MATH 534 Lie Theory I (Lie Algebras and their representations)

MATH 534 Lie Theory I (Lie Algebras and their representations)

Text:

The course will be mostly based on J. Humphreys "Lie algebras and representation theory". However, we will occasionally refer to several other sources, including: Classes: Tue, Th 11am-12:30pm in MATX1102.
My office: Math 217.
e-mail: gor at math dot ubc dot ca
Office Hours: By appointment.

Announcements

HOMEWORK

There will be approximately bi-weekly written homework assignments in addition to the list of problems for in-class discussion.
Graph paper for varous root systems on W. Casselman's page (scroll to the bottom of the page).
  • Please do Exercise 1.2.11 in Diamond-Shurman and look at problem set on Dirichlet series by November 20 (do not hand this in). -->

    Final presentations:

    The last two weeks of class will be your lectures.
    Googe doc with current information on who does what and when. Feel free to edit. Possible topics (the ones marked with ** are the ones I think someone has to choose as I'd really like them to be part of the course, and * is something I would like to see as well):
    • Isomorphism theorem (H, 14.1 and 14.2) -- we stated it in class on October 29, without proof. The goal of the presentation is to present the proof.
    • Free Lie algebra, Serre relations, existence and uniqueness theorems (Chapter 18) -- 2 people can share this and take a whole lecture. -- Ellie.
    • Proof of Poincare-Brikhoff-Witt theorem (H 17.4).
    • ** Freudenthal's character formula (H 22.3)
    • ** Weyl's and Kostant's character formulas (H 24.1-24.3) -- this is for 2 or even 3 people; see also Weyl character formula in Fulton-Harris.
    • Chevalley groups (H., Chapter 25) : Ben and Gurk; for this one needs Serre relations (Ellie).
    • * Borel subgroups and flag varieties (FH, Chapter 23) -- Toni.
    • * Jacobson-Morozov theorem; Dynkin-Kostant classification of nilpotent orbits in a Lie algebra (also for 2 people). Sources: Notes by Ana Balibanu (more to be added soon).
    • McKay correspondence -- Nina
    • Cartan involution and the compact real form -- Cindy and Taeuk.
    • Feel free to suggest topics of interest to you!

    Detailed Course outline

    Short descriptions of each lecture and relevant additional references will be posted here as we progress.
    • Tuesday September 3 : NO CLASS -- Imagine day and qualifying exams. Graduate Orientation at 4:30pm.
    • Thursday September 5 and Tuesday Sep.10. : Lectures 1-2: Lie algebras: motivation; 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

    • Thursday Sep. 12 -Tue Sep. 17 : Lectures 3-4: The radical. Nilpotent and solvable Lie algebras. Engel's Theorem. Lie's theorem. References FH: 9.2, 9.3, H: 3.1, 3.2, 3.3.

    • Thursday Sep. 19 : Lecture 5: Cartan's criterion. References: H: 4.1, 4.3 I tried to simplify the proof of Cartan's criterion and that didn't work. We have completed the easy parts; will do the hard part (Lemma in Humphreys 4.3) next class. See also a note by David Vogan (I am apparently in good company). Defined Killing form.

    • Tuesday Sep 24 : Lecture 6: Finish the proof of Cartan's criterion; decomposition of a semi-simple Lie algebra as a direct sum of simple ideals. References: FH 9.3, H: 5.1, 5.2.
    • Thursday Sep. 26 : Lecture 7: will finish sections 5.1, 5.2 of Humphreys: non-degeneracy of the Killing form for a semi-simple Lie algebra. Then talk about complete reducibility of representations. Casimir element, Weyl's theorem. References: FH 9.3, H: 5.1, 5.2, 6.1, 6.2, 6.3

    • Tuesday Oct. 1 : Lecture 8: Jordan-Chevalley decomposition. FH: appendix C.2, H:chapter 6.4.
    • Thursday Oct. 3 : HOMEWORK DISCUSSION: we discussed the first 5 problems from "the list of problems" and Problems 1-3 on linear algebra. Odds and ends from the previous lectures.

    • Tuesday October 9: Lecture 9: representations of sl(2). We proved that irreducible finite-dimensional representations of sl(2) are in bijections with the natural numbers (to every n we associate the highest weight module with highest weight n).
      H: Chapter 7, FH: 11.1
    • Thursday October 11 : Lecture 10: Root space decomposition; sl(2)-triples. H: 8.1, started 8.2.

    • Tuesday October 15 : Lecture 11: The centralizer of a maximal toral subalgebra (H: 8.2). Orthogonality properties (with respect to Killing form) for root subspaces. (H: 8.3, 8.4).
    • Thursday October 17 : Lecture 12: Associating a root system with a Lie algebra. Finished "Rationality properties" (H: 8.5) and all of chapter 9, except we skipped the Lemma on p.43 (which we will return to soon). By now, we learned how to associate a Dynkin diagram with a semi-simple Lie algebra. (see also sections 11.2, 11.3, in Humphreys, and the statement of the Theorem in 11.4). We did not discuss simple roots though, will come back to it next class.

    • Tuesday October 22: Lecture 13: Bases, action of the Weyl group; (H 10.1, 10.3, some of 10.2) Automorphisms of root systems (H 12.2). Construction of the root system from simple roots -- did not do.
      Mandatory reading: Please read H 11.1 and 12.1!
      Also, see Section 15 in notes by Prof. Casselman for a brief discussion of the algorithm for constructing roots.
    • Thursday October 24: : Lecture 14: Sketch of the 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, 11.3, 11.4, 14.1, 14.2)
      Home reading: Construction of the root systems of types A-G. (H, chapter 12)

    • Tuesday October 29 : Lecture 15: Dynkin diagrams vs. Coxeter graphs (H 11.2); long and short roots, dual root system (see H 9.2 for the definition of the dual root system). Stated the Isomorphism Theorem 14.2; then discussed Lie algebra automorphisms but deviated from Humphreys: gave an algebraic groups perspective on the group Int(g) of inner automorphisms of the Lie algebra -- it is the adjoint algebraic group with Lie algebra g (when g is semisimple); for the intrinsic definition in terms of derivations, see H 1.3, 2.3 and 5.3.
    • Thursday October 31 : Homework discussion.

    • Monday November 4 and Tuesday November 5. : Lectures 16-17. Will finish the discussion of Aut(g) and "conjugacy theorems" (chapter 14.3 and a summary of chapter 15) Borel subalgebras, Cartan subalgebras, conjugacy theorems (Chapters 15-16).
    • Thursday November 7: : Lecture 17. Universal Enveloping algebra. PBW theorem (without proof). (Chapter 17).

    • November 10-15: No class.

    • Make-up lectures, time TBA: The weight lattice (Chapter 13); standard cyclic modules, and the correpondence between dominant weights and highest weight modules (chapter 20).

    • November 17-22: Homework discussion and 2 presentations.

    • November 24-29: 4 presentations.