Timeline for Principal bundles and associated vector bundles, the case of the complex projective space (1,0)-forms
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 10, 2010 at 16:35 | comment | added | Emerton | Yes, that's right. | |
| Feb 10, 2010 at 14:37 | comment | added | Dyke Acland | One last last question: When you talk about $U(n)$ acting on $SU(n+1)$, you mean acting on the opposite side to one with respect to which the quotient action was defined? | |
| Feb 10, 2010 at 5:59 | vote | accept | Dyke Acland | ||
| Feb 10, 2010 at 2:53 | comment | added | Deane Yang | Let me support Ben's statement that one rarely actually constructs a bundle by writing down its transition functions. Transition functions are a means for defining what a vector bundle is and proving some basic results. In practice, however, you construct a specific vector bundle usually by somehow describing what its local sections look like. The transition functions are implied by this, by figuring out what happens on the intersection of two open sets on which local sections have been defined. | |
| Feb 10, 2010 at 2:39 | history | edited | Emerton | CC BY-SA 2.5 |
added 1952 characters in body
|
| Feb 10, 2010 at 1:38 | comment | added | Ben Webster♦ | I'll just note, while it's common to see the definition of what a vector bundle is using transition functions, they aren't really used much in practice. I would horrified if someone said to me "consider the vector bundle with such and such transition functions." Equivariance is expressed in terms of picking an isomorphism between the pull-back of my bundle by the action map and projection map $G\times X \to X$. You can express pull-backs easily enough in terms of transition functions (just pull back the functions) and isomorphisms too, but I'm not sure it will be enlightening. | |
| Feb 9, 2010 at 23:56 | comment | added | Dyke Acland | The extra explanation has been a great help. Thanks again. Just one last question: How is the equivariance of a $G$-action on a vector bundle expressed terms of the transition functions? | |
| Feb 9, 2010 at 19:24 | history | edited | Emerton | CC BY-SA 2.5 |
added 1359 characters in body
|
| Feb 9, 2010 at 19:03 | comment | added | Dyke Acland | Thanks for your answer. What I don't see, however, is why the adjoint rep of $U(n)$ on the the Lie algebra should give the $(1,0)$ and $(0,1)$ representations. How did you "come up" with this? | |
| Feb 9, 2010 at 15:13 | history | answered | Emerton | CC BY-SA 2.5 |