## Algebraic Groups

 Unit code: MATH42082 Credit Rating: 15 Unit level: Level 4 Teaching period(s): Semester 2 Offered by School of Mathematics Available as a free choice unit?: N

#### Requisites

Prerequisite

Students are not permitted to take MATH42082 and MATH62082 for credit in an undergraduate programme and then a postgraduate programme.

#### Aims

To introduce the students to the basics of algebraic groups and to study and the Lie algebra of an algebraic group and quotients in more detail.

#### Overview

The study of algebraic groups is a fascinating mixture of groups and algebraic geometry and is a very active field of research. In this course the students will meet the familiar themes of irreducibility, connectedness and dimension from algebraic geometry. In the case of algebraic groups these concepts behave particularly nicely. The students will also study actions of algebraic groups and an essential theorem about how algebraic groups can be embedded in general linear groups. The second half of the course will be a more detailed look at Lie algebras of algebraic groups and quotients. Various important theorems will be proved about and using these two important concepts. Many examples will be given throughout the course to aid with the students' understanding.

#### Assessment methods

• Other - 20%
• Written exam - 80%

#### Assessment Further Information

Coursework: weighted 20%

End of semester examination: three hours weighting 80%

#### Syllabus

Revision on Algebraic Geometry and definition and examples of Algebraic Groups [3]

Irreducibility, Connectedness and Dimension [4]

Actions of Algebraic Groups [2]

Linear Algebraic Groups [3]

Lie Algebra of an Algebraic Group [5]

Quotients [5]

"Linear Algebraic Groups and Finite Groups of Lie Type" by Malle and Testerman

#### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

#### Study hours

• Lectures - 22 hours
• Tutorials - 11 hours
• Independent study hours - 117 hours

#### Teaching staff

Michael Livesey - Unit coordinator