Representation Theory of Finite Dimensional Algebras (NMAG442) - information about the course in the summer semester 2024/2025.

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:

  1. 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.
  2. 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

  1. Chapters I-III and parts of Chapter IV.2 (the construction of the Auslander-Reiten translation) in the textbook [ASS],
  2. the lecture notes [Notes1], and
  3. Sections 3-5 of [Kra] (with some material omitted, the reflection functors are not used and the Coxeter functors are replaced by the Auslander-Reiten translation).

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.

1st problem set (hand in by April 16)

Solve the following problems from [ASS], sec. II.4 (pages 65-68):

  • Exercise 11 assuming that the field is algebraically closed,
  • Exercise 12,
  • Exercise 15,
  • Exercise 17, parts (a) and (b).

2nd problem set (hand in by May 7)

Solve the following problems from [ASS], sec. III.4 (pages 93-96):

  • Exercise 4, parts (b), (g) and (h),
  • Compute global dimension for the algebras from Exercise 4, parts (b), (g) and (h),
  • Exercises 6 and 8.

3rd problem set (bring to the exam)

  • Choose a quiver Q as your favorite orientation of the Dynkin diagram D5. What is the maximum dimension of an indecomposable KQ-module? Find all indecomposable modules of this maximum dimension up to isomorphism and write them down explicitly as representations of Q.
  • Let Q be an acyclic quiver of Euclidean type Ã2 (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 can be found below.

DateWhat has been taughtSource
Feb 19Representations 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 26Equivalence 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 5The 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
Mar 5Exercises: Quivers and path algebras. Idempotents, the Jacobson radical and admissible ideals in path algebras.Exercise sheet
Mar 12Lifting idempotents modulo the Jacobson radical and, local rings and a characterization of local finite dimensional algebras. Direct sum decompositions of modules, idempotents in endomorphisms rings and local endomorphism rings. The Krull-Schmidt Theorem (aka the Unique Decomposition Theorem).[ASS], Sec. I.4
Mar 12Exercises: Isomorphisms between represenations. Decompositions, indecomposability and endomorphism rings of representations.
Mar 19Projective and injective modules over a finite dimensional algebra A and their structure. Duality between mod A and mod Aop. Projective covers and injective envelopes in mod A - the existence and examples.[ASS], Sec. I.5
Mar 26Basic algebras, a characterization and examples. An associated basic algebra Ab to a finite-dimensional algebra A and a brief account on the Morita equivalence between mod A and mod Ab. Given a finite acyclic quiver Q, the trivial paths are primitive idempotents, the radical is the arrow ideal and KQ is basic. A reminder of relations and admissible ideals.[ASS], Sec. I.6, II.1 and II.2
Mar 26Exercises: Computations of simple, indecomposable projective and indecomposable injective representations.
Apr 2Bound quiver algebras and their properties. The quiver of a finite dimensional algebra (basic, over an algebraically closed field), definition and examples, Gabriel’s theorem that each such algebra is isomorphic to KQA/I for some admissible ideal I. Basic properties of Ext functors, projective and injective dimensions, global dimension. Hereditary algebras, algebras of the form KQ for Q finite and acyclic as examples.[Notes1]
[ASS], Sec. II.2, II.3, VII.1 and App. A.4.
Apr 9Structure theorem for herediraty algebras which are basic and over an algebraically closed field: they are precisely those isomorphic to KQ with Q finite and acyclic. The Grothendieck group K0(A) of the module category mod A and dimension vectors.[Notes1]
[ASS], Sec. III.3 and VII.1.
Apr 9Exercises: Computing radicals and socles of representations. Quivers and admissible ideals for matrix algebras.
Apr 16The Euler characteristic and the Euler quadratic form. Computation of the Euler characteristic via the inverse of the Cartan matrix and a direct formula for algebras of the form KQ using [ASS, Lemma III.2.12].[ASS], Sec. III.2 and III.3.
Apr 23The Coxeter matrix ΦA and Coxeter trasformation cA for an algebra of finite global dimension. Duality between proj A and proj Aop. The transpose Tr M of a module M and basic properties of the construction.[ASS], Sec. III.3 and IV.2.
Apr 23Exercises: Projective, injective and simple representations, Cartan matrices and the Euler characteristic.
Apr 30The Auslander-Reiten translations τ and τ-, their basic properties and an example. The Nakayama functor. The action of τ on dimension vectors of indecomposable modules M in the hereditary case A = KQ: then ΦA·dim M = dim τM > 0 if M is non-projective and ΦA·dim P(i) = -dim I(i) < 0 for M = P(i) projective. Dynkin and Euclidean diagrams and positive (semi-)definiteness of the associated quadratic form.[ASS], Sec. III.3 and IV.2.
[Kra], sec. 4.1, 4.2
May 7Roots for Dynkin and Euclidean diagrams and their properties. The action of the Coxeter transformation on roots (with emphasis on acyclic quivers of Dynkin and Euclidean type).[Kra], sec. 4.3, 4.4
May 7Exercises: Indecomposable representations of quivers of Dynkin type.
May 14Preprojective, preinjective and regular representations of a finite acyclic quiver. The Coxeter transformation as a composition of reflections. Gabriel’s classification of representations of quivers of Dynkin type. The defect of a representation of a quiver of Euclidean type, and the classification of preprojective and preinjective representations in that case.[Kra], sec. 3.1, 3.2, 3.4, 3.5, 4.4, and 5.1-5.3
May 21A connected hereditary algebra A = KQ is of finite representation type if and only if Q is of Dynkin type. Further results and possible directions (not included in the exam): a) other classes of examples which are understood (Nakayama algebras, special biserial/string/gentle algebras, group algebras in positive characteristic with cyclic or Klein Sylow subgroups), b) general results on the complexity of the classification problem for representations (the tame/wild dichotomy, the Brauer–Thrall conjectures), c) a glimpse into the Auslander-Reiten theory.[Kra], sec. 5.3
(and [ASS], Chapter 4)
May 21Exercises: Indecomposable representations of quivers of Euclidean type. Projective resolutions and global dimension.

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]

Some parts of the course are more complicated to extract from the these sources (especially when changing from one to the other). There are covered by the following lecture notes.

[Notes1] Crash course on homological algebra and hereditary algebras. [Full text in PDF]

The course consists of

  1. Chapters I-III and parts of IV.2 in [ASS],
  2. the lecture notes [Notes1] and
  3. Sections 3-5 of [Kra] (with some material omitted, the reflection functors are not used and the Coxeter functors are replaced by the Auslander-Reiten translation).

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.