Representations of Algebras

Lectures in Summer Semester 2025
Lecturer: Prof. Dr. William Crawley-Boevey
Exercises: Raphael Bennett-Tennenhaus

Lecture notes (as at 28.04.2025)

Contents and Literature

I plan to discuss the representation theory of algebras and quivers, including Auslander-Reiten theory, correspondences given by faithfully balanced modules, homological conjectures, representations of Dynkin and extended Dynkin quivers, tame and wild algebras, etc.

Here are some relevant books. The book by Erdmann and Holm is a good introduction, aimed at bachelor students. The book by Assem, Simson and Skowronski is a comprehensive introduction.

I. Assem and F. U. Coelho, Basic representation theory of algebras, Springer 2020, E-book.
I. Assem, D. Simson and A. Skowroński, Elements of the representation theory of associative algebras. Volume 1, Techniques of representation theory, CUP 2006, E-book.
M. Barot, Introduction to the Representation Theory of Algebras, Springer 2015, E-book.
M. Auslander, I. Reiten and S. O. Smalø, Representation theory of Artin algebras, CUP 1997, E-book.
H. Derksen and J. Weyman, An introduction to quiver representations, American Mathematical Society 2017.
K. Erdmann and T. Holm, Algebras and Representation Theory, Springer 2018, E-book.
P. Etingof et al., Introduction to representation theory, American Mathematical Society 2011, E-book.
P. Gabriel and A. V. Roiter, Representations of finite dimensional algebras, Springer 1977.
R. Schiffler, Quiver Representations, Springer 2014, E-book.
A. Skowroński and K. Yamagata, Frobenius algebras 1 Basic representation theory, European Mathematical Society 2011, E-book.
A. Skowroński and K. Yamagata, Frobenius algebras 2 Tilted and Hochschild extension algebras, European Mathematical Society 2017, E-book.

Lecture notes from some of my previous courses: Algebra I, Algebra II, Homological Algebra.

Exercises

Please note that the exercise classes will start in week 2, so the first will be on 17 April 2025.

Exercise sheets available here.

Examination

Oral examination at the end of the semester, by arrangement with the lecturer.

Notes

This course is part of a masters' sequence.

The first part of the masters' sequence was on homological algebra, see here. It is not necessary that students have attended that course, but they are expected to have some familiarity with rings and modules, and topics such as categories, projective and injective modules, and Ext groups.

The third part of the masters' sequence, in Winter Semester 2025/26, is planned to be about geometric methods for studying representations of algebras and quivers.

The content of the course is provisional, and may change, depending on the audience, and the amount of material I feel I can cover.

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Representations of Algebras

Contents and Literature

Exercises

Examination

Notes