Representations of Algebras II
Lectures in Winter Semester 2025/26
Lecturer: Prof. Dr. William Crawley-Boevey
Exercises: Raphael Bennett-Tennenhaus
Entry in ekvv
Contents and Literature
I want to cover a number of key topics in the representation theory of finite-dimensional associative algebras.
- Correspondences given by faithfully balanced modules, and applications to Auslander algebras and homological conjectures. (Originally planned for the previous semester, but carried over, since there was not enough time.)
- Tilting and tau-tilting theory, including equivalences of derived categories.
- Geometric methods for studying representations of algebras, including relevant facts about varieties and schemes without proofs. (There will be less time for this than originally planned.)
- If time, possibly preprojective algebras and Kleinian singularities.
Some relevant books:
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I. Assem and F. U. Coelho, Basic representation theory of algebras, Springer 2020, E-book.
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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.
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H. Derksen and J. Weyman, An introduction to quiver representations, American Mathematical Society 2017.
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P. Gabriel and A. V. Roiter, Representations of finite dimensional algebras, Springer 1977.
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A. Skowroński and K. Yamagata, Frobenius algebras 2 Tilted and Hochschild extension algebras, European Mathematical Society 2017, E-book.
Exercises
Details to come.
Examination
Oral examination at the end of the semester, by arrangement with the lecturer.
Notes
This course is the third part of a masters' sequence. The first part was on homological algebra, see here, and the second part was on Representations of Algebras, see here.
The content of the course is provisional, and may change, depending on the audience, and the amount of material I feel I can cover.
Lecture notes from some of my previous courses:
Algebra I,
Algebra II,
Homological Algebra,
Representations of Algebras.
