# Connections on principal bundles

• I
Silviu
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!