Math 261 - Lie groups and Lie algebras

Instructor: Allen Knutson, allenk@math.berkeley.edu
Lectures: Monday, Tuesday, Thursday 5 PM, room 2 Evans Hall.
Course Control Number: 55065
Office: 1033 Evans Hall, 642-4319

Text: Fulton and Harris, "Representation Theory"
There are some notes available online in PostScript and PDF, including but possibly exceeding the following:

  • general1 - classifying connected groups with given Lie algebra
  • compact2 - classifying compact Lie groups
  • symplectic3 - Morse theory on coadjoint orbits and conjugacy of tori
  • complex4 - complexifications of groups and K/T = G/B; also groups with nilpotent Lie algebra
  • Weylchambers5 - the Weyl group acts simply transitively on the set of Weyl chambers
  • BorelWeil6 - most of the Borel-Weil theorem
  • Bruhat7 - the Bruhat decomposition
  • BottSamelsons8 - Bott-Samelson manifolds
  • Demazure9 - Demazure modules and the Demazure character formula

    • While the topics we'll cover make it look like this will be independent of last term, I will frequently say "of course, we know this story well in the U(n) case". For example, we'll breeze rather quickly through the highest-weight classification of irreps of a compact group K, making note only of the exceedingly few differences from the U(n) case.

    Counting the days, and trying to guess how long things will take:
    Mon Apr 1 Tues 2 Thurs 4 Demazure and Littelmann character formulae
    Mon 8 Tues 9 Thurs 11 Littelmann tensor product formula
    Mon 15 Tues 16 Thurs No class Universal enveloping algebras, Verma modules
    Mon 22 Tues 23 Thurs 25 Asymptotic representation theory and Duistermaat-Heckman measure
    Mon 29 Tues 30 Thurs May 2 McKay correspondence
    Mon 6 Tues 7 Thurs 9 Construction of complex Lie algebras
    Mon 13 Tues 14 Conjugacy classes in compact groups

    Planned topics (perhaps not quite in this order):

  • general Lie group theory
  • Killing form
  • Cartan criterion for semisimplicity
  • automorphism group
  • Lie subalgebras <--> connected immersed subgroups
  • center vs. pi_1
  • compact Lie groups
  • implied by Killing form negative definite
  • investigate topology of K/T
  • all tori are conjugate
  • compact is product of torus and simples, up to finite factor
  • compact groups can be complexified, and K"ahler geometry of K/T
  • representation theory and Borel-Weil
  • conjugacy classes
  • Bott-Samelson manifolds and their application to representation theory
  • Bott-Samelson manifolds
  • Demazure modules
  • the Demazure character formula (probably just statement)
  • the Littelmann character formula
  • the Littelmann tensor product formula: statement
  • universal enveloping algebras
  • Verma modules, and their singular vectors (another proof of Kostant mult)
  • the center of an enveloping algebra
  • Poincare-Birkhoff-Witt
  • the Poisson structure on g^*
  • other topics
  • the cohomology of G and BG
  • coadjoint orbits and asymptotic representation theory (statements, mostly)
  • affine Lie groups and the affine Grassmannian
  • noncompact groups