Timeline for Transition Functions of the Principal Bundle $SU(2) \to \mathbb{CP}^1$
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 4, 2010 at 3:41 | comment | added | Pavel Etingof | $U(n)$ is not contained in $SU(n)\times U(1)$; rather, it is a quotient of the latter by a diagonally embedded $\Bbb Z/n$. | |
| Feb 3, 2010 at 22:42 | comment | added | Dyke Acland | No need for another post, just use the polar decomposition of the minor of $a_{i1}$ to formulate a direct generalisation of the $\mathbb{CP}^1$ case. | |
| Feb 3, 2010 at 21:21 | comment | added | Dyke Acland | Yes, this is this problem I was talking about in my earlier comment. As I said before I think the best approach is to map $V_i$ to $U(n)$ viewed as contained in $SU(n) \times U(1)$, but I'm not too sure how to do so exactly. As you said this is really a linear algebra problem (I can't see continuity being a problem) and I might re-post it as such | |
| Feb 3, 2010 at 20:59 | comment | added | Pavel Etingof | yes. So let $V_1$ be the first chart, where the 1-st coordinate is not zero. Then to a point $z=(z_1,...,z_n)$ of this chart one should attach a unitary matrix $u(z)$ with the first row proportional to $z$, and do so continuously in $z$ when $z$ is in $V_1$. This is a linear algebra problem: basically you need to construct a frame continuously depending on $z$ whose first vector is a multiple of $z$. This is not hard to do, but I don't really know the best way of doing this. And then you do similarly for $V_i$, $i>1$. | |
| Feb 3, 2010 at 20:42 | comment | added | Dyke Acland | But surely I need the trivializations to define the sections. | |
| Feb 3, 2010 at 20:30 | comment | added | Pavel Etingof | Well, there are many different ways to define these trivializations. Basically, for each of the standard charts $V_1,...,V_n$ on the projective space, choose sections $s_i: V_i\to U(n)$, and then compute the transition functions as ratios of these sections on intersections of the charts. | |
| Feb 3, 2010 at 20:24 | vote | accept | Dyke Acland | ||
| Feb 3, 2010 at 20:13 | comment | added | Dyke Acland | What about the general bundle $SU(n) \to \mathbb{CP}^{n-1}$ with fibre $U(n-1)$? What are the trivialisations $\alpha_k:\pi^{-1}(U_k) \to U_k \times U(n-1)$. My guess is that one embeds $U(n)$ into $SU(n) \times U(1)$, and then maps $(a_{ij})$ to $(h^k_{a_{ij}},\frac{a_{k1}}{|a_{k1}|})$. I 'feel' that $h^k_{a_{ij}}$ should somehow be related to the minor of $a_{i1}$ but can't see how to define it. | |
| Feb 3, 2010 at 19:54 | vote | accept | Dyke Acland | ||
| Feb 3, 2010 at 20:10 | |||||
| Feb 3, 2010 at 18:04 | history | answered | Pavel Etingof | CC BY-SA 2.5 |