attachment:notes_noel.txt of msri07/notes

   1 11:30 a.m.
   2 Structure and Representations of Real Reductive Lie Groups: A Computational Approach
   3 Alfred Noel
   4 
   5 http://atlas.math.umd.edu/
   6 
   7 This software is very specific, not general at all.  Requires knowledge of Lie Theory.
   8 The software LIE describes complex Lie groups.
   9 
  10 Look at the complex matrix A:=<<a,d>|<c,b>>;  Determinant(A) = 1.  We'll call this a Sample Lie Group.
  11 SL_2(CC) -> sl_2(CC) = <<a,c>|<b,-a>>
  12 
  13 Basis of sl_2(CC): h = <<1,0>|<0,-1>>, k=<<0,0>|<1,0>>, g=<<0,1>|<0,0>>
  14 [h,x] = 2x.
  15 [h,y] = 2y.
  16 
  17 Look upon h as constructing a 1-dimensional space.  Define:  e_1(h) = 1.  e_2(h) = -1.  Then, the map [h,x] can be viewed as (e_1-e_2)(h)(x).  We can then define the Lie Algebra entirely by these functional e_1 - e_2.
  18 
  19 Complex Lie Groups:  (A,B,C,D) are Matrix groups.  Exceptional groups:  E_6, E_7, E_8, F_4, G_2.
  20 sl_2(CC) = A_1   (*)e_1 - e_2
  21 sl_3(CC) = A_2   (*)e_1 - e_2 ------- () e_1-e_3
  22 Dynkin Diagrams!!
  23 
  24 Weyl Group:  S_n  group of n symmetries.  |S_n| = n!
  25 Simple Roots  PI: alpha_1,..., alpha_l
  26 Roots alpha:  \sum lamba_ialpha_i   where lambda_i are either nonnegative or nonpositive (splits roots into two cases).
  27 
  28 SL_n(CC) --rho--> Aut(V)
  29  ^                  ^
  30 exp                exp
  31  |                  |
  32 sl_n(CC) --drho-> End(V)
  33 
  34 sl_2 :  CC h + CC x + CC y
  35 g = gg (+) sum( XX_{alpha} )     gg is a Cartan subalgebra
  36 alpha in h^* come from the Cartan subalgebra
  37 
  38 g_{CC} = g_{RR} + ig_{RR}   the "complexification of the Lie Algebra" is not unique up to g_{RR}
  39 E_8 is the "benchmark," the biggest group.
  40 
  41 The software tells you how many different root systems you can create by your choices of real Cartan subalgebrae.
  42 
  43 When we move to the calculating the real Weyl group, things get more complicated.
  44 
  45 This software can compute representations.