Representation Theory of Finite Dimensional Algebras (NALG022) - information about the lecture in the summer semester 2009/2010.

## 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. For an outline of the course and basic information see the description in the Student Information System.

## Literature

• I. Assem, D. Simson, A. Skowroński, Elements of the representation theory of associative algebras, Vol. 1, Cambridge University Press, 2006.
• M. Auslander, I. Reiten, S. Smalø, Representation theory of Artin algebras, Cambridge University Press, 1997.

## What has been lectured

The references to chapters are for the book by Assem, Simson and Skowroński:

• quivers and path algebras (Chapter II.1),
• representations of quivers (Chapter III.1),
• the relation between representations and modules (Chapter III.1),
• relations, admissible ideals, representations of bound quivers (Chapters II.2 and III.1),
• the Jacobson radical of an algebra, Nakayama's lemma (Chapter I.1),
• idempotents, decomposition, connected algebras, local algebras (Chapter I.4),
• Unique decomposition theorem (a.k.a. the Krull-Schmidt theorem) (Chapter I.4),
• representation type of an algebra - finite/infinite (Chapter I.4),
• a basic algebra associated to a finite dimensional algebra (Chapter I.6),
• the quiver of a finite dimensional algebra over an algebraically closed field (Chapter II.3),
• projective covers, injective envelopes, and the duality between finitely generated left and right modules (Chapter I.5),
• irreducible and almost split morphisms (Chapter IV.1),
• the Jacobson radical of the module category (Appendix A.3),
• almost split sequences (a.k.a. Auslader-Reiten sequences) and several ways to characterize them (Chapter IV.1),
• the transpose and the Auslander-Reiten translation (Chapter IV.2),
• the Auslander-Reiten formulas (Chapter IV.2),
• the existence of almost split sequences (Chapter IV.3),
• the Auslander-Reiten quiver of a finite dimensional algebra over an algebraically closed field (Chapter IV.4).