Math 223a: Lie Algebras
Fall 2011



What is new: Introduction and referenced added to Complete lecture notes.

Course information

Lecture Notes
 
Dates of lectures Description Notes Comments
  9/7     Basic concepts   Sec 1   (unchanged)
  9/12     Ideals   Sec 2  
  9/14     Nilpotent and solvable algebras   Sec 3   Promised revision not ready.
  9/19     Jordan decomposition and Cartan's criterion   Sec 4   correct proof that y in M added.
  9/21     Semisimple algebras and the Killing form   Sec 5  
  9/26     Casimir operator and Weyl's Theorem   Sec 6  
  9/28     Representations of sl(2,F)   Sec 7 footnote   plus a part of sec 6
  10/3     Root space decomposition   Sec 8  
  10/5     Root strings   Sec 8b   some misprints corrected
  10/10     Abstract root systems   Sec 9   misprints corrected.
      Examples in R3   A3, B3, C3   animation by Sonja Igusa.
      Updated animation   newB3, newC3   simple roots in green, negative root behind wall
  10/12     Weyl chambers   Sec 10   corrected
  10/19     Classification part a   Sec 11   Dn, Cn
  10/24     Exceptional algebras   Sec 12  
  10/26     Isomorphism Theorem   Sec 14  
  10/31     Cartan subalgebras   Sec 15  
  11/02     Conjugacy of CSA's   Sec 16a   using topology!
  11/07     Conjugacy of Borel subalgebras   Sec 16b  
  11/09     Universal enveloping algebra   Sec 17a  
      Proof of PBW   Sec 17b   We will skip this.
  11/14     Weights and maximal vectors   Sec 20   final version
  11/16     Finite dimensional modules   Sec 21   final version
  11/21     Formal characters   Sec 22a   half of this section
  11/28     Weyl character formula and Schur polynomials   Sec 22b  
  11/30     Schur functors   Sec 23, Worksheet   notes extended
  12/05     First character formula   Sec 24a   beginning of the proof
  12/07     Weyl character formula   Sec 24b   end of the proof
      Combined notes   Sec 1-24   complete!



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Department of Mathematics  Phone:  (781) 736-3062  Fax: (781) 736-3085  Started:  August 27, 2011
Brandeis University Email:  igusa@brandeis.edu   Last modified:  Dec 22, 2011