Lie algebra study guide

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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 Rn or S1. One example of a vector bundle over Rn is to just attach a copy of Rn to each point. A less trivial example is to attach copies of R to each point of S1 in a way such that you can define a section of the total space to be the Moebius band (take S1 as the meridian circle of the band). Basically, just draw lines through each point of S1 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 S1.
An example of a principal fiber bundle over a manifold is to, instead of attaching copies of R to S1, attach copies of S1 itself. If this is done in a trivial manner, you get a space homeomorphic to S1xS1 = T2. Hopf attached copies of S1 to each element of S2 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 S1 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)
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)