Representation Theory of Finite Dimensional Algebras (NMAG442) - information about the course in summer semester 2018/2019.

## Basic information

The aim of the course is to give an introduction to representation theory of finite dimensional algebras and illustrate the lectured concepts on examples. The contents of the course and other basic information are available in the Student Information System.

The schedule (to be found also in the Student Information System):

• Thursday, 2.00pm-3.30pm and
• Thursday, 5.20pm-6.50pm

in the seminar room of the Department of Algebra. The problem sessions (with Jakub Kopřiva) take place once in two weeks from 5.20pm to 6.50pm.

## Credit

The credit will be granted for solved exercise problems. There will be three sets of the problems which will appear here. Requested are at least 50 % of successfully solved problems handed in within the deadlines.

Problem set #1 (hand in by April 25):
Exercises 7, 13, 15 and 17 from Section II.4 Exercises (p. 65-68) in the monograph by Assem, Simson, Skowroński.
Problem set #2 (hand in by May 16):
Exercise 4(a), (b), (g), (h) and exercises 7, 8, 15 from Section III.4 Exercises (p. 93-96) in the monograph by Assem, Simson, Skowroński.
Problem set #3:
Describe the roots of the Dynkin diagram D5 in Z5. Fix any orientation Q of the diagram D5 and describe all the indecomposable representations of Q.

## What has been lectured

A brief overview of what has been taught in individual lectures, including references to the literature, can be found below.

February 28, 2019
Quivers and their representations in the category of vectors spaces, some motivating problems (e.g. configurations of subspaces of a given space). Additive and K-linear categories. Equivalence between the categories of representations of a quiver and module categories over the corresponding path algebras ([ASS], Ch. I, Sec. II.1, Sec. III.1).
March 14, 2019
The category of representations of Q bound by a set relations is equivalent to modules over the corresponding quotient of KQ. Admissible ideals of path algebras. The Jacobson radical of a ring and the Nakayama lemma. Idempotent elements and direct sum decomposition of a ring to right ideals, finite dimensional algebras have complete sets of primitive orthogonal idempotents ([ASS], Ch. I, Sec. II.1 - II.2, Sec. III.1).
March 21, 2019
Central idempotents, connected rings, an interpretation for algebras of the form KQ/I for an admissible ideal I. Local rings and a characterization of local finite dimensional algebras. Idempotents in endomorphism rings of modules and the relation to direct sum decomposition ([ASS], Sec. I.4, Sec. II.1 - II.2).
March 28, 2019
The Krull-Schmidt theorem on the uniqueness of direct sum decomposition, the Jacobson radical of a module, projective covers ([ASS], Sec. I.3 - I.5).
April 4, 2019
Existence of projective covers, duality between the categories of left and right modules, simple and injective modules ([ASS], Sec. I.5).
April 11, 2019
The socle of a module and its properties, the existence and the structure of injective envelopes, basic algebras, Morita equivalence, the quiver of a (suitable) finite dimensional algebra, Gabriel's theorem on that every finite dimensional algebra over an algebraically closed field is Morita equivalent to a path algebra modulo an admissible ideal ([ASS], Sec I.5, I.6, II.3, III.2).
April 25, 2019
Proof of Gabriel's structure theorem, hereditary algebras, crash course on the Ext functors ([ASS], Sec. II.3, VII.1).
May 2, 2019
Finite dimensional hereditary algebras over an algebraically closed field are Morita equivalent to path algebras without relations, the Grothendieck group of a path algebra and dimension vectors, the Euler bilinear form, symmetric bilinear and quadratic forms associated with finite graphs ([ASS], Sec. VII.1, III.3, [Kra], Sec. 3.2, 4.1).
May 9, 2019
The classification of connected graphs with positive definite quadratic form (Dynkin diagrams) and positive semi-definite quadratic form (Euclidean diagrams), roots and reflections in Zn, reflection functors and their properties, the induced bijections between indecomposable representations of Q and σiQ with the exception of the simples at vertex i ([Kra], Sec. 3.3, 4.2, 4.3).
May 16, 2019
Admissible ordering of vertices of a quiver, the Coxeter transformation and Coxeter functors ([Kra], Sec. 3.1, 3.4, 4.4).
May 23, 2019
Gabriel's theorem on finite representation type (a finite acyclic connected quiver is of finite representation type if and only if its underlying graph is a Dynkin diagram and in this case the indecomposable representations bijectively correspond to the positive roots), preprojective and preinjective representation of arbitrary finite acyclic quivers, the defect of a representation of a Euclidean quiver and how it determines the type of the representation (preprojective/preinjective/regular), classification of the indecomposable representations of the Kronecker quiver ([Kra], Sec. 3.5, 4.4, 5.1 - 5.3 and briefly also 9.1 - 9.3)

## Literature

The lectured material will be mostly based on 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 roughly covers the first three chapters of [ASS] (with a short venture to Section VII.1 because of hereditary algebras) and Sections 3 to 5 in [Kra] (possibly with a brief account on Section 9 about the Kronecker quiver as well).

Several other monographs on the topic of representation theory of finite dimensional algebras appeared recently. There is, however, also an older one which is certainly worth mentioning:

 [ARS] M. Auslander, I. Reiten, S. Smalø, Representation theory of Artin algebras, Cambridge University Press, 1997.