Math 207A - Lie algebras and category 𝒪 (Fall 2023)

Lecture times and location: MWF 10-10:50AM, APM B412
Textbook: James Humphreys, Representations of semisimple Lie algebras in the BGG category 𝒪
Instructor: Steven Sam email
Instructor office hours: Friday 9-10AM (shared with 184, I will split if it becomes a problem) or by appointment
Discord server (for discussion about the content): link

Course description

We will discuss highest weight representations of semisimple Lie algebras and category 𝒪, introduced by Bernstein, Gelfand, and Gelfand.
I will review all of the core results about semisimple Lie algebras without proof. Prior knowledge is helpful, but not mandatory.
I expect participants to have completed Math 200 (first-year graduate abstract algebra) or its equivalent.
One of my goals is to construct the BGG resolution of a finite-dimensional representation. Along the way we will study the structure of Verma modules.
Time permitting, we might discuss highest weight categories, introduced by Cline, Parshall, and Scott, which generalize the above to other settings.

The lectures will be podcasted. You can find them here or on Canvas.

Homework

Homework is required for undergraduates and masters students taking this course for credit.

Homework 1

My notes for the course

Notes (last updated 12/8/23)

Schedule

This table will contain what we covered in each lecture. Sections refer to the notes above.

Week 1
Sep 29	Basic definitions (1.1) Basic operations (1.2)
Oct 2	Representations of sl₂ (1.3) Roadmap (1.4)
Oct 4	Enveloping algebras, definition (2.1) PBW basis (2.2)
Oct 6	Induction (2.3) Verma modules for sl₂ (3.1)
Week 2
Oct 9	Spherical harmonics (3.2) Semisimple Lie algebras (4.1) Classical series (4.2)
Oct 11	Roots and weights (4.3) Root systems (4.4)
Oct 13	4.4 continued
Week 3
Oct 16	Formal characters (4.5) Borel subalgebras (4.6) Highest weight representations (4.7)
Oct 18	Definition of category 𝒪 (4.8)
Oct 20	Central characters (5.1) Linked weights (5.2)
Week 4
Oct 23	Harish-Chandra's theorem (5.3) 𝒪 is artinian (5.4) Grothendieck group (5.5)
Oct 25	Yoneda Ext (6.1) Blocks (6.2)
Oct 27	Subcategories 𝒪_χ (6.3)
Week 5
Oct 30	Dominant and antidominant weights (6.4) Duality (6.5)
Nov 1	6.5 continued Projective objects (7.1)
Nov 3	7.1 continued
Week 6
Nov 6	Projective covers (7.2)
Nov 8	Injective objects (7.3) Standard filtrations (7.4)
Nov 10	Holiday - no class
Week 7
Nov 13	Resolutions (8.1) Chevalley-Eilenberg complex (8.2)
Nov 15	Weights in the relative Chevalley-Eilenberg complex (8.3)
Nov 17	8.3 continued
Week 8
Nov 20	Bott's theorem (8.4)
Nov 22	Complements of BGG complex (8.5) Translation functors (8.6)
Nov 24	Holiday - no class
Week 9
Nov 27	Kazhdan-Lusztig polynomials (9.1-9.4)
Nov 29	Highest weight categories (10.1)
Dec 1	10.1 continued
Week 10
Dec 4	10.1 continued
Dec 6	Path algebras (10.2)
Dec 8	Quasi-hereditary algebras (10.3)