FD-Atlas (= Atlas of finite-dimensional algebras)
- FD-Atlas (pdf file, 342 pages, 06.07.22)
The representation theory of finite-dimensional algebras is a relatively young area of mathematics. Its big bang or rather big bangs were Gabriel's classification of representation-finite quivers in 1970 and Auslander and Reiten's discovery of almost split sequences (aka Auslander-Reiten sequences) in 1975.
There is quite a large zoo of classes of finite-dimensional algebras which people study for various reasons. Many of these classes have a beautiful representation theory and often provide a link to other areas of mathematics or mathematical physics.
Part 1 of the FD-Atlas is a compilation of short notes on the most important classes. (I identified about 120 of these up to now.) Usually, I will briefly define a class, give some examples, mention a few important results, and provide literature recommendations for further reading.
Part 2 contains a recollection of some fundamental results and techniques from the representation theory of finite-dimensional algebras. This includes an overview of the categories and subcategories which are frequently studied. I also give a list of general conjectures, e.g. the classical homological conjectures. Many more conjectures can be found in the various more specialized sections of Part 1.
There will be regular updates.