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 MATH 204.
My office: Math 217.
e-mail: gor at math dot ubc dot ca
Office Hours: By appointment.
• (This document is consistent with the university policies and contains all the course information regarding marking, etc.)

### Announcements

• December "presentation marathon" - December 21, 10am-2:30pm or so, in Math 126.

### HOMEWORK

There will be approximately bi-weekly written homework assignments in addition to
• the list of problems for in-class discussion.

• ### Detailed Course outline

Short descriptions of each lecture and relevant additional references will be posted here as we progress.
• Tuesday September 6 and Thursday Sep.8. : Lectures 1-2: Lie algebras: motivation; overview; the basic definitions (including ideals, homomorphisms, centre, radical). The notions of nilpotent, solvable, simple and semisimple Lie algebras; examples (the classical Lie algebras); proof that sl(2) is simple. References: H: 1.1, 1.2, 2.1, 3.1. FH: 9.1, some of 10.1

• Tuesday Sep. 13 -Thurs Sep. 15 : 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.

• Tuesday Sep. 20 : Lecture 5: Review and preview: some discussion of Jordan canonical form; discussion of what 'semi-simple' means; started Cartan's criterion. References: H: 4.1, 4.3 Just started talking about Cartan's criterion; will do the hard part (Lemma in Humphreys 4.3) next class. See also a note by David Vogan .

• Thursday Sep 22 : SORRY, NO CLASS, sick.

• Tuesday Sep. 27 -Thursday Sep. 29 : 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.
Discussed Problems 1-2 on the List of problems .

• Tuesday Oct. 4 : Lecture 8 (Longer lecture -- make-up class): Complete reducibility of representations. Casimir element, Weyl's theorem. References: FH 9.3, H: 5.1, 5.2, 6.1, 6.2, 6.3 Jordan-Chevalley decomposition. FH: appendix C.2, H:chapter 6.4.
• Thursday October 6: Lecture 9: 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 11 - Thurs Oct 13 : Lectures 10-11: Root space decomposition. H: 8.1. Worked out the 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).

• October 18-20 : Lectures 12-13. Orthogonality properties (with respect to Killing form) for root subspaces. (H: 8.3, 8.4). Associating a root system with a Lie algebra. Finished "Rationality properties" (H: 8.5) and all of Chapter 9.

• Tuesday October 25: Lecture 14 (Longer lecture). 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.
Bases, action of the Weyl group; (H 10.1, some of 10.2)

• Thursday October 27: : Lecture 15: Cartan matrix of a root system; Dynkin diagrams vs. Coxeter graphs (H 11.1, 11.2). Sketch of the proof of the classification theorem for irreducible root systems; Reducible root systems. (H: 11.3, 11.4)
Home reading: 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.

• November 1-4 : Lectures 16-17. 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). 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 skipped), plus some perspective involving algebraic groups.

• Tuesday November 8 : Homework discussion.

• Thursday November 10 -- Tuesday November 22 : 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) ). More specifically:
• finished the discussion of the Isomorphism Theorem (without proof, which is postponed till the presentations), and of the group of automorphisms Aut(g). (chapter 14.3)
• Discussed a summary of chapter 15 -- Cartan subalgebras, conjugacy theorems. We did 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 did not prove that CSAs are the same as maximal toral subalgebras when G is semi-simple. Instead, we talked a little about the context of algebraic groups; discussed that the Weyl group is isomorphic to N(T)/T for a maximal torus in G (where Lie(G)=g and Lie(T) is our CSA. )
• 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.
• Tuesday November 15 there no class (sick).

• Thursday November 24. Universal Enveloping algebra. (Chapter 17).

• November 28 (Monday) - make-up class. PBW theorem (without proof); 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.

• November 29 -December 1. - Standard cyclic modules, and the correpondence between dominant weights and highest weight modules (chapter 20); the weight diagrams (Chapter 21).
December 1: Discussion of all the remaining problems on the problem list.

• Tuesday December 6. Presentations:
• The existence theorem (H: Chapter 18) (Pamela).
• The E_8 lattice. (Andreas)

• Presentations on December 21 (10am to approximately 2:30pm, with lunch break), in Math 126.
• Isomorphism theorem (Yuve)
• D_4 and triality (Shubhrajit)
• Borel subgroups and flag varieties (Elizabeth)
• Weyl character formula (Zhenheng)
• Infinitesimal characters and Harish-Chandra's theorem (Simone and Devang).