Rolf Farnsteiner

Representations of Associative Algebras

Organization

Lecturer

Rolf Farnsteiner

Secretary

Gudrun Christiansen

Class Times

Mo, Th 12:30 - 14:00, LMS4-325

Tutorial
Daniel Bissinger

Contents (tentative)

This course intends to provide an introduction to the structure- and representation theory of associative algebras. The motivating problem is the determination of the so-called indecomposable modules and the linear maps between them. One instance, where we have a complete understanding of these data is given by a truncated polynomial ring k[X]/(Xn), whose indecomposable modules correspond to the Jordan canonical forms of nilpotent matrices of nilpotency class at most n.   

Starting point of the course, which only requires Algebra I and Algebra II as prerequisites, is the classical theory of semisimple algebras, which was set forth by Wedderbum in his thesis from 1907. We will present Artin's generalization and move on to basic features of the structure theory, as developed by Brauer-Nesbitt-Nakayama, beginning in the late1930s. 

With the advent of homological algebra in the mid 1940s new techniques were introduced that, among other aspects, furnished a combinatorial method to describe algebras and their modules. This will lead us to representations of quivers and Auslander-Reiten theory, which constitutes an effective method to organize the module category of an algebra, even when the aforementioned classification of indecomposable modules is beyond reach.  

  • Basics
    1. Artinian and noetherian rings and modules
    2. Semisimplicity and the Jacobson radical
    3. The Theorem of Wedderburn-Artin
    4. Projective modules, idempotents and Cartan invariants
    5. Injective modules and self-injective algebras
    6. The block decomposition
  • Tools from homological algebra
    1. Schanuel's Lemma
    2. The Heller operator and the stable module category
    3. Extensions, the Gabriel quiver and connectedness
  • Equivalences of categories
    1. Definitions and basic properties
    2. Morita equivalence
    3. Stable equivalence and finite representation type
    4. Triangular matrix rings
    5. Algebras with radical square zero
  • Almost split sequences
    1. The definition
    2. Pull-backs and push-outs
    3. The Nakayama functor and the dual transpose
    4. Self-injective algebras revisited
    5. The defect of an exact sequence
    6. The existence theorem

 

Assignments

  • Assignment 1 pdf
  • Assignment 2 pdf
  • Assignment 3 pdf
  • Assignment 4 pdf
  • Assignment 5 pdf
  • Assignment 6 pdf
  • Assignment 7 pdf
  • Assignment 8 pdf
  • Assignment 9 pdf
  • Assignment 10 pdf
  • Assignment 11 pdf 

 

References

  • M. Auslander, I. Reiten, S. Smalo: Representation Theory of Artin Algebras. Cambridge studies in advanced mathematics 36. Cambridge University Press, 1995.
  • I. Assem, D. Simson, A. Skowronski: Elements of the Representation Theory of Assciative Algebras. London Mathematical Society Student Texts 65, Cambridge University Press, 2006.
  • D. Benson: Representations and Cohomology, I. Cambridge studies in advanced mathematics 30. Cambridge University Press, 1991.