Lie Theory Through Examples 3
Posted by John Baez
We spent last week catching up with the notes. I decided to spend this week’s seminar explaining how the concept of weight lattice, so important in representations of simple Lie groups and Lie algebras, connects to what we’ve been doing so far. My approach follows that of Frank Adams:
- J. Frank Adams, Lectures on Lie Groups, University of Chicago Press, Chicago, 2004.
This book puts the representation theory of Lie algebras in its proper place: subservient to the Lie groups! At least, that’s the right way to get started. Groups describe symmetries; a Lie algebra begins life as a calculational tool for understanding the corresponding Lie group. Only later, when you become more of an expert, should you dare treat Lie algebras as a subject in themselves.
- Lecture 3 (Oct. 21) - Representations of Lie groups. The weight lattice of a simply-connected compact simple Lie group.
I’m busy preparing a calculus midterm and two talks I’ll be giving later this week at the University of Illinois at Urbana–Champaign, so no pretty pictures this time. Eventually the weights of representations will give us beautiful pictures… but not today!
(In Illinois I’ll be visiting Eugene Lerman and also the topologist Matt Ando. I’ll speak about the number 8 and also higher gauge theory and the string group. Regular customers know both these talks, but I decided to tweak the second one a bit, to make it easier to understand — mainly by leaving stuff out. So, if it was incomprehensible last time you tried, check it out now.)
Re: Lie Theory Through Examples 3
Hi John,
So, can I expect that the most dense lattice, in relation to the Z^{n} lattice density (p. 37), is E_{8}? I mean, there are no exceptional groups bigger than E_{8}.
In (p. 35), you said that D_{8} allows an extra space, so that you can use a denser lattice. But, in (p. 37), you give the number for denser lattices in 6 and 7 than you implied 2 pages before! Why?
Also, I read your exposition about “24”, but it was not clear why that was the densest. You didn’t show any comparative table as in the exposition about the “8”.