resources:books:baez_muniain
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
resources:books:baez_muniain [2017/12/27 12:32] jakobadmin [Solutions to Exercises] |
resources:books:baez_muniain [2018/05/05 17:17] (current) jakobadmin ↷ Page moved from resources:baez_muniain to resources:books:baez_muniain |
||
|---|---|---|---|
| Line 1: | Line 1: | ||
| ====== Baez, Muniain: Gauge Fields, Knots and Gravity ====== | ====== Baez, Muniain: Gauge Fields, Knots and Gravity ====== | ||
| + | |||
| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| <WRAP imageshadow> | <WRAP imageshadow> | ||
| Line 11: | Line 11: | ||
| | | ||
| - | <tabbox Reading Notes> | + | <tabbox Reading Notes/Summaries> |
| - | * http://michaelnielsen.org/blog/yang_mills.pdf | + | * A great summary of Gauge, Fields, Knots and Gravity was [[http://michaelnielsen.org/blog/yang_mills.pdf|published by Michael Nielsen]] |
| Line 56: | Line 56: | ||
| --> Exercise 1# | --> Exercise 1# | ||
| + | **Q:**Given a Lie group $G$, define its identity component $G_0$ to be the connected component | ||
| + | containing the identity element. Show that the identity component of any Lie group is a subgroup, and a Lie | ||
| + | group in its own right. | ||
| + | **A:** Suppose we have a path from the identity to $g\in G_0$. Now map this | ||
| + | path to a new path by multiplying each element by $h\in G_0$. This path starts at $h$ and since the mapping is continuous, must remain in $G_0$. (Otherwise, smoothly mapping the group manifold to $\R^n$ would show a | ||
| + | discontinuity at some point.) Thus $hg\in G_0$ for all $h,g\in G_0$. | ||
| + | |||
| + | There's a certain tension between having a smooth manifold with disconnected pieces --- given a map from the | ||
| + | the manifold to itself, one must take care that it does not have a discontinuous action, mapping some points in one component to another component. When this map is an element of the group, this requirement makes | ||
| + | $G_0$ into a subgroup. | ||
| <-- | <-- | ||
resources/books/baez_muniain.1514374329.txt.gz · Last modified: 2017/12/27 11:32 (external edit)
Page Tools
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International
