Lie algebras. (Math 534, Term I 2011)

Announcements:

• New written homework is due December 7.
• Next homework problems discussion -- On Wednesday December 7, 3:30pm. Next written homework due December 7!

• Please fill out the doodle poll for your avialability for the last four presentations (instead of the final exam), December 5-9. It is the easiest to use "calendar view" to fill out this poll. Do not forget to check your exam invigilation assignments and other after-the-end-of-term duties before you fill it out. We need to find two 1.5-hour slots. I hope it works.
• Midterm canceled :-).
• Make-up lecture: Monday November 21, 3-5pm, in our usual room Math 202.
• NO CLASS on Tuesday, November 1.
• NO CLASS on Tuesday October 18.
• Please fill out the doodle poll for the scheduling of make-up lectures. (Please ignore the dates -- the poll is about your availability at pretty much all possible times on Mondays, Wednesdays, Thursdays and Fridays. If we find a slot that works for everyone, we will have to meetings on two different weeks at that time). When filling out Thursdays, please assume that it applies to a Thursday when there is no Number Theory Seminar.
• No class on September 13 and 15. Two make-up classes will be sheduled later. Next class - Tuesday Sept. 20. Please note that it will start with the discussion of the problems from the growing list of problems .
• The first written homework is due Thursday September 22.

Textbook:

• James E. Humphreys, "Introduction to Lie algebras and representation theory".
• Also useful (but you do not have to buy it) : W. Fulton and J. Harris "Representation theory. A first course".
• Some interesting links: (under construction)
  • The Atlas project page.
  • Introduction to representation theory by P. Etingof et al.
  • Notes on Algebraic groups by W. Casselman. Relevant for us: Sections 10-12, which discuss the Weyl group, and the algorithm for constructing roots.
  • For fun:
    • No theory of Everything

  Prerequisites

  The only prerequisites are: good understanding of linear algebra (including Jordan canonical form of a matrix), and a first course in group theory.

  Marking

  There will be a 1-hour written midterm exam (in class), everyone will be expected to do either an in-class presentation or a short essay (in place of the final exam) - whether it will be essays of presentations will depend on class size and everyone's interests, we'll decide during the first week of class; there will be some written homework problems, but most homework will be dicussed in class. The in-class homework discussions will happen roughly once in two weeks, the dates will be posted here.

  Homework

• To be discussed in class: a growing list of problems . The list of problems will grow, and they will be discussed as necessary. The ones discussed already will be marked with a check. The next discussion date: the last day.
• To be handed in:
  • Problem set 1 . Due Thursday September 22.
  • Problem set 2. Due Tuesday October 11. Some useful Maple code can be found at Carmen Bruni's page.
  • Problem set 3. Due Tuesday October 25.
  • Note: for information about tensor products and symmetric powers, see Appendix A in FH; also useful: Section 1 of the notes by P. Etingof et al. , or a chapter in the book "Linear algebra and geometry" by Kostrikin and Manin (let me know if you'd like to borrow the book).
  • Problem set 4. You might find Professor Casselman's "Graph paper for Coxeter groups" helpful: click here and scroll down to find it.

  Tentative schedule of presentations:

  • Monday Nov. 21, 3-5pm:
    • Lance, Classification of the semisimple orbits (Chevalley's Theorem);
    • XinYu, Classification of nilpotent orbits.
  • Tuesday Nov 22:
    • Adrian and Carmen, Lie algebra cohomology.
    • Carmen's notes
  • Thursday Nov 24:
    • Ehsan, Lie groups and the exponential map.
    • Pang, Topology of the classical Lie groups.
  • Tuesday Nov 29:
    • Vasu, Schur-Weyl duality.
    • Li Zheng, Lie algebras over the reals.
  • Thursday Dec 1:
    • Asif, Harish-Chandra modules (beginnings of representation theory for gl(2, R) ).
    • Saman, The symplectic structure on nilpotent orbits.
      Saman's excellent reference list:
      • Lectures on the Orbit Method (book by A.A. Kirillov);
      • "Merits and demerits of orbit method", a survey article by A.A. Kirillov;
      • Surveys in Modern Mathematics (The Independent University of Moscow Seminars)
      • "Geometric Quantization for Nilpotent Co-adjoint Orbits", article by William Graham and David A. Vogan, Jr (appeared in this volume )
      • Slides of the talk "The Orbit Method for Reductive Groups" by D. Vogan;
      • Lectures on Calogero-Moser Systems by P. Etingof;
      • Formality Conjecture by M. Kontsevich.
    • The "half-marathon" on Tuesday December 6 .
      • 11:00 - 11:45 Atsushi, Kac-Moody algebras. Atsushi's notes.
      • 11:50 - 12:35 Pooya, McKay correspondence. Pooya's notes
    • The "marathon" on Wednesday December 7 .
      • 1:00-2:00 Heidar, Kirillov's orbit method. (in Math 126)
      • 2:00-3:30 Arman and Shuhang, Freudenthal's, Weyl's and Kostant's formulas. (in Math 126)
      • 3:30-4:15 (room TBA) The last homework discussion.
      • 4:15-5:00 Hang Yu, Chevalley groups. (room TBA)

    Approximate detailed syllabus:

    Descriptions of lectures, homework problems, and interesting links will appear here as we go along. In references, "H" is Humphreys, "FH" is Fulton-Harris, and numbers are the section numbers.
    
  • September 6-8: Lie algebras: an overview; the basic definitions (including ideals, homomorphisms, centre, radical, ...) The notions of nilpotent, solvable, simple and semisimple; examples (the classical Lie algebras); the 2-dimensional Lie algebras; adjoint representation; proof that sl(2) is simple. References: H: 1.1, 1.2, 2.1, 3.1, 3.2; 1.4, 2.2. FH: 9.1, some of 10.1
  • September 13-15: No class. Please fill out the doodle poll for scheduling the make-up classes.
  • September 20-22: The radical. Nilpotent and solvable Lie algebras. Engel's Theorem, Lie's theorem. Cartan's criterion. References FH: 9.2, 9.3, H: 3.1, 3.2, 3.3, 4.1, 4.3.
  • September 27-29: Discussion of Jordan decomposition (however, we have not yet proved that the semisimple and nilpotent parts of every element of g are in g). Complete reducibility of representations: the Killing form, Casimir element, Weyl's theorem. References: FH 9.3, H: 5.1, 5.2, 6.1, 6.2, 6.3
  • October 4-6: Jordan-Chevalley decomposition; representations of sl(2). Root space decomposition. (FH: appendix C.2, H:chapters 6.4, and 7 and 8.1, 8.2)
  • October 11: Root space decomposition continued (H: 8.3, 8.4, 8.5).
  • October 13: Root systems; the definition of the Dynkin diagram, statement of the classification theorem. (we did not discuss simple roots though, will come back to it next class) (H: chapter 9, also sections 11.2, 11.3, statement of the Theorem in 11.4)
  • October 17-20: Bases, action of the Weyl group; (H 10.1, 10.3, some of 10.2) Automorphisms of root systems. Construction of the root system from simple roots. See Sections 10-12 in notes by Prof. Casselman for a discussion of the algorithm for constructing roots. October 20: homework discussion.
  • October 25-27: Proof of the classification theorem for irreducible root systems; Reducible root systems. Proof of the classification theorem for the semisimple Lie algebras. (H: 10.4, 11.4, 14.1, 14.2) Home reading: Construction of the root systems of types A-G. (H, chapter 12)
  • No Class on Tuesday November 1.
  • November 3: Automorphisms of Lie algebras, Cartan subalgebras, inner automorphisms. (H, 14.4, chapter 15 -- home reading, skipped; 16.1, also section 2.3).
  • Monday November 7 -- make-up lecture.
  Borel subalgebras; conjugacy of Borel subalgebras. Automorphisms of Lie algebras.(Chapter 16, except for 16.4, which we replaced by the "proof" using the adjoint group.). Sketch of: The universal enveloping algebra, PBW theorem (H, 17.1-17.2).
  • November 8: Homework discussion; Continuation fo PBW theorem. H, 17.3.
  • November 10: Existence theorem, Serre's relations (quick summary of Chapter 18). For free Lie algebras, see for example, online book by S. Sternberg, Section 11.
  • November 15-17: Representation theory: the weight lattice; weight spaces; standard cyclic modules; classification of the finite-dimensional representations. (H. Chapters 13, 20, and 21).
  • November 21-24: Make-up class: November 21. Presentations start.
  • November 29-Dec 1: Presentations.
  • The remaining presentations and the last homework discussion -- the week of December 5 -- 9.