Algebraic Groups - MAT00003M

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  • Department: Mathematics
  • Credit value: 10 credits
  • Credit level: M
  • Academic year of delivery: 2022-23

Related modules

Pre-requisite modules

Co-requisite modules

  • None

Prohibited combinations

  • None

Additional information

Pre-requisite knowledge for MSc students: A second course on Group Theory; a course on Algebraic Geometry.

Module will run

Occurrence Teaching period
A Spring Term 2022-23

Module aims

The aims of the module are to: introduce the notion of an algebraic group as a group which also carries the structure of an affine variety; to study particular algebraic groups such as special linear groups, also solvable algebraic groups, connected algebraic groups and tori; to develop some of the general properties of these groups via a small amount of representation theory.

Module learning outcomes

  • The concept of an algebraic group as a group and an affine variety or algebraic set in affine space.

  • How the group operations viewed as homomorphisms of algebraic sets influence the group structure of an algebraic group.

  • Common examples of groups, such as general linear groups, special linear groups and groups of upper triangular matrices in the context of algebraic group theory.

  • The notion of the connected component of an algebraic group and be able to compute it in simple examples.

  • Diagonalisable groups.

Module content

Syllabus:

  • The definition of a linear algebraic group, homomorphisms, classical examples.
  • The linearization procedure.
  • The character group and diagonalizable groups.
  • The identity component.

Indicative assessment

Task % of module mark
Closed/in-person Exam (Centrally scheduled) 100

Special assessment rules

None

Indicative reassessment

Task % of module mark
Closed/in-person Exam (Centrally scheduled) 100

Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

Indicative reading

Early chapters from

  • T. A. Springer, Linear Algebraic Groups, Birkhauser
  • J E Humphreys, Linear Algebraic Groups, Springer
  • G. Malle and D. Testerman, Linear Algebraic Groups and Finite Groups of Lie Type, Cambridge University Press
  • M. Geck, An Introduction to Algebraic Geometry and Algebraic Groups, Oxford University Press