Quivers

Course manual 2020/2021

Course content

The goal of the course is to introduce basic techniques from representation theory, homological algebra and Lie theory and apply them to quiver representations.

A quiver is an oriented graph. A representation of a quiver is the assignment of a vector space to each vertex and the assignment of a linear map to each directed edge, with the domain and codomain of the linear map being the vector spaces attached to the source and target vertex of the directed edge. So roughly speaking, a quiver representation is a finite collection of matrices with particular size restrictions, naturally organized by an oriented graph.

Quivers nowadays play an important role in representation theory, geometry and Lie theory. The emphasis of the course is to introduce basic techniques and concepts from representation theory and homological algebra and apply them to quiver representations. Many of these techniques and concepts will reappear in master courses within the track Algebra, Geometry and Mathematical Physics. First year master students within this track are strongly encouraged to follow this course.

Topics that will be discussed are: abelian categories, indecomposable projective and injective modules, modules over path-algebras, projective and injective resolutions, ext groups, semisimple algebras and modules, bounded quiver algebras, and Gabriel’s theorem on the classification of quivers of finite representation type.

Study materials

Literature

• 1. Ralf Schiffler, "Quiver representations", CMS Books in Mathematics, Springer.

  2. I. Assem, D. Simson, A. Skowronski, "Elements of the representation theory of associative algebras. Vol. 1, Techniques of Representation Theory", London Math. Soc. Student Texts 65, Cambridge University Press.

Syllabus

• Syllabus text will be made available during the course

Objectives

• has learned basic techniques from representation theory and homological algebra
• can apply these techniques to representations of quivers and of path algebras.

Teaching methods

• Lecture
• Self-study

There will be a weekly meeting of 2 hours. A substantial part will be lectures, but we will also occasionally discuss and/or work on exercises. Outside the lectures the student is expected to practice by making homework exercises and recommended exercises and by studying the material that has been treated during the lectures. Occasionally the student will be asked to study a small part from the book by him/herself.

Learning activities

Attendance

This programme does not have requirements concerning attendance (TER-B).

Assessment

 Homework exercises are discarded if the mark for the exam is less than 5. Homework exercises are also discarded when the student takes the re-exam.

Assignments

Homework exercises

• individual, contributes 20% to final mark if mark exam is at least 5, will be corrected and returned to the students during the lecture.

Fraud and plagiarism

The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl

Course structure

Timetable

The schedule for this course is published on DataNose.

Contact information

Coordinator

• prof. dr. Jasper Stokman