A rough overview
March 22, 2007  Posted by John Armstrong  Atlas of Lie Groups
7 Comments »
Leave a Reply Cancel reply
About this weblog
This is mainly an expository blath, with occasional highlevel excursions, humorous observations, rants, and musings. The mainline exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).
I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.

Recent Posts
 The Submodule of Invariants
 More New Modules from Old
 New Modules from Old
 Reducible Modules
 Irreducible Modules
 Lie Algebra Modules
 All Derivations of Semisimple Lie Algebras are Inner
 Decomposition of Semisimple Lie Algebras
 Back to the Example
 The Radical of the Killing Form
 The Killing Form
 Cartan’s Criterion
 A Trace Criterion for Nilpotence
 Uses of the JordanChevalley Decomposition
 The JordanChevalley Decomposition (proof)
Blogroll
Art
Astronomy
Computer Science
Education
Mathematics
 A Dialogue on Infinity
 A Singular Continuity
 Ars Mathematica
 Carter and Complexity
 Curious Reasoning
 Fightin’ the resistance of matter
 God Plays Dice
 Good Math, Bad Math
 Gowers's Weblog
 Gyre&Gimble
 Intrinsically Knotted
 Low Dimensional Topology
 Mathematics and physics
 Mathematics under the Microscope
 Michi’s blog
 Really Hard Sums
 Rigorous Trivialities
 Secret Blogging Seminar
 Sketches of Topology
 Steven Strogatz
 Sumidiot
 The Everything Seminar
 The Museum of Mathematics
 The nCategory Café
 The Universe of Discourse
 Theoretical Atlas
 This Week's Finds in Mathematical Physics
 Topological Musings
 What's new
Me
Philosophy
Physics
Politics
Science
RSS Feeds
Feedback
Got something to say? Anonymous questions, comments, and suggestions at Formspring.me!Subjects
Archives
Site info
The Unapologetic MathematicianThe Andreas04 Theme. Create a free website or blog at WordPress.com.
I am not a mathematician but still I try to comprehend. Are there 2 or 3 dimensional diagrams of these simple Lie groups?
Comment by eliza  March 24, 2007 
Well, sort of. The lower ones down, at least. For example, B1 is the collection of all rotations of threedimensional space. In general, Bn is made up of rotations in (2n+1)dimensional space, and the D series gives rotations in evendimensional spaces. Unfortunately, due to a technicality, rotations in the plane aren’t considered a simple Lie group.
If you’re thinking of a diagram like the one that ran alongside all the news reports, it’s actually not the Lie group they’re talking about. It’s a sort of tool used in Cartan’s classification called a “root system”, and the picture is a 2dimensional rendering of an 8dimensional (for E8) shape. There are pictures like these for all Lie groups, and John Baez has a bunch of them in his most recent column.
Comment by John Armstrong  March 24, 2007 
Can you please dumb it down further and explain any possible practical application for this? Maybe cite something Star Trek or Star Wars and a maybe a reference to a weapon or some cool space ship? Or even cooler some invading alien force? I understand what your saying but I have no real frame of reference for it because I am not a mathematician but I do understand stuff like “fundamental underlying principle to teleporation” or “fundamental underlying principle to big explosions”. Or even “fundamental underlying principle for making space craft fly”.
I just don’t have a frame of reference for this formula or how it matters.
Comment by Kenny Coffin  March 25, 2007 
Kenny, that’s a great question and it deserves its own post. I’m going to mull it over and write it up in the next day or so.
Comment by John Armstrong  March 25, 2007 
Cool! Thanks for the explanation. Seems like the press could certainly have explained that. So can this be applied to computer graphics?
Comment by SomeGuy  March 26, 2007 
I’m actually not sure what this can directly apply to. I just like it ’cause it’s pretty (in an intellectual sense). I’ve just put up another post linking to other sketches by the people directly involved, and my thoughts on why (in a realworld sense) we care about this.
Comment by John Armstrong  March 26, 2007 
[…] I gave a quick overview of the idea of a Lie group, a Lie algebra, and a representation; a rough overview for complete neophytes of what, exactly, had been calculated; and an attempt to explain why we […]
Pingback by Lie Groups in Nature « The Unapologetic Mathematician  January 11, 2010 