bundles are mathematical tools used to describe the parallel transport of a fiber bundle, which represents the symmetries of a space or system. They are used to study the curvature and holonomy of the bundle.

3. How do connections on principal bundles relate to gauge theories?

Connections on principal bundles are essential in the formulation of gauge theories, which describe the interactions between fundamental particles in physics. The gauge fields in these theories are described by the connections on the principal bundle.

4. What is the difference between a principal bundle and a vector bundle?

A principal bundle has a group acting as the structure group, while a vector bundle has a vector space acting as the fiber. Additionally, principal bundles are used to study symmetries and gauge theories, while vector bundles are used to describe vector fields and their bundles.

5. What are some applications of connections on principal bundles?

Connections on principal bundles have many applications in mathematics and physics. They are used in differential geometry to study differential equations, in topology to classify manifolds, and in theoretical physics to describe gauge theories and the interactions between particles.

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