Iordan Ganev

Introduction to Algebraic Groups

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)

Lecture Notes

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.

Week-by-week Schedule and Exercises

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.

Week 1

• 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).

Week 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).

Week 3

• 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).

Week 4

• 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.

Week 5

• 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).

Week 6

• 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).

Week 7

• 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).

Week 8

• Lecture notes (17 Dec): Lie algebras (pdf).


• Exercise sheet here. Due 4 January.

Week 9

Week 10

• 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).

Week 11

• 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).

Week 12

• Lecture notes (14 Jan): Reductive groups (pdf). Root systems (pdf).


• Exercise sheet here. Due 25 January.

Week 13

• Lecture notes (21 Jan): Weyl Groups and Root Data (pdf).


• Optional exercises: 7.4.7(1), (2), (3), (4), (5).

Week 14

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

Final Exam

• Final exam (pdf). Due 21 February.