Basic information
There is a change in the schedule: The lecture/exercise session was moved from Thursday evening to Friday morning (starting at 9am in room K12).
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 lecture room K9,
- Friday 9-10.30am in lecture room K12.
Exercise sessions take place once in two weeks (with Chiara Sava).
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.
1^{st} problem set (hand in by April 18)
Solve the following problems from [ASS], sec. II.4 (pages 65-68):
- Exercise 11 (assume that the field is algebraically closed),
- Exercise 14,
- Exercise 15,
- Exercise 17, part (c).
2^{nd} problem set (hand in by May 16)
Solve the following problems from [ASS], sec. III.4 (pages 93-96):
- Exercise 10,
- A variant of Exercise 10: consider the path algebra of the same quiver Q, but I = ⟨βα⟩. What is the global dimension of A = KQ/I now?
Solve also the following problems from [ASS], sec. VII.6 (pages 298-299):
- Exercise 10,
- Exercise 11.
3^{rd} problem set (bring to the exam)
- Let Q be an acyclic quiver of Euclidean type Ã_{2} (i.e. a triangle with a non-cyclic orientation). Describe all indecomposable preprojective representations and their dimension vectors.
Program of the course
A brief overview of what has been taught will appear here.
Date | What has been taught | Sources |
---|---|---|
Feb 14 | Representations of quivers and algebras (introduction), motivating problems, categories of representations as additive, K-linear abelian categories, path algebras. | [ASS], sec. II.1 and III.1 |
Feb 21 | Examples of path algebras. Ideals in path algebras: powers of the arrow ideal, relations. Representations bound by relations. K-linear equivalence between Mod KQ/I and Rep_{K}(Q,I). | [ASS], sec. II.1, II.2 and III.1 |
Feb 23 | Exercises: Path algebras, subalgebras of the full matrix rings, direct sum decomposition of representations. | Exercise sheet 1 |
Feb 28 | Admissible ideals of path algebras. The Jacobson radical, computation of rad(KQ/I) with I admissible, the Nakayama lemma, nilpotence of rad(A) for A finite dimensional. Reminder of the Wedderburn-Artin theorem. | [ASS], sec. I.1 – I.3 and II.2 |
Mar 2 | Idempotents and decompositions of A as a right module and as a ring. Complete sets of primitive orthogonal idempotents, vertex idempotents as such sets for KQ/I with I admissible. Local rings, lifting idempotents modulo nilpotent ideals. | [ASS], sec. I.4, II.1 and II.2 |
Mar 7 | Characterization of local rings. Direct sum decompositions and idempotents in endomorphisms rings, local endomorphism rings. The Krull-Schmidt Theorem (aka the Unique Decomposition Theorem). Finite and infinite representation type. | [ASS], sec. I.4 |
Mar 9 | Exercises: Admissible ideals, endomorphism ring and indecomposable representations. | Exercise sheet 2 |
Mar 14 | Simple and projective modules over a finite dimensional algebra, the radical of a module and its properties, projective covers, their characterization and uniqueness. | [ASS], sec. I.3 and I.5 |
Mar 16 | Exercises: Simple and projective representations. The radical of representations, projective covers of simple representations. | Exercise sheet 3, [ASS], sec. III.2 |
Mar 21 | Construction of projective covers. Vector space duality between the left and modules. Injective modules, the socle of a module, injective envelopes. | [ASS], sec. I.3 and I.5 |
Mar 23 | Injective envelopes as duals of projective covers. Basic algebras, some non-examples, but KQ/I is basic for I admissible. The basic algebra associated to a general finite dimensional algebra. Algebraically closed fields have no proper finite dimensional central division ring extensions. | [ASS], sec. I.5 and I.6 |
Mar 28 | The quiver Q_{A} of a finite dimensional algebra A. Each basic finite dimensional algebra A over an algebraically closed field is isomorphic to KQ_{A}/I for some admissible ideal I. | [ASS], sec. II.1 and II.3 |
Mar 30 | Basic properties of Ext functors, global dimension. Hereditary algebras and their structure (after [ARS, Prop. III.1.4 and Lemma III.1.11 – Prop. III.1.13]): a basic hereditary finite dimensional algebra A over an algebraically closed field has an acyclic quiver and is isomorphic to KQ_{A}. | [ASS], app. A.4, [ARS], sec. III.1 |
Apr 4 | Numerical invariants of representations: dimension vectors, the Grothendieck group, the Euler characteristic and related concepts (the Cartan matrix, the Coxeter transformation). | [ASS], sec. III.3 |
Apr 6 | Exercises: Injective representations, the socle of a representation, quivers of matrix algebras. | Exercise sheet 4 |
Apr 11 | The Euler characteristic (aka Euler form) for A=KQ. Dynkin and Euclidean diagrams and positive (semi)definiteness of associated quadratic forms. | [Kra], sec. 3.2, 4.1 – 4.2 |
Apr 18 | Proof of the positive (semi)definitness, roots (properties and examples), admissible ordering of vertices of a quiver, the Coxeter transformation via reflections | [Kra], sec. 3.1 – 3.2, 4.2 – 4.4 |
Apr 20 | Exercises: Reflections, the Coxeter transformation, roots, admissible orderings of vertices. | Exercise sheet 5 |
Apr 25 | The Coxeter transformations via reflections, construction of reflection functors. | [Kra], sec. 3.3 – 3.4, 4.4 |
May 2 | Properties of reflection functors, construction of Coxeter functors. | [Kra], sec. 3.3 – 3.4 |
May 5 | Exercises: Reflection functors, Coxeter functors. | Exercise sheet 6 |
May 9 | Properties of Coxeter functors. Preprojective, preinjective and regular representations. | [Kra], sec. 3.4 – 3.5 |
May 12 | Classification of indecomposable modules for quivers of Dynkin type. The defect of a representation in the Euclidean type. | [Kra], sec. 5.1 – 5.2 |
May 16 | Preprojective and preinjective representations in the Euclidean case, characterization of when KQ is of finite representation type. | [Kra], sec. 5.3 |
May 19 | Exercises: Classification of representations of quivers of Dynkin type. | Exercise sheet 7 |
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 2021/22.
- Home page of the course in the academic year 2020/21 (in Czech).