Basic information
The aim of the course is to give an introduction to the representation theory of finite dimensional algebras over a field. This brings insight into various demanding linear algebra problems on one hand, but it also allows to employ basic linear algebra for understanding concrete examples of abstract concepts in module theory on the other hand. The highlights are the following two results due to Peter Gabriel:
- Given any finite dimensional algebra A over an algebraically closed field K, there is a finite quiver Q and a finite set of relations R such that Mod A is equivalent to the category of representations of Q in vector spaces over K bound by the relations in R. Therefore, the problem of understanding A-modules is essentially a linear algebraic one, about finite diagrams of vector spaces.
- A precise charaterization of which finite quivers without oriented cycles have only finitely many indecomposable representations up to isomorphism, and a description of the indecomposable representations in these cases.
Basic information about the course can be also found in the Student Information System. Schedule (which also can be found in the Student Information System):
- Wednesdays 10.40am-12.10pm in the lecture room K10C,
- Wednesdays 2-3.30pm in the seminar room of the Department of Algebra.
Exercise sessions take place once in two weeks (with Kateřina Fuková).
Exam
The exam will be oral, please contact me to agree on a time. The required knowledge is covered by the first three chapters of the textbook [ASS] and Sections 3 to 5 in the paper [Kra].
Credit
The credit will be granted for solved exercise problems. There will be three sets of the problems which will appear here and solutions are to be handed over or sent by e-mail to the lecturer. Requested are at least 65% of points from solved problems.
Program of the course
A brief overview of what has been taught can be found below.
| Date | What has been taught | Source |
|---|---|---|
| Feb 19 | Representations of quivers, motivating problems from linear algebra, path algebras, a K-linear equivalence of categories of representations and categories modules over the corresponding path algebras. Relations and factors of path algebras. | [ASS], Sec. I.1, II.1, III.1 and App. A.1, A.2 |
| Feb 26 | Equivalence of categories of representations and modules for a bound quiver (Q,I). The Jacobson radical of a ring, the Nakayama lemma and the nilpotence of rad(A) for a finite dimensional algebra. A description of rad(KQ) for a finite acyclic quiver. | [ASS], Sec. I.1, I.2 and III.1 |
| Mar 5 | The Jacobson radical of a module and its properties. Simple and semisimple modules, composition series, the length of a module. Idempotents and direct sum decompositions. | [ASS], Sec. I.3 and I.4 |
Literature
The lectured material will be mostly covered by the following sources:
| [ASS] | I. Assem, D. Simson, A. Skowroński, Elements of the representation theory of associative algebras, Vol. 1, Cambridge University Press, 2006. |
| [Kra] | H. Krause, Representations of quivers via reflection functors, arXiv:0804.1428. [Full text in PDF] |
The course consists of the first three chapters of [ASS] and Sections 3 to 5 of [Kra].
Several other monographs on representation theory of finite dimensional algebras from various points of view appeared recently. Here we list on the other hand some more classical sources, which are only complementary as far as this course is concerned, but they are worth mentioning:
| [ARS] | M. Auslander, I. Reiten, S. Smalø, Representation theory of Artin algebras, Cambridge University Press, 1997. |
| [Ben] | D. J. Benson, Representations and cohomology I, Basic representation theory of finite groups and associative algebras, Second edition, Cambridge University Press, Cambridge, 1998. |
Basic facts about modules over general rings can be found also in the monograph
| [AF] | F. W. Anderson, K. R. Fuller, Rings and categories of modules, 2nd edition, Springer-Verlag, New York, 1992. |
Links
- Computations with finite dimensional algebras and their finite dimensional representations can be done with help of a computer. If you give a representation to a computer, you can have automatically computed e.g. its projective cover or a basis of the homomorphism space to another finite dimensional representation. Such computations have been implemented in the QPA package for a freely available software GAP. Up-to-date information is available on the home page of Øyvind Solberg who maintains the QPA package.
- Home page of the course in the academic year 2023/24 (in Czech),
- Home page of the course in the academic year 2022/23.