Warning: This page concerned the lectures given in summer semester 2021/22. Please consult the homepage for information about current lectures.
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. Basic information about the course can be also found in the Student Information System.
Schedule (to be found also in the Student Information System):
- Tuesdays 2-3.30pm in the seminar room of the Department of Algebra,
- Thursdays 9-10.30am in lecture room K7.
Exercise sessions take place once in two weeks (with J. Kopřiva).
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 J. Šťovíček. Requested are at least 50% of points from solved problems.
1^{st} problem set (hand in by April 19)
Solve the following problems from [ASS], sec. II.4 (pages 65-68):
- Exercise 7,
- Exercise 11 (assume that the field is algebraically closed),
- Exercise 15,
- Exercise 17, part (b).
2^{nd} problem set (hand in by May 12)
Solve the following problems from [ASS], sec. III.4 (pages 93-96):
- Exercise 4, parts (a), (b), (g), (h),
- Exercise 5,
- Exercise 12 for the algebras from parts (a), (b), (g), (h) of Exercise 4.
3^{rd} problem set (bring to the exam)
- Find a root of the Dynkin diagram E_{6} which has number 3 in some component.
- Fix an orientation of the Dynkin diagram E_{6} and describe an indecomposable representation corresponding to the chosen root as concretely as possible.
Program of the course
A brief overview of what has been taught can be found below.
Date | What has been taught | Sources |
---|---|---|
Feb 17 | Algebras over a field, the Jacobson radical, examples. | [ASS], sec. I.1 |
Feb 18 (substitute) |
Modules over an algebra, the Nakayama Lemma, nilpotence of the radical of a finite dimensional algebra. Homomorphisms of modules and their kernels, cokernels and images. Direct sums and indecomposable modules. Schur's Lemma, the Wedderburn-Artin Theorem, Maschke's Theorem. | [ASS], sec. I.2 and I.3 |
Feb 22 | The radical and the top of a module and their properties. Idempotents and a relation to decomposition of a ring to a direct sum of right ideals. | [ASS], sec. I.3 and I.4 |
Feb 24 | Exercises: Equivalences of categories, simple and semisimple modules, the Wedderburn-Artin Theorem. | exercise sheet |
Mar 1 | Central idempotents and products of rings. Lifting idempotents modulo nilpotent ideals. Indecomposable summands of the regular module have a simple top. Local rings, statement of equivalent conditions and examples. | [ASS], sec. I.4 |
Mar 3 | Characterization of local rings (proof). Finite dimensional indecomposable modules have local endomorphism rings. The Krull-Schmidt Theorem on unique decomposition to indecomposable modules. | [ASS], sec. I.4 |
Mar 8 | Free and projective modules, projective precovers, presentations and resolutions. Superfluous submodules and projective covers. Uniqueness of projective covers and their existence for finite dimensional algebras. | [ASS], sec. I.5 |
Mar 10 | Exercises: Idempotents, local algebras, and projective covers. | exercise sheet |
Mar 15 | A proof of the existence of projective covers, the duality between left and right modules, injective envelopes. | [ASS], sec. I.5 |
Mar 17 | Basic algebras and their characterization, a basic algebra associated to a general finite dimensional algebra. Equivalence of the module categories of a finite dimensional algebra and its associated basic algebra. Quivers, path algebras, examples. | [ASS], sec. I.6 and II.1 |
Mar 22 | Ring-theoretic notions specialized to path algebras (a complete set of primitive orthogonal idempotents, the arrow ideal, the radical, connected path algebras). The category of representations of a quiver. | [ASS], sec. II.1 and III.1 |
Mar 24 | Exercises: Representations of quivers and their properties. | exercise sheet |
Mar 29 | Relations in quivers, factors of path algebras, representations bound by relations. Equivalences between categories of modules and representations. | [ASS], sec. II.2 and III.1 |
Mar 31 | Admissible ideals and factors over them, the radical, connectedness and cosets of trivial paths as a complete set of primitive orthogonal idempotents. The socle and radical in the language of representations. | [ASS], sec. II.2 and III.2 |
Apr 5 | Indecomposable projective and injective representations, examples. | [ASS], sec. III.2 |
Apr 7 | Exercises: Simple, projective and injective representations, endomorphisms of representations, properties of bound path algebras modulo an admissible ideal. | exercise sheet |
Apr 12 | The quiver Q_{A} of a basic finite dimensional algebra A over an algebraically closed field K. Each such algebra is isomorphic to KQ_{A}/I for some admissible ideal I. Examples. | [ASS], sec. II.3 |
Apr 14 | Hereditary algebras and their structure (after [ASS, sec. VII.1] a [ARS, Lemma III.1.11]): a basic hereditary finite dimensional algebra A over an algebraically closed field has an acyclic quiver and A ≅ KQ_{A}. Dimension vectors and the Grothendieck group. |
[ASS], sec. III.3 and VII.1 [ARS], sec. III.1 |
Apr 19 | The Cartan matrix of a path algebra and its properties. The Euler characteristic and its homological interpretation. | [ASS], sec. III.3 |
Apr 21 | Exercises: Simple, projective and injective representations, endomorphisms of representations, Gabriel's theorem and hereditary algebras. | exercise sheet |
Apr 26 | The Euler characteristic (aka Euler form) for hereditary algebras. The Coxeter transformation. Changing orientation of a quiver and admissible orderings of vertices. |
[ASS], sec. III.3, [Kra], sec. 3.1, 3.2 |
Apr 28 | Reflection with respect to a vertex of a quiver and construction of reflection functors. Properties of reflection functors and their action on indecomposable modules. Coxeter functors. | [Kra], sec. 3.2-3.4 |
May 3 | Properties of Coxeter functors, their relation to the Coxeter transformation. |
[Kra], sec. 3.4 and 4.4 |
May 5 | Exercises: Cartan matrices, reflection functors. | exercise sheet |
May 10 | Preprojective and preinjective modules. Dynkin and Euclidean diagrams and positive (semi)definiteness of associated quadratic forms. | [Kra], sec. 3.5-4.2 |
May 12 | Roots and their properties for Dynkin and Euclidean diagrams. Further results on the Coxeter transformation. Classification of indecomposable finite dimensional representations of quivers whose undelying graph is a Dynkin diagram by positive roots. | [Kra], sec. 4.3-5.1 |
May 19 | Exercises: Representations of quivers with non-Dynkin underlying graphs (infinite representation type). | exercise sheet |
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
- Home page of the course in the academic year 2020/21.
- Home page of the course in the academic year 2019/20.
- Home page of the course in the academic year 2018/19.
- 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.