Welcome! This is the homepage for the course "Introduction to Algebraic Groups", taught at the Weizmann Institute of Science in the first semester of the 2020-2021 academic year.

The course syllabus is here (pdf). Video recordings of the lectures are collected in a YouTube playlist, and also appear in this folder on Panopto. The course roughly follows the first seven chapters of:

- T.A. Springer, Linear algebraic groups (2nd edition), Modern Birkhäuser Classics, 1988 (link)

The notes are in a github repository here.

- Introduction to the class.
- Algebraic geometry.
- First properties of linear algebraic groups.
- Jordan decomposition.
- Semisimple and unipotent elements of SL_2.
- Commutative algebraic groups.
- Clarifications on unipotent and commutative groups.
- Diagonalizable groups and tori.
- Derivations and differentials.
- Lie algebras.
- Clarifications on Lie algebras.
- The adjoint action.
- Parabolic subgroups.
- Reductive groups.
- Root systems.
- Weyl groups and root data.
- Vistas toward geometric representation theory.

Some exercises will be eligible for submission, others are just for practice. Any exercises from a lecture that you decide to submit will be due eleven days later at the start of the TA session.

- Lecture notes (29 Oct): Introduction to the class (pdf). Algebraic Geometry (pdf).
- Exercise eligible for submission (due 9 Nov): 1.7.2(1) in Springer.
- Optional exercises: 1.1.4(1)-(4), 1.3.4(3), 1.4.4(1)-(2), 1.7.2(2), 1.7.5(1)-(2).

- Lecture notes (5 Nov): First Properties of Linear Algebraic Groups (pdf), pages 1-11.
- Exercises eligible for submission (due 17 Nov): 2.1.5(3)(a)-(c), 2.2.2(4).
- Optional exercises: 2.1.5(1)(a), 2.1.5(5), 2.2.2(1), 2.2.2(2).

- Lecture notes (12 Nov): First Properties of Linear Algebraic Groups (pdf), pages 12-23. Jordan Decomposition (pdf), pages 1-7.
- Exercises eligible for submission (due 23 Nov): 2.3.4(2), 2.3.9(1).
- Optional exercises: 2.3.4(1), 2.3.4(3), 2.3.9(2).

- Lecture notes (19 Nov): Jordan Decomposition (pdf), pages 8-17.
- Optional exercise: fill in the details of the proofs of 2.4.2 and 2.4.3.

- Lecture notes (26 Nov): Jordan Decomposition (pdf), pages 18-24. Semisimple and Unipotent Elements of SL_2 (pdf). Commutative Algebraic Groups (pdf).
- Exercises eligible for submission (due 7 Dec): 2.4.10(2), 2.4.15.
- Optional exercises: 2.4.10(1), 2.4.10(3).

- Lecture notes (3 Dec): Clarifications on unipotent and commutative groups (pdf). Diagonalizable groups and tori (pdf).
- Exercises eligible for submission (due 14 Dec): 3.2.10(3), 3.2.10(4). You can assume char(k) = 0.
- Optional exercises: 3.2.10(2), 3.2.10(6).

- Lecture notes (10 Dec): Derivations and differentials (pdf).
- Exercises eligible for submission (due 21 Dec): 4.1.9(3).
- Optional exercises: 4.1.9(1), 4.1.9(2), 4.1.9(4), 4.2.5(1), 4.2.5(2).

- Lecture notes (31 Dec): Clarifications on Lie algebras (pdf). The adjoint action (pdf).
- Exercises eligible for submission (due 11 Jan): 4.4.11(7), 4.4.15(3).
- Optional exercises: 4.4.11(4), 4.4.15(1), 4.4.15(2), 4.4.15(5).

- Lecture notes (7 Jan): Parabolic subgroups (pdf).
- Exercises eligible for submission (due 18 Jan): 5.5.9(3), 6.2.11(1).
- Optional exercises: 5.5.9(7)(a), 5.5.9(8), 6.2.11(2).

- Lecture notes (14 Jan): Reductive groups (pdf). Root systems (pdf).
- Exercise sheet here. Due 25 January.

- Lecture notes (21 Jan): Weyl Groups and Root Data (pdf).
- Optional exercises: 7.4.7(1), (2), (3), (4), (5).

- Lecture notes (28 Jan): Vistas toward geometric representation theory (pdf).

- Final exam (pdf). Due 21 February.