MATH 534 Lie Theory I (Lie Algebras and their representations)
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
Syllabus and course outline
(This document is consistent with the university policies and contains all
the course information regarding marking, etc.)
- December "presentation marathon" - December 21, 10am-2:30pm or so, in
There will be approximately bi-weekly written homework assignments in
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.
Thursday November 24.
Universal Enveloping algebra.
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
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. -
cyclic modules, and the correpondence between dominant weights and
highest weight modules (chapter 20); the weight diagrams (Chapter 21).
- Tuesday September 6 and Thursday
Lie algebras: motivation; overview; the basic definitions (including
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
- Tuesday Sep. 13 -Thurs Sep. 15
Lectures 3-4: The radical. Nilpotent and solvable Lie algebras. Engel's
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
- 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).
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
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
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 :
Orthogonality properties (with respect to Killing form)
subspaces. (H: 8.3, 8.4).
Associating a root system with a Lie algebra.
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
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)
Mandatory reading: Please read H
11.1 and 12.1!
- Thursday October 27: :
Cartan matrix of a root system; Dynkin diagrams vs. Coxeter graphs (H
Sketch of the proof of the classification theorem for
Reducible root systems. (H: 11.3, 11.4)
I will skip
the construction of the root system from simple
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 :
The Weyl group (H 10.3); Automorphisms of root systems (H 12.2).
long and short
(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,
- 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
- Tuesday November 8 :
- Thursday November 10 -- Tuesday
November 22 :
Discussed the topics of Chapters 15-16, skipping some proofs and
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), 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).
December 1: Discussion of all the remaining problems on the problem list.
Tuesday December 6.
Presentations on December 21 (10am to
approximately 2:30pm, with lunch break), in Math 126.
- The existence theorem (H: Chapter 18) (Pamela).
- The E_8 lattice. (Andreas)
- 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