Algebraic Groups

Lectures at FU Berlin, Summer Term 2016

Lei Zhang



Introduction

Group schemes are "groups" in the category of schemes, just as the usual groups are "groups" in the category of sets and Lie groups are "groups" in the category of smooth manifolds. An algebraic group is a group scheme whose underlying space is of finite type. In this course we are going to study algebraic groups in certain generality. Given a smooth algebraic group \(G\), we first split it into the connected part and the étale part using the connected-étale sequence. For a connected algebraic group we can split it further into an affine algerbaic group and an abelian variety using the theorem of Chevalley. Then we can find a maximal connected normal solvable subgroup of the affine algebraic group so that the quotient is semisimple. This normal subgroup is called the radical. There is also a maxiaml connected unipotent normal subgroup which is called the unipotent-radical. The aim of the course is to understand how can one decompose a smooth algebraic group in this way. We will introduce all the notions involved and prove most of theorems used.

Prerequisites

The prerequests for this course is a first course in algebraic geometry.

Exam

If you want to get a grade, please contact me via l.zhang@fu-berlin.de. There will be oral exams.

Course outline

You can find the course outline here.
Eine deutsche Version finden Sie hier.
Please send questions and comments to me at l.zhang@fu-berlin.de.

Exercises

Every Tursday there will be a new exercise sheet. You can try to solve them, and if you have any questions please contact me at l.zhang@fu-berlin.de.

  1. Exercise sheet April 21, 2016.
  2. Exercise sheet April 28, 2016.
  3. Exercise sheet May 5, 2016.
  4. Exercise sheet May 12, 2016.
  5. Exercise sheet May 19, 2016.
  6. Exercise sheet May 26, 2016.
  7. Exercise sheet June 3, 2016.
  8. Exercise sheet June 9, 2016.
  9. Exercise sheet June 16, 2016.
  10. Exercise sheet June 24, 2016.
  11. Exercise sheet June 30, 2016.
  12. Exercise sheet July 15, 2016.

Other Information

Place (course): SR E.31/A7 (Arnimallee 7)

Place (exercise): SR 210/A3 Seminarraum (Arnimallee 3)

Date: Wednesday 10:00-12:00 and 14:00-16:00

First Appointment: 20.04.2016

Course Language: English