Course Identification

Introduction to Algebraic groups

Lecturers and Teaching Assistants

Dr. Iordan Ganev
Elyasheev Leibtag

Course Schedule and Location

First Semester
Thursday, 14:15 - 16:00

Monday, 10:15 - 11:00,

Field of Study, Course Type and Credit Points

Mathematics and Computer Science: Lecture; Elective; Regular; 2.00 points




Some familiarity with algebraic geometry or commutative algebra. If you do not have this, but want to take the course, please contact the lecturer.



Language of Instruction


Attendance and participation

Expected and Recommended

Grade Type

Pass / Fail

Grade Breakdown (in %)


Evaluation Type

Final assignment

Scheduled date 1


Estimated Weekly Independent Workload (in hours)



The theory of algebraic groups is central to various subjects in mathematics, ranging from number theory to mathematical physics. While the subject emerged from the analytic theory of Lie groups, the modern theory of algebraic groups is rooted in the principle that many groups of interest (including finite groups, classical groups, etc.) can be defined and understood using the powerful language of algebraic geometry. This principle provides a uniform approach to studying a wide variety of groups over arbitrary fields beyond the real and complex numbers encountered in the case of classical Lie groups.

In this introductory course, we exposit the rich theory of algebraic groups and supply useful tools to those interested in group theory, representation theory, number theory, algebraic geometry, and related subjects. The focus will be on linear (affine) algebraic groups over a field of characteristic zero, while touching on several other pertinent topics.

Specific material will include:

-- Review of basic algebraic geometry and commutative algebra

-- Structure of algebraic groups: Borel subgroups, parabolic subgroups, Jordan decomposition

-- Homogeneous spaces and quotients

-- Derivations, differentials, and Lie algebras

-- Roots, weights, Weyl groups, and root datum

-- Reductive groups and their classification in terms of root data

-- Commutative algebraic groups and abelian varieties (time permitting)


Learning Outcomes

Upon successful completion of the course the students will be able to:

Gain familiarity with the structure of linear algebraic groups, as well as the fundamental results of the subject.

Reading List

The course will follow T.A. Springer,  Linear algebraic groups (2nd edition), Modern Birkhäuser Classics, 1988 (

We may cover or reference material from the following textbooks:

-- Armand Borel, Linear Algebraic Groups
-- James Humphreys, Linear Algebraic Groups
-- J. S. Milne course notes