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

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The course will be mostly based on J. Humphreys "Lie algebras and representation theory". However, we will occasionally refer to several other sources, including:

• W. Fulton and J. Harris "Representation Theory (A first course)" (available online at the library website).
• Introduction to representation theory (Notes by P. Etingof et al).

Classes: Tue, Th 11am-12:30pm in MATH 126.
My office: Math 217.
e-mail: gor at math dot ubc dot ca
Office Hours: By appointment.

• Syllabus and course outline (This document is consistent with the university policies and contains all the course information regarding marking, etc.)

Announcements

• Note that the room is different from the official course schedule! The course meets in Math 126.
• December "presentation marathon" - Monday December 9, 10am-2:30pm in Math 126.

HOMEWORK

There will be approximately bi-weekly written homework assignments in addition to

the list of problems for in-class discussion. Next discussion: December 9.

• Problem set 1 (due September 19)
• Problem set 2 (due Thursday October 10).
• Problem set 3 (Due November 5 or so).
• Problem set 4 (Due December 23 on Canvas).
  You might like to use Graph paper for various root systems on W. Casselman's page (scroll to the bottom of the page).
• Problems on linear algebra (DO NOT hand in). For now, mostly Section 1 is relevant. More of these problems will become relevant as the course goes on.

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Final presentations schedule:

• 10am-10:35am Stella, more on finite-dimensional irreducible representations
• 10:40am-11:15am: Henri, Universal Casimir elements
• 11:20am - 12pm Souptik, The Laplacian as a Casimir element
• 12:05pm -12:40pm: Kin Ming, Freudenthal's Formula
• 12:40-- 1pm Lunch (from Gobble, will be delivered).
• 1pm -- 1:35pm Andy, Kostant character formula.
• 1:40pm - 2:15pm Jonathan, Borel subalgebras and flag varieties.

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Detailed Course outline

Short descriptions of each lecture and relevant additional references will be posted here as we progress. For the dates in the future, this is an approximate plan.

• Thursday Sep.5. : Lecture 1. Lie algebras: motivation; overview; the basic definitions (including ideals, homomorphisms, lower central series and derived series). The notions of nilpotent, solvable, simple and semisimple Lie algebras; examples (the classical Lie algebras). References: H: 1.1, 1.2, 2.1, 3.1. FH: 9.1, some of 10.1


• Week 2: Tuesday Sep. 10 -Thurs Sep. 12 : Lectures 2-3: Proof that sl(2) is simple. 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.


• Week 3: Sep. 17-19. Lectures 4-5. Lecture 4: Finished the proofs of Engel's Theorem and Lie's Theorem and its corollaries. H: 3.3 and 4.1. Lecture 5: some discussion of Jordan canonical form; discussion of what 'semi-simple' means; Cartan's criterion. References: H: 4.2, 4.3 See also a note by David Vogan .


• Week 4, Sep. 24-26. : Lectures 6-7: Finished the proof of Cartan's criterion (H 4.3). Non-degeneracy of the Killing form for a semi-simple Lie algebra. Decomposition of a semi-simple Lie algebra as a direct sum of simple ideals. References: FH 9.3, H: 5.1, 5.2.
  Started complete reducibility of representations. Casimir element, Weyl's theorem. References: FH 9.3, H: 5.1, 5.2, 6.1, 6.2. Spent some time dicussing the meaning of Problem 7 on the list of problems.


• Tuesday Oct. 1 : Lecture 8: Finished the proof of Weyl's theorem, H: 6.3. A side note (not required reading): found a write-up of the same proof of Weyl's theorem in terms of homological algebra: a note by Christian Schnell (he uses the full universal enveloping algebra; you can substitute our associative algebra constructed from $\rho$).
• Thursday October 3: Jordan-Chevalley decomposition FH: appendix C.2, H:chapter 6.4. Representations of sl(2). We proved that irreducible finite-dimensional representations of sl(2) are in bijection with the natural numbers (to every n we associate the highest weight module with highest weight n).
  H: Chapter 7, FH: 11.1


• Tuesday October 8 - Thurs Oct 10 : Lectures 10-11: Root space decomposition. H: 8.1. Worked example -- root system for sl(3). A calculation of everything about the root system for sl(3) .
  The centralizer of a maximal toral subalgebra (H: 8.2).
  Thursday Oct 10: the discussion of some of the problems from the list.


• October 15-17 : Lectures 12-13. Orthogonality properties (with respect to Killing form) for root subspaces; integrality properties. (H: 8.3, 8.4). Associating a root system with a Lie algebra. Almosty finished "Rationality properties" (H: 8.5).


• October 22-24 Finished 8.5 and went over Chapter 9 quickly; please read it. (except for now we skipped the Lemma on p.43, to which we will return next week). The key point: we associated 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).
  Defined bases and the Weyl group; (H 10.1, some of 10.2)
  Defined Cartan matrix of a root system; discussed Dynkin diagrams vs. Coxeter graphs (H 11.1, 11.2). Sketched the proof of the classification theorem for irreducible root systems.


• October 29-31. Mandatory reading: Please read H 11.1 and 12.1. I will skip the construction of the root system from simple roots. Please read the construction of the root systems of types A-G (H, chapter 12) Also, see Section 15 in notes by Prof. Casselman for a brief discussion of the algorithm for constructing roots.
October 29, Lecture 16: Reducible root systems. (H: 11.3, 11.4) The Weyl group (H 10.3); Automorphisms of root systems (H 12.2). long and short roots, dual root system (see H 9.2 for the definition of the dual root system).
October 31: Discussion of the problems from the list. Check out a note by Charlotte Chan that talks about many of the things I left for the problem list; thus it has many hints to some of the problems.


• November 5-7 : Stated the Isomorphism Theorem 14.2; then discussed Lie algebra automorphisms (14.3) 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. For the discussion of the Lie groups with a given Lie algebra, see FH 7.3 and also FH 8.3 for the discussion of the exponential map.
  Irreducible roots systems correspond to simple Lie algebars see (H 10.4, H. 11.3, 14.1).


• In summary, by now we finished everything in Humphreys up to Chapter 11, skipped Chapter 12 as home reading, skipped Chapter 13 for now (will return to it later), and covered Chapter 14 (without the proof of the Isomorphism theorem, which we postponed until the presentations), plus some perspective involving algebraic groups. The next goal is to finish the proof that the root system attached to a semi-simple Lie algebra does not depend on the choice of the maximal toral subalgebra (this is the last remaining piece for the Lie algebra ---> root system association).


• Tuesday November 12: UBC break, NO CLASS.
• Thursday November 14: Will finish the discussion of the group of automorphisms Aut(g). (chapter 14.3) Will start a summary of chapter 15 -- Cartan subalgebras, conjugacy theorems. We will not go into discussion of the auxilliary group $\mathscr E(g)$ (which later turns out to be isomorphic to Int(g) when g is semi-simple). We will not prove that CSAs are the same as maximal toral subalgebras when G is semi-simple.
• The rest of the course:
  • Discussed the topics of Chapters 15-16, skipping some proofs and occasionally replacing them with arguments involving a group with the given Lie algebra (e.g., can take G=Int(g) ), finished the discussion of the Isomorphism Theorem (without proof, which is postponed till the presentations)
  • Borel subalgebras; conjugacy of Borel subalgebras (again we used the group G=Int(g) and Borel's fixed point theorem instead of the argument in Chapter 16.)
  • Discussed the independence of the root system from any choices.
  • Defined the Universal Enveloping algebra via the universal property. Discussed its interpretation as the algebra of left-invariant differential operators on a group G with the givel Lie algebra (e.g. G=Int(g) ). Discussed its analogy with the group ring of a group. These topics are outside of Humphreys.
  • Stated PBW theorem (without proof); Only stated the version that defines the PBW-bases. (Chapter 17).
  • The weight lattice (Chapter 13); saturated sets of weights. The fundamental group of a root lattice. Also had an extended "aside" discussion of the different Lie groups (over C) or algebraic groups with a given Lie algebra. See Chapter 4, Section 2.8 of Onischik and Vinberg "Lie groups and algebraic groups" for a summary of the facts about the weight lattice and the fundamental group. See also Section 3.5 in Chapter 4 for the discussion of the algebraic groups, the centre of the simply connected group, and the fundamental group.
  • Standard cyclic modules, Verma modules, and the correpondence between dominant weights and highest weight modules (chapter 20); the weight diagrams (Chapter 21).
  • Discussed the centre of the universal enveloping algebra and stated Harish-Chandra's Theorem (see 23.2, 23.3). More precisely, defined the infinitesimal character and proved the easy part (see Proposition in 23.2 on p.129 and corollaries from it): when the highest weights are associate, the infinitesimal characters coincide. The converse will not be proved. However, discussed Chevalley map at length (and there will be more in the presentations) (see H. 23.1 and Appendix at the end of Chapter 23).
  
  
  • Tuesday December 3. Review of the last half of the course; finished Chevalley map discussion; discussed Problems 12,13,16, 18 and some of 24 from the list.
  • Thursday December 6. Presentations:
    • The isomorphism theorem (H: Chapter 14) (Peilin).
    • The Poisson algebra. (Hao)
  
  
  • Presentations on December 9 (10am to approximately 3pm, with lunch break), in Math 126.
    • Details TBA.