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Representation theory of finite-dimensional algebras

Schedule

Tuesday, 1 October 2019 to Sunday, 31 May 2020
Total hours: 40
Hours of lectures: 40

Examination procedure

  • oral exam

Prerequisites

Prerequisites: linear algebra, elementary algebra.

Suggested audience: fourth-fifth year students.

Syllabus

The aim is to study moduls (=representations) of a finite-dimensional algebra A, using the language of quivers, that is, sets ofvector spaces and linear maps.
There are strong connections to homological algebra and the representation theory of Kac-Moody algebras.

Program:

  • Elements of category theory: categories, functors, adjoint functors (Hom and tensor).
  • Gabriel's theorem: representations of finite-dimensional algebras are equivalent to representations of quivers.
  • Theorem of Krull-Remak-Schmidt: each representation has an essentially unique decomposition into indecomposable representations.
  • Classification of representations of ADE quivers: description of all indecomposiable representations, Auslander-Reiten quiver.
  • Hall algebras: connection between representation theory of quivers and Kac-Moody algebras.

Bibliographical references

  • Ibrahim Assem, Daniel Simson, Andrzej Skowronski: Elements of the Representation Theory of Associative Algebras: Volume 1, Cambridge University Press.
  • Ralf Schiffler: Quiver representations, Springer Verlag.
  • Andrew Hubery: Ringel-Hall algebras, web page.
  • Olivier Schiffmann: Lectures on Hall algebras, arxiv:math/0611617.