Connections on principal bundles

[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!

 

[martinbn]

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.