Lie algebra study guide
Subject classification: this is a mathematics resource. |
Subject classification: this is a physics resource. |
Contents
Resources[edit]
MIT 18.755 Introduction to Lie Groups
Study hints[edit]
What is useful for me is to start by thinking of the most simple Lie Group that I can think of which is a translation left and right. Imagine a group G, whose elements are all "shifts left and right."
The Lie Algebra which corresponds to the Lie group is just a unit vector pointing left, and a unit vector pointing right.
Once I've gotten some initution regarding this, then I make the group a little more complicated by allowing for arbritrary translations.
Help wanted[edit]
I think I have a good picture in my mind of what a fiber bundle is, but I need some one to illustrate what a principle fiber bundle looks like.
Roadrunner 14:16, 22 August 2006 (UTC)
- A fiber bundle is like a generalized vector field. Instead of attaching vectors to each point of a space, we attach fibers, which allows us to talk about different types of behavior. In technical terms, a fiber bundle is a triple (B, T, p) where B is the base space, the space we are interested in, T is the total space: the space created by attaching fibers to each element of B, and p is the surjective projection function that maps fibers in T onto points in B such that the pre-image of a given neighborhood N of each point in B is homeomorphic to the space NxB. This assures us that the bundle "behaves nicely and each fiber gets along with the other fibers" locally.
- For example, in differential geometry, we study vector bundles, a fiber bundle where the fiber is a vector space. It is usually the case that the base space is a smooth manifold, such as R^{n} or S^{1}. One example of a vector bundle over R^{n} is to just attach a copy of R^{n} to each point. A less trivial example is to attach copies of R to each point of S^{1} in a way such that you can define a section of the total space to be the Moebius band (take S^{1} as the meridian circle of the band). Basically, just draw lines through each point of S^{1} so that the lines vary continuously and you can trim the lines to resemble the Moebius band. A trivial vector bundle is the cylinder over S^{1}.
- An example of a principal fiber bundle over a manifold is to, instead of attaching copies of R to S^{1}, attach copies of S^{1} itself. If this is done in a trivial manner, you get a space homeomorphic to S^{1}xS^{1} = T^{2}. Hopf attached copies of S^{1} to each element of S^{2} to accomplish the amazing Hopf fibration. If you're attaching finite groups to the space, you can imagine little polygons attached to each point.
- I'm only giving one (inexperienced) opinion of the mathematical picture, though. I'm not sure what you use these for physically. :-)
- PS. Mathworld gives good examples.
- PPS. Actually, now that I think about it, Feynman describes photons in his QED as particles with a rapidly rotating vector attached (dependent on energy). Perhaps one can attach copies of S^{1} to each point in space and once one chooses a path, a section of that bundle would correspond to the vector orientations for a photon of a certain energy, and one can integrate over the section to get the probability vector! Eh, overkill. :D
Ron 12:41, 23 August 2006 (UTC)
- I'd like to know what SU(m,n) generators look like. --HappyCamper 19:23, 22 August 2006 (UTC)
- What does look like mean? Are you looking for their definitions?--Hillgentleman|User talk:hillgentleman 06:46, 22 November 2006 (UTC)
- I think what I am looking for is the representation of that particular lie algebra... --HappyCamper 01:14, 23 April 2007 (UTC)
Principal bundles[edit]
A trivial example: principal G-bundle over a point[edit]
Let G be any group of your choice. G acts on itself simply transitively by left translation. Let X be the set of elements in G, forgetting the group operation. Then the set-theoretic map X-->point is a principal G-bundle over a point, with G acting on X on the left.--Hillgentleman|User talk:hillgentleman 06:41, 22 November 2006 (UTC)
Frame bundle[edit]
GL(n) acts simply transitively on the set of frames in a n-dimensional vector space. Take any rank n vector bundle, such as the tangent bundle of an n-sphere. One can choose n sections of the vector bundle, such that in each fibre their restrictions form a basis (frame). The collection of all such choices form a principal GL(n) bundle.
Principal GL(n) bundle over a point[edit]
Let the base be a point. Take the vector space R^n. Then the map R^n-->point is a vector bundle over a point. The set of bases in R^n form a principal GL(n,R) bundle over a point. -Hillgentleman|User talk:hillgentleman 06:41, 22 November 2006 (UTC)