Tuesday, 1 October 2019 to Sunday, 31 May 2020
Total hours: 40
Hours of lectures: 40
Prerequisites: linear algebra, elementary algebra.
Suggested audience: fourth-fifth year students.
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.
- 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.
- 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.