Charlotte Chan

Math 612: Lie Algebras, Winter 2022

Instructor: Charlotte Chan, charchan@umich.edu
Lectures: MW, 11:30a-1:00p, EH 3088; Lecture Capture Recordings (log-in required)
Office Hours: M, 1:00p-3:00p, EH 4844

Texts

[H] Humphreys, Introduction to Lie Algebras and Representation Theory

There will also be some content from:
[BtD] Brocker and tom Dieck, Representations of Compact Lie Groups
[B] Bourbaki, Lie Algebras, Chapters 4-6
[Ha] Hall, Lie Groups, Lie Algebras, and Representations

Grading

Homework - 90%
Attendance - 10%

Homeworks

Homework is due at the start of class. I generally do not accept late homework, but please email me if something comes up.

On each homework 1-2 problems will be graded for correctness. The remaining problems will be graded on completeness.

Schedule

M 1/5 - (P)Review of Lie groups; manifolds, tangent spaces; definition of Lie algebra; [BtD Ch1, S1-2]
W 1/10 - One parameter subgroups; computing Lie algebras of Lie groups; adjoint representation; Lie bracket; [BtD Ch1, S2] [Ha, S2.4] [H, Ch1]
W 1/12 - Derivations; derived and lower central series; solvable, nilpotent, abelian; reductive, semi simple, simple; [H,Ch1-3]; HW1 posted
M 1/17 - No class
W 1/19 - sl2 is simple, gl2 is reductive; nilpotent endomorphisms, ad-nilpotent elements; Lie subalgebras of gl consisting of nilpotent endomorphisms have a simultaneous eigenvector; [H, 3.2-3.3] ** Special time: OH 1-3p in EH 4844 **
M 1/24 - Engel's theorem, Lie's theorem; [H, 3.3-4.1]
W 1/26 - Jordan decomposition, Cartan's solvability criterion; [H, 4.2-4.3]; HW1 due; HW2 posted
M 1/31 - Killing form, semisimplicity; [H, 5]
W 2/2 - Schur's lemma, Casimir operator, Weyl's theorem (complete reducibility of finite-dimensional representations of semisimple Lie algebras); [H, 6.1,6.2,6.3]
M 2/7 - finish proof of Weyl's theorem, preservation of Jordan decomposition [H, 6.3,6.4]; **REVISED OH FOR THIS WEEK: M 2p-3p + Tu 2p-3p**
W 2/9 - representations of sl2; [H, 7]; HW2 due
F 2/11 - HW3 posted
M 2/14 - maximal toral subalgebras and roots; [H, 8.1-8.2]
W 2/16 - ** No class **
M 2/21 - more on roots; [H, 8.3-8.4]
W 2/23 - root systems, axioms, examples; [H, 9]
F 2/25 - HW3 due
M 2/28 - No class (Winter Break)
T 3/1 - HW4 posted
W 3/2 - No class (Winter Break)
M 3/7 - bases and Weyl chambers, simple roots [H, 10.1-10.2]; **REVISED OH FOR THIS WEEK: M 2p-3p + Tu 2p-3p**
W 3/9 - Weyl groups, irreducible root systems [H, 10.3-10.4]
M 3/14 - Cartan matrix and classification of irreducible root systems [H, 11]
W 3/16 - construction of irreducible root systems [H, 12]; HW4 due; HW5 to be posted
M 3/21 - semisimple Lie algebras are distinguished by their root systems [H, 14]
W 3/23 - universal enveloping algebra, PBW theorem [H, 17]
M 3/28 - highest weight modules [H, 20]
W 3/30 - Verma modules and the irreducible highest weight modules [H, 20.3]; conditions on finite-dimensionality [H, 21] HW5 due; HW6 to be posted
M 4/4 - weight diagrams [H, 13 and 21.3]
W 4/6 - universal Casimir element, composition series [H, 22, 24.1]
M 4/11 - formal characters [H, 22.5, 24.1]
W 4/13 - Weyl character formula; HW6 due
M 4/18 - Kazhdan--Lusztig conjectures: big-picture summary of Humphreys' "Representations of Semisimple Lie Algebras in the BGG Category O"

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